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+{"text":"\\section{Introduction}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarProjectionDetails:a}\n\t\t\\includegraphics[height=2.6in]{\"Figures\/326_53858c_case1\"}}\n\t\\hspace{-0.19in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarProjectionDetails:b}\n\t\t\\includegraphics[height=2.6in]{\"Figures\/157_53354r_case1\"}}\n\t\\hspace{-0.19in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarProjectionDetails:c}\n\t\t\\includegraphics[height=2.6in]{\"Figures\/912_55586l_case1\"}}\n\t\\hspace{-0.19in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarProjectionDetails:d}\n\t\t\\includegraphics[height=2.6in]{\"Figures\/780_55195l_case1\"}}\n\t\\caption{Examples of projecting lidar points to images before (top row) and after (bottom row) ego-motion correction. Misalignment of lidar points and features in the images due to ego-motion distortion can be clearly seen for a traffic sign in (a), a parked car and a building behind in (b), a tree trunk in (c), and lastly a walking pedestrian in (d). The figures in the bottom row show the improved results in the projection of ego-motion corrected lidar points using the proposed approach. The ego-motion correction uncertainty is available for each projected point, but not shown in the figures due to size constraints. The projected lidar points are colour-coded by range.}\n\t\\label{fig:LidarProjectionDetails}\n\\end{figure}\n\nTo navigate through any environment, a mobile robot platform is required to perceive the environment and achieve some level of understanding of the surroundings. In many sophisticated systems, this requires the combining of information from heterogeneous on-board sensors.\nLidars and cameras are complementary sensors that are extensively used in various robotic systems. Each sensor has a different strength---lidars offer precise range and reflective intensity measurements that are registered in 3D space, while cameras provide rich information of colour, texture and other visual features only in 2D. A considerable proportion of autonomous driving solutions proposed and developed by the automotive industry and research institutes rely on multiple cameras and lidars, in particular multi-beam lidars, to capture the activity of road users in the vicinity and to build contextual information---pedestrians, cyclists, other vehicles, traffic signs, lane markings, the road itself, etc.---in a traffic scene. Through the fusion of information from the two sensors of different modalities, we are able to transfer relevant data from the lidar to the camera domain, and vice versa, providing a better understanding of the surroundings' structure \\cite{paper:ChienKlette2016}. It is thus of great importance to achieve accurate and robust perception by fusing camera and lidar information in a consistent manner.\n\nAlthough often over-simplified in many applications as being measured at a single time, each point contained in a lidar scan is in fact captured at a slightly different time due to the laser firing cycles. The motion of the egocentric robot platform causes distortion in the lidar measurements as the sensor coordinate system moves along with the platform during the period of a scan. In theory, every 3D point is measured from a temporally unique frame of reference. The lidar points therefore must be compensated for ego-motion and transformed into a common reference coordinate system before further point cloud and sensor fusion related processing can take place. This can include, for instance, lidar feature extraction, projection to image frames, transferring segmentation results from the image space onto the 3D point cloud, or 3D mapping. The ego-motion correction becomes more essential for higher speed motion of the robot system, where the distortion caused by ego-motion tends to be more severe. Examples presented in Figure \\ref{fig:LidarProjectionDetails} illustrate misalignment of lidar and visual features in the environment when projecting uncorrected lidar points to images, which can cause degraded performance in the sensor fusion. Interested readers can refer to \\cite{paper:RiekenMaurer2016} for quantitative analyses of the time-related effects of moving scanning sensors on different perception tasks for multiple sensor systems.\n\n\n\nDepending on the way the underlying ego-motion estimation of the sensor platform is conducted, there are two main categories of existing approaches proposed in the literature to correct the 3D lidar point cloud distortion due to ego-motion of the platform. In the first type, such as \\cite{paper:SchneiderHimmelsbach2010, paper:HimmelsbachMuller2010, paper:MerriauxDupuis2017}, lidar scans are corrected by exploiting information from motion estimation sensors, such as IMU and odometry measurements. More complicated work presented in \\cite{paper:VargaCostea2017, paper:ByunNa2015} obtains vehicle pose translation and rotation by fusing precise GNSS and IMU measurements.\nHowever, high end GNSS\/INS units are costly,\nand their desired performance would not be achieved in GNSS denied environments. The other type relies on lidar based odometry estimation \\cite{paper:TangYoon2018}, which eliminates the requirement of additional hardware. It can be further decomposed to simultaneous localization and mapping (SLAM) based approaches \\cite{paper:MoosmannStiller2011, paper:ZhangSingh2014} that estimate the ego-motion by comparing point cloud features, and iterative closest point (ICP) based methods \\cite{paper:HongKo2010} which infer the ego-motion through matching of consecutive scans. The SLAM based approaches are preferred in 3D map construction, yet loop closure is not achievable in some environments. ICP based methods are prone to errors brought by moving objects in the scene, such as pedestrians and vehicles \\cite{paper:VargaCostea2017}.\nOverall, it requires a substantial computational overhead for extracting features from a lidar point cloud and comparing lidar scans in these approaches.\n\n\nNone of the above approaches provides a way to estimate the uncertainty associated with each of the 3D lidar points and\/or 2D image points in ego-motion correction process. We stress that there is always some uncertainty in 3D space associated with each motion corrected lidar point brought by errors in ego-motion estimation, regardless of which odometry sensors and\/or estimation frameworks are adopted. Likewise, the motion corrected points when projected into a camera coordinate system also contain uncertainty in 2D image coordinates. The uncertainty is often considerable under many circumstances and has to be estimated along with the ego-motion correction process. Thus, the perception system can benefit from the uncertainty estimates in the subsequent point cloud and sensor fusion processing pipeline.\n\nA probabilistic approach was proposed by \\cite{paper:LeGentilVidalCalleja2018} that includes the correction due to motion distortion in 3D point clouds using IMU data considering measurement uncertainty. Yet, the approach mainly focuses on recovery of extrinsic calibration parameters of a lidar-IMU tightly coupled system, which does not produce explicit estimation uncertainty for corrected lidar points. The more recent work \\cite{paper:CharikaShan2019} presents a probabilistic approach to estimate the uncertainty in the lidar-to-camera projection process. It employs a Jacobian based uncertainty model to estimate for each projected lidar point the combined uncertainty (in 2D) resulting from noise in ego-motion correction and errors in sensors' extrinsic and intrinsic calibration. Nevertheless, the uncertainty estimation for ego-motion corrected lidar points themselves (in 3D) is not supported by the approach.\n\nThe paper examines probabilistic ego-motion correction of lidar 3D point clouds to an arbitrary reference timestamp and projection to 2D image frames considering uncertainty in ego-motion estimation of the moving platform. On top of ego-motion correction outcomes, the proposed approach provides uncertainty estimates separately for each ego-motion corrected 3D lidar point and each projected 2D pixel point. Besides, the proposed approach considers additional uncertainty brought by time jitter in sensor data timestamps, which is a practical issue in many robotics systems. The proposed approach is validated using real-world data collected on an electric vehicle platform. Simulation results are also presented to quantitatively evaluate the proposed approach and assess the estimator credibility.\n\nThe remainder of the paper is organised as follows. Section \\ref{sec:approach} presents the details of the proposed approach, including the probabilistic lidar ego-motion correction and the projection to an image frame. The experiment outcomes are presented in Section \\ref{sec:results}, where simulation results are also included. Lastly, Section \\ref{sec:conlusions} concludes the paper.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Proposed Approach}\n\\label{sec:approach}\n\nIn this section, we elaborate on the calibration of multiple cameras and a lidar in our experimental platform, and the methodology to estimate uncertainties as a result of probabilistic lidar ego-motion correction and projection.\n\n\n\n\n\n\\subsection{Calibration}\n\nThe cameras located on the electric vehicle platform used in the experiment have a lens of $100^\\circ$ horizontal field of view, which is classified as a fisheye lens. We have calibrated the cameras by using a variation of the ROS package \\textit{camera\\_calibration} \\cite{ros_camera_calib} that uses a generic camera model \\cite{fish_eye_model}. The camera intrinsic parameters for this model consist of the focal length, principal points and 4 fisheye equidistant distortion coefficients. These values are critical for lidar-to-camera projection and the subsequent sensor fusion.\n\nThe extrinsic camera calibration is challenging when working with wide angle cameras due to the significant distortions in the lens. The extrinsic calibration in our electric vehicle platform was conducted as specified in the previous work \\cite{SurabhiITSC}. This process uses a checkerboard from which the same features are extracted by both the camera and the lidar. The features are the centre point and the normal vector of the board. These features are fed to a genetic algorithm which is in charge of optimising the geometrical extrinsic parameters of the 3D transformation \\(T_{cam}^{ld}\\) between the two sensors.\n\n\n\\subsection{Probabilistic Lidar Ego-Motion Correction and Projection to Image Frame}\n\n\n\n\nUsually within a lidar scan, lidar measurements with similar timestamps are grouped into a single lidar packet, with a common timestamp assigned to the grouping for convenience of processing. For instance, the Velodyne VLP-16 software driver used in our electric vehicle platform produces 76 packets for each full revolution scan. Each packet covers an azimuth angle of approximately \\(4.74^{\\circ}\\). Alternatively, processing can be based on each individual lidar point with its own precise timestamp, though this comes at a significantly higher computational cost.\n\nEach of the lidar packets is transformed based on the estimated delta translation and rotation of the vehicle platform between the packet's timestamp and the reference timestamp \\(t_{ref}\\), as illustrated in Figure \\ref{fig:time_line}. The proposed approach makes use of the unscented transform (UT) to propagate the uncertainties from the ego-motion estimation to corrected 3D lidar points and then to projected pixel coordinates in each camera image. The entire process can be divided into three cascaded stages, namely vehicle ego-motion estimation, lidar motion correction, and lidar-to-camera projection, each can be fitted into the UT pipeline.\n\nThe reference time \\(t_{ref}\\) is usually chosen to be the timestamp corresponding to a common frame of reference where sensor fusion or subsequent processing happens. In scenarios where camera-lidar sensor fusion is desired, rectification of the lidar points have to be matched with the timestamp of the associated camera frame before the lidar-to-camera projection can be carried out \\cite{paper:VargaCostea2017}. For instance, the \\(t_{ref}\\) can be set to coincide with the timestamp of the most recent or closest image.\n\n\\begin{figure}[!t]\n\\vspace{2mm}\n\\centerline{\n\\includegraphics[width=0.9\\columnwidth]{Figures\/time_line.png}\n}\n\\caption{Lidar point cloud motion correction process.}\n\\label{fig:time_line}\n\\end{figure}\n\n\nWe assume a lidar scan is comprised of a set of \\(N\\) packets and their timestamps denoted as\n\\begin{equation}\n\\left\\{pk_{i}, t_{i}^{lpk}\\right\\}_{i=0}^{N-1},\n\\end{equation}\nwhere \\(pk_{i}\\) contains a set of \\(M\\) 3D lidar measurement points \\(\\left\\{\\bm{z}_{i,j}^{ld}\\right\\}_{j=0}^{M-1}\\), and \\(\\bm{z}^{ld} = \\begin{bmatrix} x^{ld} & y^{ld} & z^{ld} & 1 \\end{bmatrix}^{T}\\).\n\nBefore we proceed, the UT state decomposition and recovery functions are presented in Table \\ref{table:UT_Decompose} and Table \\ref{table:UT_Restore} respectively for the convenience of subsequent discussion, where \\(\\lambda = \\alpha^2\\left(d+\\kappa\\right)-d\\), \\(d = dim\\left(\\textbf{x}\\right)\\) is the dimension of state \\(\\textbf{x}\\), scaling parameters \\(\\kappa \\ge 0\\), \\(\\alpha \\in \\left(0, 1\\right]\\), and \\(\\beta = 2\\) for Gaussian distribution, \\(\\left(\\sqrt{\\bm{\\Sigma}}\\right)_i\\) is to obtain the \\(i^{th}\\) column of the matrix square root \\(\\textbf{R} = \\sqrt{\\bm{\\Sigma}}\\), which can be computed by Cholesky decomposition such that we have \\(\\bm{\\Sigma} = \\textbf{R}\\textbf{R}^{T}\\).\n\n\\subsubsection{Vehicle Ego-Motion Estimation}\n\nA proper way to estimate the ego-motion of the moving sensor platform is required to address its changing poses when perceiving the environment. In the electric vehicle platform used in our experiments, instantaneous linear and angular velocities are read from onboard wheel encoders and an IMU, respectively, at a rate of 100 Hz. Based on a sequence of monotonically increasing packet timestamps \\(\\bm{t}_{0:N-1}^{pk} = \\left\\{t_{i}^{pk}\\right\\}_{i=0}^{N-1}\\), it is reasonable to construct a sequence of linear velocity vectors \\(\\bm{z}_{0:N-1}^{v} = \\left\\{\\bm{z}_{i}^{v}\\right\\}_{i=0}^{N-1}\\) corresponding to \\(\\bm{t}_{0:N-1}^{pk}\\), and likewise a sequence of angular velocity vectors \\(\\bm{z}_{0:N-1}^{\\omega} = \\left\\{\\bm{z}_{i}^{\\omega}\\right\\}_{i=0}^{N-1}\\).\n\n\\begin{table}[!t]\n\\vspace{2mm}\n\t\\centering\n\t\\caption{Algorithm: State Decomposition in Unscented Transform}\n\t\\label{table:UT_Decompose}\n\t\\scalebox{1.0}{\n\t\t\\begin{tabular}{cl}\n\t\t\t\\toprule\n\t\t\t\\multicolumn{2}{l}{\\(\\left\\{\\bm{\\mathcal{X}}_{i}, w_{i}^{m}, w_{i}^{c}\\right\\}_{i=0}^{2d} \\leftarrow UTD\\left(\\bar{\\textbf{x}}, \\bm{\\Sigma}\\right)\\)}\\\\\n\t\t\t\\midrule\n\t\t\t1: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\t\\bm{\\mathcal{X}}_{0} = \\bar{\\textbf{x}}\n\t\t\t\\)\\\\\n\t\t\t2: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\t\\bm{\\mathcal{X}}_{i} = \\bar{\\textbf{x}} + \\left(\\sqrt{\\left(d+\\lambda\\right)\\bm{\\Sigma}}\\right)_i\\ \\text{for}\\ i=1,\\cdots,d\n\t\t\t\\)\\\\\n\t\t\t3: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\t\\bm{\\mathcal{X}}_{i} = \\bar{\\textbf{x}} - \\left(\\sqrt{\\left(d+\\lambda\\right)\\bm{\\Sigma}}\\right)_i\\ \\text{for}\\ i=d+1,\\cdots,2d\n\t\t\t\\)\\\\\n\t\t\t4: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\tw_{0}^{m} = \\frac{\\lambda}{d+\\lambda}\n\t\t\t\\)\\\\\n\t\t\t5: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\tw_{0}^{c} = \\frac{\\lambda}{d+\\lambda} + \\left(1-\\alpha^2+\\beta\\right)\n\t\t\t\\)\\\\\n\t\t\t6: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\tw_{i}^{m} = w_{i}^{c} = \\frac{1}{2\\left(d+\\lambda\\right)}\\ \\text{for}\\ i=1,\\cdots,2d\n\t\t\t\\)\\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}}\n\\end{table}\n\n\\begin{table}[!t]\n\t\\centering\n\t\\caption{Algorithm: State Recovery in Unscented Transform}\n\t\\label{table:UT_Restore}\n\t\\scalebox{1.0}{\n\t\t\\begin{tabular}{cl}\n\t\t\t\\toprule\n\t\t\t\\multicolumn{2}{l}{\\(\\bar{\\textbf{x}}, \\bm{\\Sigma} \\leftarrow UTR\\left(\\left\\{\\bm{\\mathcal{X}}_{i}, w_{i}^{m}, w_{i}^{c}\\right\\}_{i=0}^{2d}\\right)\\)}\\\\\n\t\t\t\\midrule\n\t\t\t1: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\t\\bar{\\textbf{x}} = \\sum_{i=0}^{2d}{w_{i}^{m} \\bm{\\mathcal{X}}_{i}}\n\t\t\t\\)\\\\\n\t\t\t2: & \\hspace{-10pt}\n\t\t\t\\(\n\t\t\t\\bm{\\Sigma} = \\sum_{i=0}^{2d}{w_{i}^{c} \\left(\\bm{\\mathcal{X}}_{i}-\\bar{\\textbf{x}}\\right) \\left(\\bm{\\mathcal{X}}_{i}-\\bar{\\textbf{x}}\\right)^T}\n\t\t\t\\)\\\\\n\t\t\t\\bottomrule\n\t\\end{tabular}}\n\\end{table}\n\nEach \\(\\bm{z}_{i}^{v}\\) is a column vector with linear velocity readings along with \\(x\\), \\(y\\), and \\(z\\) and each \\(\\bm{z}_{i}^{\\omega}\\) is a column vector with the angular velocity measurements in \\(roll\\), \\(pitch\\), and \\(yaw\\) in the local frame of reference of the vehicle. In cases where odometry data and lidar packets are asynchronous, \\(\\bm{z}_{i}^{v}\\) and \\(\\bm{z}_{i}^{\\omega}\\) can be well approximated using those with the closest timestamps to \\(t_{i}^{pk}\\), respectively, so long as the assumption that the vehicle kinematic state does not change dramatically during the time difference holds. Also, \\(\\bm{z}_{i}^{v}\\) and \\(\\bm{z}_{i}^{\\omega}\\) are assumed to contain independently and identically distributed zero-mean Gaussian noises with their covariance matrices denoted as \\(\\bm{\\Sigma}^{v}\\) and \\(\\bm{\\Sigma}^{\\omega}\\), respectively. The timing jitter in \\(t_{i}^{pk}\\) is modelled as zero-mean Gaussian noise with its standard deviation \\(\\sigma_{t}\\). Please note that one can choose to use other off-the-shelf ego-motion estimation methods depending on the sensor configurations on the target platforms, as long as the method can provide robust and consistent uncertainty estimates.\n\nGiven \\(\\bm{t}_{0:N-1}^{pk}\\), \\(\\bm{z}_{0:N-1}^{v}\\), \\(\\bm{z}_{0:N-1}^{\\omega}\\), and \\(t_{ref}\\), the vehicle ego-motion estimation is required to find out the sequence of Gaussian variables \\(\\left\\{\\textbf{x}_{i}^{veh} \\sim \\mathcal{N}\\left(\\bar{\\textbf{x}}_{i}^{veh}, \\bm{\\Sigma}_{i}^{veh}\\right)\\right\\}_{i=0}^{N-1}\\) representing the estimated vehicle egocentric poses at \\(\\bm{t}^{lpk}\\) with respect to that at \\(t_{ref}\\). Let \\(\\textbf{x}_{ref}^{veh}\\) denote the Gaussian variable representing the vehicle egocentric state at \\(t_{ref}\\).\n\\begin{equation}\n\\textbf{x}_{ref}^{veh} \\sim \\mathcal{N}\\left(\\bar{\\textbf{x}}_{ref}^{veh}, \\bm{\\Sigma}_{ref}^{veh}\\right),\n\\end{equation}\nwhere we set the mean vector \\(\\bar{\\textbf{x}}_{ref}^{veh} = \\bm{0}\\) and the covariance matrix \\(\\bm{\\Sigma}_{ref}^{veh}\\) to a diagonal matrix with close to zero elements, since we are performing ego-motion estimation within the local coordinate system of the vehicle at \\(t_{ref}\\).\n\nIf \\(t_{ref} > t_{0}^{lpk}\\), then backward ego-motion estimation is performed by first initialising intermediate variables:\n\\begin{align} \\label{eq:egomotion_prediction_init}\nt_{*} &\\leftarrow t_{ref} & \\bar{\\textbf{x}}_{*}^{veh} &\\leftarrow \\bar{\\textbf{x}}_{ref}^{veh} & \\bm{\\Sigma}_{*}^{veh} &\\leftarrow \\bm{\\Sigma}_{ref}^{veh}.\n\\end{align}\n\nThen for \\(i = \\max\\left( \\left\\{p : t_{p}^{pk} \\in \\bm{t}^{pk} \\wedge t_{p}^{pk} < t_{ref} \\right\\} \\right), \\cdots, 0\\), an augmented state vector is constructed by concatenating intermediate vehicle egocentric state \\(\\textbf{x}_{*}^{veh}\\) and kinematic measurements at \\(t_{i}^{pk}\\).\n\\begin{equation} \\label{eq:egomotion_prediction_start}\n\\textbf{x}_{*}^{a}\n\\sim\n\\mathcal{N}\\left(\n\\bar{\\textbf{x}}_{*}^{a},\n\\bm{\\Sigma}_{*}^{a}\n\\right),\n\\end{equation}\nwhere\n\\(\n\\bar{\\textbf{x}}_{*}^{a} =\n\\begin{bmatrix}\n\\left(\\bar{\\textbf{x}}_{*}^{veh}\\right)^{T} & \\left(\\bm{z}_{i}^{v}\\right)^{T} & \\left(\\bm{z}_{i}^{\\omega}\\right)^{T} & t_{i}^{pk} & t_{*}\n\\end{bmatrix}\n\\), and\n\\(\n\\bm{\\Sigma}_{*}^{a} =\n\\begin{bmatrix}\n\\bm{\\Sigma}_{*}^{veh} & \\bm{0} & \\bm{0} & 0 & 0 \\\\\n\\bm{0} & \\bm{\\Sigma}_{v} & \\bm{0} & 0 & 0 \\\\\n\\bm{0} & \\bm{0} & \\bm{\\Sigma}_{\\omega} & 0 & 0 \\\\\n0 & 0 & 0 & \\sigma_{t}^{2} & 0 \\\\\n0 & 0 & 0 & 0 & \\sigma_{t}^{2} \\\\\n\\end{bmatrix}\n\\).\n\nThe backward motion estimation goes from a later timestamp \\(t_{*}\\) to an earlier \\(t_{i}^{pk}\\), resulting in a negative time difference considered in the kinematic model.\n\nThe augmented state mean and covariance matrix are decomposed through UT into a set of sigma points.\n\\begin{equation}\n\\left\\{\\bm{\\mathcal{X}}_{j}^{a}, w_{j}^{m}, w_{j}^{c}\\right\\}_{j=0}^{2d} \\leftarrow UTD\\left(\\bar{\\textbf{x}}_{*}^{a}, \\bm{\\Sigma}_{*}^{a}\\right).\n\\end{equation}\n\nFor \\(j = 0,\\cdots,2d\\), motion estimation is conducted backward in time.\n\\begin{equation}\n\\bm{\\mathcal{Y}}_{j}^{veh} = f_{km}\\left(\\bm{\\mathcal{X}}_{j}^{a}\\right),\n\\end{equation}\nwhere \\(f_{km}\\left(\\cdot\\right)\\) is the vehicle kinematic model that predicts vehicle pose based on a given pose and kinematic velocities over a given time duration.\n\nThe estimated vehicle egocentric state at timestamp \\(t_{i}^{pk}\\) is recovered by\n\\begin{equation}\n\\bar{\\textbf{x}}_{i}^{veh}, \\bm{\\Sigma}_{i}^{veh} \\leftarrow UTR\\left(\\left\\{\\bm{\\mathcal{Y}}_{j}^{veh}, w_{j}^{m}, w_{j}^{c}\\right\\}_{j=0}^{2d}\\right).\n\\end{equation}\n\nThe results also serve as intermediate variables for the next iteration:\n\\begin{align} \\label{eq:egomotion_prediction_end}\nt_{*} &\\leftarrow t_{i}^{pk} & \\bar{\\textbf{x}}_{*}^{veh} &\\leftarrow \\bar{\\textbf{x}}_{i}^{veh} & \\bm{\\Sigma}_{*}^{veh} &\\leftarrow \\bm{\\Sigma}_{i}^{veh}.\n\\end{align}\n\nIf \\(t_{ref} \\leq t_{N-1}^{pk}\\), then forward vehicle ego-motion estimation is carried out by initialising intermediate variables as in \\eqref{eq:egomotion_prediction_init}, and for \\(i = \\min\\left(\\left\\{p : t_{p}^{pk} \\in \\bm{t}^{pk} \\wedge t_{p}^{pk} \\geq t_{ref}\\right\\}\\right),\\cdots,N-1\\), using the same set of equations \\eqref{eq:egomotion_prediction_start}-\\eqref{eq:egomotion_prediction_end}, except that in this case\n\\(\n\\bar{\\textbf{x}}_{*}^{a} =\n\\begin{bmatrix}\n\\left(\\bar{\\textbf{x}}_{*}^{veh}\\right)^{T} & \\left(\\bm{z}_{i}^{v}\\right)^{T} & \\left(\\bm{z}_{i}^{\\omega}\\right)^{T} & t_{*} & t_{i}^{pk}\n\\end{bmatrix}\n\\), and in every iteration the motion estimation starts from an earlier timestamp \\(t_{i}^{pk}\\) to a later \\(t_{*}\\).\n\n\\subsubsection{3D Lidar Points Motion Correction}\n\nWith a sequence of estimated vehicle egocentric poses \\(\\left\\{\\textbf{x}_{i}^{veh} \\sim \\mathcal{N}\\left(\\bar{\\textbf{x}}_{i}^{veh}, \\bm{\\Sigma}_{i}^{veh}\\right)\\right\\}_{i=0}^{N-1}\\) at \\(\\bm{t}^{pk}\\) obtained from the vehicle ego-motion estimation stage, motion correction is applied for each corresponding packet of 3D lidar measurement points.\n\nFor \\(i = 0,1,\\cdots,N-1\\), the estimated state mean and covariance matrix are decomposed into a set of sigma points:\n\\begin{equation}\n\\left\\{\\bm{\\mathcal{X}}_{i,k}^{veh}, w_{i,k}^{m}, w_{i,k}^{c}\\right\\}_{k=0}^{2d} \\leftarrow UTD\\left(\\bar{\\textbf{x}}_{i}^{veh}, \\bm{\\Sigma}_{i}^{veh}\\right).\n\\end{equation}\n\nA set of \\(4\\times 4\\) homogeneous transformation matrices \\(\\left\\{\\mathcal{T}_{i,k}^{veh}\\right\\}_{k=0}^{2d}\\) are constructed based on the rotation and translation in each \\(\\bm{\\mathcal{X}}_{i,k}^{veh}\\).\n\nFor \\(j = 0,\\cdots,M-1\\), and for \\(k = 0,\\cdots,2d\\), a motion corrected sigma point is calculated as\n\\begin{equation}\n\\bm{\\mathcal{Z}}_{i,j,k}^{cld} = (T_{veh}^{ld})^{-1} \\cdot \\mathcal{T}_{i,k}^{veh} \\cdot T_{veh}^{ld} \\cdot \\bm{z}_{i,j}^{ld},\n\\end{equation}\nwhere the lidar point is translated to the vehicle's base frame by the rigid transform \\(T_{veh}^{ld}\\), followed by transformation that encapsulates delta ego-motion in vehicle base frame. Lastly, the point is translated back to the lidar coordinate system.\n\nAt this stage, a motion corrected lidar point \\(\\bm{z}_{i,j}^{cld}\\) within lidar packet \\(pk_{i}\\) can be recovered to a Gaussian variable through\n\\begin{equation}\n\\bar{\\bm{z}}_{i,j}^{cld}, \\bm{\\Sigma}_{i,j}^{cld} \\leftarrow UTR\\left(\\left\\{\\bm{\\mathcal{Z}}_{i,j,k}^{cld}, w_{i,k}^{m}, w_{i,k}^{c}\\right\\}_{k=0}^{2d}\\right).\n\\end{equation}\n\nIn the end, the process produces a set of motion corrected sigma points for lidar points denoted by \n\\begin{equation}\n\\bm{\\Omega} = \\left\\{\\left\\{\\left\\{\\bm{\\mathcal{Z}}_{i,j,k}^{cld}\\right\\}_{j=0}^{M-1}, w_{i,k}^{m}, w_{i,k}^{c}\\right\\}_{k=0}^{2d}\\right\\}_{i=0}^{N-1}.\n\\end{equation}\n\nThe lidar ego-motion correction with uncertainty completes at this stage. Further transformation can be applied to \\(\\bm{\\Omega}\\) for lidar-to-camera projection with ego-motion uncertainty. Please refer to next section for details.\n\n\n\\subsubsection{Lidar-to-Camera Projection}\n\nThis stage is only for systems that require camera-lidar sensor fusion. It takes a motion corrected 3D lidar point cloud as input and project the lidar points to a given camera coordinate system. Essentially, the timestamp of the image for projection has been used as the reference time \\(t_{ref}\\) in the motion correction process, in pursuance of bringing about an accurate camera-lidar projection. Before projection happens, a 3D lidar point needs to be translated from lidar frame to camera frame given the extrinsic calibration between both camera and lidar sensors represented as the transformation matrix \\(T_{cam}^{ld}\\):\n\\begin{equation}\n\\bm{z}^{cam} = T_{cam}^{ld} \\bm{z}^{ld},\n\\end{equation}\nwhere \\(\\bm{z}^{cam} = \\begin{bmatrix} x^{cam} & y^{cam} & z^{cam} & 1 \\end{bmatrix}^{T}\\) is the 3D lidar point translated to camera frame.\n\nThe generic lidar-to-camera projection function is defined as\n\\begin{equation}\n\\begin{bmatrix} u \\\\ v \\end{bmatrix} =\nf_{proj}\n\\left(\\bm{z}^{cam}\\right),\n\\end{equation}\nwhich is to find the pixel coordinates \\(u\\) and \\(v\\) in the image frame corresponding to a 3D lidar point \\(\\bm{z}^{cam}\\) in the camera frame by using the camera model and its intrinsic parameters.\n\nThe function first makes use of the generic pinhole camera-image projection equations, which states\n\\begin{align}\na &=  \\frac{x^{cam}}{z^{cam}} & b &=  \\frac{y^{cam}}{z^{cam}} \\label{eq_pc1}\n\\end{align}\n\\begin{align}\nr &= \\sqrt{a^{2}+b^{2}} & \\theta &= \\textup{atan}(r) \\label{eq_pc2}.\n\\end{align}\n\nSince our cameras have fisheye lenses, we need to apply the distortion established by the camera model to find the corresponding pixel in the image \\cite{opencv}. The distortion of the lens is calculated as follows:\n\\begin{equation} \\label{eq_pc3}\n   \\theta_d = \\theta(1+k_1\\theta^2+k_2\\theta^4+k_3\\theta^6+k_4\\theta^8),\n\\end{equation}\n\nwhere $k_1$, $k_2$, $k_3$ and $k_4$ are the lens' distortion coefficients. Then we compute the distorted point coordinates as\n\\begin{align}\nx' &=  (\\theta_d\/r)a & y' &=  (\\theta_d\/r)b.\n\\end{align}\n\nThe definite pixel coordinates vector \\(\\begin{bmatrix} u & v \\end{bmatrix}^{T}\\) in the image frame of a 3D lidar point can be calculated as\n\\begin{align}\nu &= f_x \\cdot (x' + e y')+c_x & v &= f_y \\cdot y'+c_y,\n\\label{eq_pc4}\n\\end{align}\n\nwhere \\(e\\) is the camera's skew coefficient, $[c_x,c_y]$ the principal point offset and $[f_x, f_y]$ are the focal lengths expressed in pixel units.\n\nIn order to produce projected points in the image frame with uncertainty information, and also to avoid unnecessary UT, this stage works directly on \\(\\bm{\\Omega}\\), which is the set of sigma points for each 3D lidar point as the output of the 3D lidar points motion correction stage.\n\nWe can combine translation of each sigma point \\(\\bm{\\mathcal{Z}}_{i,j,k}^{cld} \\in \\bm{\\Omega}\\) from lidar frame to camera frame using \\(T_{cam}^{ld}\\) and the projection to the image frame by\n\\begin{equation} \\label{eq_pc0}\n\\left\\{\\bm{\\mathcal{K}}_{i,j,k}^{cam} : \\left(\\exists \\bm{\\mathcal{Z}}_{i,j,k}^{cld} \\in \\bm{\\Omega}\\right)\n\\left[\n\\bm{\\mathcal{K}}_{i,j,k}^{cam} = f_{proj}\\left(T_{cam}^{ld} \\bm{\\mathcal{Z}}_{i,j,k}^{cld}\\right)\n\\right]\n\\right\\}.\n\\end{equation}\n\nFor \\(i = 0,\\cdots,N-1\\) and for \\(j = 0,\\cdots,M-1\\), the image pixel projected from the lidar point \\(\\bm{z}_{i,j}^{cld}\\) within lidar packet \\(pk_{i}\\) can be recovered with its mean values and covariance matrix by\n\\begin{equation}\n\\begin{bmatrix} \\bar{u}_{i,j} \\\\ \\bar{v}_{i,j} \\end{bmatrix}, \\bm{\\Sigma}_{i,j}^{uv} \\leftarrow UTR\\left(\\left\\{\\bm{\\mathcal{K}}_{i,j,k}^{cam}, w_{i,k}^{m}, w_{i,k}^{c}\\right\\}_{k=0}^{2d}\\right).\n\\end{equation}\n\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Experiment Results}\n\nWe implemented the proposed approach in C++ under ROS Melodic release and tested it in the USyd Dataset \\cite{usyd_dataset, USYD_Segmentation_2019}, which was obtained with the electric vehicle platform shown in Figure \\ref{fig:platform}. The vehicle is equipped with a Velodyne VLP-16 lidar and five fixed lens gigabit multimedia serial link (GMSL) cameras, each covers a \\(100^{\\circ}\\) horizontal field of view. The camera images have a resolution of \\(1920 \\times 1208\\) and are captured at 30 FPS by an onboard NVIDIA DRIVE PX2 automotive computer. The extrinsic camera calibration is conducted relative to the lidar sensor frame, and both are registered to the local frame of reference of the vehicle. The platform also contains wheel encoders and an IMU, which produce odometry measurements at 100 Hz. The constant turn rate and velocity (CTRV) \nkinematic model is adopted for the vehicle.\n\nIn the experiment, we use the proposed approach to correct the lidar point cloud using the timestamp of the last lidar packet as \\(t_{ref}\\). We also project the latest lidar point cloud to the most recent image frames from three front facing cameras, in which case the timestamps of the image frames are chosen as \\(t_{ref}\\). Only the \\(x\\) component of the linear velocity measurements is used with the standard deviation of its noise set to 0.05 m\/s. The measurements of angular velocities in \\(roll\\), \\(pitch\\), and \\(yaw\\) are used with 2 deg\/s set as the standard deviation of their noise, which takes into account the IMU's thermo-mechanical white noise, and the noise from mechanical vibration when the vehicle is moving. \\(\\sigma_{t}\\) is set to 0.0006 s. Each corrected point cloud is published as a ROS \\textit{sensor\\_msgs\/PointCloud2} message, where the data fields of every 3D lidar point are augmented with its covariance in 3D, its projected image coordinates, and their associated covariance in 2D.\n\nAn example of lidar ego-motion correction can be found in Figure \\ref{fig:LidarCorrectionLarge}, which clearly demonstrates the corrected lidar point cloud with uncertainty estimates, and is compared with the uncorrected point cloud. The ego-motion distortion can often cause issues in lidar feature extraction. This can, for instance, manifest as a ghost image of the lidar features which is often observed in the overlapping area of the first and last packets of a lidar scan, as shown in the left column of Figure \\ref{fig:LidarCorrectionDetails}. As illustrated in the right column of Figure \\ref{fig:LidarCorrectionDetails}, the issue can be effectively rectified using the proposed approach, which also provides uncertainty estimate for each lidar point.\n\n\\begin{figure}[!t]\n\\vspace{2mm}\n\\centerline{\n\\includegraphics[width=0.9\\columnwidth]{Figures\/platform.png}\n}\n\\caption{Experimental platform equipped with five cameras (two side cameras and an arrange of three cameras front facing), one Velodyne VLP-16 lidar, wheel encoders and an IMU that contains gyroscopes, accelerometers and magnetometers. }\n\\label{fig:platform}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\vspace{2mm}\n\t\\centering\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionLarge:a}\n\t\t\\includegraphics[trim={1.5cm 0 1.5cm 0.9cm},clip, width=0.9\\columnwidth]{\"Figures\/784_uncor\"}}\n\t\\hspace{-0.0in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionLarge:b}\n\t\t\\includegraphics[trim={1.5cm 0 1.5cm 0.9cm},clip, width=0.9\\columnwidth]{\"Figures\/784_cortd_x\"}}\n\t\\hspace{-0.0in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionLarge:c}\n\t\t\\includegraphics[trim={1.5cm 0 1.5cm 0.9cm},clip, width=0.9\\columnwidth]{\"Figures\/784_cortd_y\"}}\n\t\\hspace{-0.0in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionLarge:d}\n\t\t\\includegraphics[width=2.0in]{\"Figures\/colorbar\"}}\n\t\\caption{Lidar point cloud before and after ego-motion correction. (a) shows the point cloud before ego-motion correction. (b) and (c) present the corrected point cloud coloured by the standard deviation in \\(x\\) and \\(y\\) directions, respectively. As lidar scans in the clockwise direction, the older points tend to have a higher level of uncertainty due to ego-motion. The uncertainty in \\(z\\) is found less significant and thus not shown.}\n\t\\label{fig:LidarCorrectionLarge}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\vspace{2mm}\n\t\\centering\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionDetails:a}\n\t\t\\includegraphics[trim={0 0.15cm  0 0.2cm},clip,width=1.6in]{\"Figures\/320_uncor\"}}\n\t\\hspace{-0.1in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionDetails:b}\n\t\t\\includegraphics[trim={0 0.15cm  0 0.2cm},clip,width=1.6in]{\"Figures\/320_cortd_x\"}}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionDetails:c}\n\t\t\\includegraphics[trim={0 0.15cm  0 0.4cm},clip,width=1.3in]{\"Figures\/799_uncor\"}}\n\t\\hspace{-0.1in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionDetails:d}\n\t\t\\includegraphics[trim={0 0.15cm  0 0.4cm},clip,width=1.3in]{\"Figures\/799_cortd_x\"}}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionDetails:e}\n\t\t\\includegraphics[trim={0 0.15cm  0 0.1cm},clip,width=1.6in]{\"Figures\/653_uncor\"}}\n\t\\hspace{-0.1in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionDetails:f}\n\t\t\\includegraphics[trim={0 0.15cm  0 0.1cm},clip,width=1.6in]{\"Figures\/653_cortd_x\"}}\n\t\\hspace{-0.0in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarCorrectionDetails:g}\n\t\t\\includegraphics[width=2.0in]{\"Figures\/colorbar\"}}\n\t\\caption{Lidar features before and after ego-motion correction. (a), (c), and (e) illustrate the lidar points of a traffic sign, a pillar, and a pedestrian, respectively, before ego-motion correction. (b), (d), and (f) depict the motion corrected points coloured by the standard deviation in \\(x\\) direction. The standard deviations in \\(y\\) and \\(z\\) directions are available but not shown here.}\n\t\\label{fig:LidarCorrectionDetails}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\vspace{2mm}\n\t\\centering\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarProjectionLarge:a}\n\t\t\\includegraphics[trim={0.2cm 0.15cm 0.2cm 0.15cm},clip, height=2.37in]{\"Figures\/635_54770l_case1\"}}\n\t\\hspace{-0.13in}\n\t\\subfigure[]{ \n\t\t\\label{fig:LidarProjectionLarge:b}\n\t\t\\includegraphics[trim={0.2cm 0.15cm 0.2cm 0.15cm},clip, height=2.37in]{\"Figures\/635_54770l_case2\"}}\n\t\\caption{An example of lidar-to-camera projection using the proposed approach. The projection of the raw point cloud to an image from the front-left camera is shown in top figure of (a), where the misalignment of lidar points and visual features is apparent. A finer overlapping between the points and the image can be observed in the projection of motion corrected lidar points in bottom figure of (a). In addition, (b) shows the uncertainty estimates of each projected point as a result of the proposed approach. Every ellipse in (b) covers a 95\\% confidence area. The projected lidar points are colour-coded by range.}\n\t\\label{fig:LidarProjectionLarge}\n\\end{figure}\n\n\nBesides the results previously presented in Figure \\ref{fig:LidarProjectionDetails}, more results of lidar-to-camera projection can be found in Figure \\ref{fig:LidarProjectionLarge}. It can be clearly seen that precision of the projection improves significantly through the use of the proposed approach. The uncertainty estimates for each projected lidar point on the image are illustrated in Figure \\ref{fig:LidarProjectionLarge:b}. It is important to note that in each lidar-to-camera projection figure presented here, the lidar can partially see behind objects seen by the camera. The lidar viewpoint is slightly different to the camera as the sensors are not co-located, and as a result objects observed by one sensor can block the visibility of the other. This is due to the physical setup of cameras and the lidar in the vehicle being mounted in different positions with the aim of providing wide coverage using an array of cameras. In this case, the cameras and lidar perceive the environment from different vantage points.\nFurther processing is required to address this occlusion problem \\cite{paper:SchneiderHimmelsbach2010}. \n\n\n\\subsection{Simulation Results}\n\nThe proposed approach is also assessed quantitatively using simulation, as the ground truth is not available for the experiments with the real vehicle. The simulation is setup as close as possible to the vehicle platform used in the experiment. In every simulation episode the vehicle is moving with ground truth constant linear velocity \\(v_{x}\\) in the vehicle's \\(x\\) direction of travel and angular velocity \\(\\omega_{yaw}\\) in \\(yaw\\) randomly drawn from uniform distributions \\(\\mathcal{U}\\left(2, 10\\right)\\) m\/s and \\(\\mathcal{U}\\left(-60, 60\\right)\\) deg\/s, respectively. As the vehicle moves, the lidar scans for one revolution in 0.1 s, generating 76 lidar packets at different rotational angles. Each packet contains one pair of elevation angle and range data drawn from uniform distributions \\(\\mathcal{U}\\left(-15, 15\\right)\\) deg and \\(\\mathcal{U}\\left(1, 100\\right)\\) m, respectively. Linear and angular velocity measurements are polluted with additive Gaussian noise \\(\\mathcal{N}\\left(0, 0.1^{2}\\right)\\) m\/s and \\(\\mathcal{N}\\left(0, 5^{2}\\right)\\) deg\/s, respectively. The timestamp of every piece of sensory measurement contains jitter modelled as Gaussian noise \\(\\mathcal{N}\\left(0, 0.0003^2\\right)\\) s. Those parameters are chosen to produce a clear result for visualisation. As soon as the lidar finishes one revolution of scanning, the front camera takes an image, to which the lidar point cloud are then projected. Here, the timestamp of the image is used as the reference time. The same intrinsic and extrinsic calibration parameters in the experiment vehicle platform are used in the simulation.\n\nThe simulation results are analysed based on 200 Monte Carlo runs, which in total generate over 15,000 3D lidar points and 4,000 2D projected points. Figure \\ref{fig:SimulationLidar} presents the ego-motion corrected lidar point cloud and the comparison with ground truth and uncorrected point clouds from one of the simulation runs, and Figure \\ref{fig:SimulationImage} illustrates the same point cloud projected to the image. The ground truth linear and angular velocities in this particular case are 2.81 m\/s and -56.2 deg\/s (negative means turning right), respectively. Due to the constraint of figure size, we only show the uncertainty of a corrected 3D lidar point and a 2D projected point in Figure \\ref{fig:SimulationLidar:b} and Figure \\ref{fig:SimulationImage:b}, respectively.\n\nNormalised estimation error squared (NEES) is adopted in the test as the metric of consistency for the proposed lidar ego-motion correction approach. The NEES value for a given 3D lidar sample or 2D projected sample \\(\\mathcal{N}\\left(\\bar{\\bm{z}}_{i}, \\bm{\\Sigma}_{i}\\right)\\) and its ground truth \\(\\bm{z}_{i}\\) is calculated by\n\\begin{equation}\n\\epsilon\\left(i\\right) = \\left(\\bar{\\bm{z}}_{i} - \\bm{z}_{i}\\right)^{T} \\bm{\\Sigma}_{i}\\left(\\bar{\\bm{z}}_{i} - \\bm{z}_{i}\\right).\n\\end{equation}\n\nThen the \\(\\epsilon\\left(i\\right)\\) will have a \\(\\chi^{2}\\) (chi-square) distribution with \\(dim\\left(\\bm{z}_{i}\\right)\\) degrees of freedom, under the hypothesis that the tested estimator is consistent and approximately linear and Gaussian\n\\cite{paper:BahrWalter}. The state estimation errors are considered consistent with the calculated covariances if \n\\(\n\\epsilon\\left(i\\right) \\in \\left[\\chi^{2}_{dim\\left(\\bm{z}_{i}\\right)}\\left(0.025\\right),\\chi^{2}_{dim\\left(\\bm{z}_{i}\\right)}\\left(0.975\\right)\\right]\n\\), \nwhere \\(dim\\left(\\bm{z}_{i}\\right) = 3\\) for a 3D point, and \\(dim\\left(\\bm{z}_{i}\\right) = 2\\) for a 2D point. This interval associates bounds for the two-sided \\(95\\%\\) probability. The estimation tends to be optimistic if the \\(\\epsilon\\) for all motion corrected lidar points rises significantly higher than the upper bound, while if it stays below the lower bound for a majority of time, the estimator is considered conservative \\cite{paper:BaileyNieto}. The consistency check results are presented in Figure \\ref{fig:NEES}.\n\n\\begin{figure}[!t]\n\\vspace{2mm}\n\t\\centering\n\t\\subfigure[]{ \n\t\t\\label{fig:SimulationLidar:a}\n\t\t\\includegraphics[width=3.3in]{\"Figures\/simulation_lidar\"}}\n\t\\subfigure[]{ \n\t\t\\label{fig:SimulationLidar:b}\n\t\t\\includegraphics[width=3.1in]{\"Figures\/simulation_ellipsoid\"}}\n\t\\caption{The comparison of motion corrected lidar point cloud with uncorrected and ground truth point clouds. In (a), as the lidar rotates in clockwise direction, the correction is found more evident to those older points, which have timestamps further from the reference time. The uncertainty of a 3D lidar point after correction is represented as an ellipsoid in (b), which covers 95\\% volume of confidence.}\n\t\\label{fig:SimulationLidar}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\vspace{2mm}\n\t\\centering\n\t\\subfigure[]{ \n\t\t\\label{fig:SimulationImage:a}\n\t\t\\includegraphics[width=3.1in]{\"Figures\/simulation_image\"}}\n\t\\subfigure[]{ \n\t\t\\label{fig:SimulationImage:b}\n\t\t\\includegraphics[width=3.1in]{\"Figures\/simulation_ellipse\"}}\n\t\\caption{The projection of motion corrected lidar point cloud to the image compared with that of uncorrected and ground truth point clouds. lidar scans from left to right in the image in (a), where the correction is found more evident to those points at the left side, which have timestamps further from the reference time. The uncertainty of a 2D projected point after correction is represented as an ellipse in (b), which covers a 95\\% confidence area.}\n\t\\label{fig:SimulationImage}\n\\end{figure}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\subfigure[]{ \n\t\t\\label{fig:NEES:a}\n\t\t\\includegraphics[trim={0.9cm 0 0.9cm 0.3cm},clip,width=3.2in]{\"Figures\/simulation_lidar_nees\"}}\n\t\\subfigure[]{ \n\t\t\\label{fig:NEES:b}\n\t\t\\includegraphics[trim={0.9cm 0 0.9cm 0.3 cm},clip,width=3.2in]{\"Figures\/simulation_image_nees\"}}\n\t\\caption{NEES consistency test for the ego-motion correction results. The in-bound rate for the motion corrected 3D lidar points is about 90.92\\%, while the in-bound rate of the projection results in the image frame is found to be 94.29\\%. Both results indicate consistent estimation of uncertainties.}\n\t\\label{fig:NEES}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Conclusions And Future Work}\n\\label{sec:conlusions}\n\nIn this paper, a novel probabilistic approach is proposed for lidar ego-motion correction and lidar-to-camera projection with robust uncertainty estimation. The approach accounts for the main error sources which include noise in ego-motion estimation and time jitter in sensor measurements due to practical and theoretical limitations.\n\n\n\nThe proposed approach considers a sequence of lidar packets, calculates the vehicle ego-motion estimation results for the given packet timestamps, applies the motion correction to the lidar packets against an arbitrarily chosen reference timestamp, and projects the motion corrected lidar points to camera coordinate system. The chain of the above three cascaded stages is formulated into an unscented transform pipeline. Essentially, the corrected and projected points are produced with ego-motion uncertainty information preserved for subsequent processing.\n\nThe experimental results demonstrate the accuracy of the ego-motion correction for lidar points, and the projection to the image frame. This was tested on an electric vehicle platform driven in a university campus environment. The simulation results further validate the consistency of the uncertainty estimation in motion correction and lidar-to-camera projection.\n\nEssentially, the capability of producing robust and consistent uncertainty estimates incorporating the lidar ego-motion correction process makes the proposed approach one of the first of its kind to have the potential to be integrated into perception applications that require uncertainty information. The proposed approach associates 3D lidar point and the 2D image coordinates in a probabilistic manner. This is particularly useful in applications that involve probabilistic camera-lidar sensor fusion, where information can be transferred from lidar point to image domain and vice versa with the relevant uncertainty considered.\n\nThe future work includes the probabilistic fusion of ego-motion corrected lidar points with semantically labelled images, which combines the heuristic uncertainty associated with a labelled image and the uncertainty from the ego-motion correction of LiDAR point clouds. In this case, the value of the semantic label retrieved from the corresponding pixel in an image frame can be included probabilistically into a point cloud as an additional information field for each 3D point. This helps pave the way to a higher level understanding of the scene, which can be used to enable context based algorithms for collision avoidance and navigation.\n\n \n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nBy the 1940's, physicists had identified two classes of ``elementary\" particles with widely different group behavior, bosons and fermions. The prototypic boson is the photon which generates electromagnetic forces;  electrons, the essential constituents of matter, are fermions which satisfy Pauli's exclusion principle. This distinction was quickly extended to Yukawa's particle (boson), the generator of  Strong Interactions, and to nucleons (fermions). A compelling characterization followed: matter is built out of fermions, while forces are generated by bosons.  \n\nEinstein's premature dream of unifying {\\em all} constituents of the physical world should have provided a clue for that of fermions and bosons; yet it took physicists a long time to relate them by symmetry. This fermion-boson symmetry is called ``{\\em Supersymmetry}\".  \n\nSupersymmetry, a necessary ingredient of string theory, turns out to have further remarkable formal properties when applied to local quantum field theory, by restricting its ultraviolet behavior, and  providing unexpected insights into its non-perturbative behavior. It  may also play a  pragmatic role as the glue that  explains the weakness of the elementary forces within the  Standard Model of Particle Physics at short distances.\n\n \n\n\\section{Early Hint}\nIn 1937, Eugene Wigner, with some help from his brother-in-law, publishes one of his many famous papers\\cite{Wigner} ``On Unitary Representations of the Inhomogeneous Lorentz Group\". He was then at the University of Wisconsin at Madison, a refugee from Princeton which had denied him tenure. It was not an easy paper to read, but its results were very simple: there were five types of representations labelled by the values of  $P^2\\equiv p^\\mu p_\\mu=m^2$, one of the Poincar\\'e group's Casimir operator. \n \n All but two representations describe familiar particles found in Nature. Massive particles come with momentum $\\bf p$, spin $j$, and $2j+1$ states of polarization, e.g.  electrons and nucleons with spin $1\/2$. There are also four types of massless representations  with spin replaced by helicity (spin projection along the momentum). The first two describe massless particles with a single helicity (photons with helicity $\\pm1$), or half-odd integer helicity, such as   ``massless\" neutrinos with helicity $+1\/2$.\n \nThe last two representations $O(\\Xi)$ and $O'(\\Xi)$ describe states which look like massless ``objects\", particle-like in the sense that they have four-momentum, but with bizarre helicities: each representation contains an infinite tower of helicities, one with  integer helicities, the other with half-odd integer helicities. These have no analogues in Nature\\footnote{``Infinite spin\" representations do not appear in the Poincar\\'e decomposition of the conformal group}.   \n \nPhysicists were slow in recognizing the importance of  group representations, even though Pauli's provided the first solution of the quantum-mechanical Hydrogen atom using  group-theory.  Wigner's paper does not seem to have moved any mountains, and infinite spin representations were simply ignored, except of course by Wigner. \n\nYet, $O(\\Xi)$ and $O'(\\Xi)$ contained important information: they are ``supersymmetric partners'' of one another!\n\n\\section{Hadrons \\& Mesons}\n\\label{sec:2}\nSymmetries were gaining credence among physicists, not as a simplifying device but as a guide to the organization of Nature. Wigner and St\\\"uckelberg's ``supermultiplet model\" unified $SU(2)$ isospin and spin. Once Gell-Mann and Ne'eman generalized isospin to $SU(3)$, it did not take long for \n Feza G\\\"ursey and Luigi Radicati\\cite{Gursey}, as well as Bunji Sakita\\cite{Sakita}, to propose its unification with spin into $SU(6)$.  Pseudoscalar and vector mesons (bosons)  were found in the $\\bf {35}$ of SU(6), while the hadrons (fermions) surprisingly lived in the $\\bf{56}$, not in the $\\bf{20}$\\cite{Sakita}, as expected by the statistics of the time. This non-relativistic unification\n proved very successful, both experimentally and conceptually, since it led to the hitherto unsuspected {\\em color} quantum number. \n \nIn 1966, Hironari Miyazawa\\cite{Miyazawa}  proposed further unification. His aim was to assemble the fermionic $\\bf{56}$ and the bosonic $\\bf{35}$ into one mathematical structure such as $SU(9)$, but at the cost of disregarding spin-statistics.\n\nIn order to explain the bounty of strange particle discovered in the 1950's, Sakata had proposed to explain mesons as $T\\overline T$  bound states of the spin one-half triplet \n \n$$T~=~(\\,p,\\,n,\\,\\Lambda\\,).$$ \nMiyazawa adds  a {\\em pseudoscalar} triplet  \n\n$$t~=~(\\,K^+_{},  K^0_{},\\,\\eta\\,),$$\nto the Sakata spinor triplet. The hadron octet would then be described by another bound state, $T\\bar t$, but he could not describe the spin three-half baryons decimet in the ${\\bf 56}$. \n \nHe introduces a toy model with two fundamental constituents, a spin one-half and a spin zero particle, $ {\\bf p}=(\\alpha_\\uparrow,\\alpha_\\downarrow,\\gamma)$. The nine currents\n \n$$ {\\bf p}^\\dagger\\lambda^{}_i{\\bf p}=\\cases{F^{}_i,\\quad i=0,1,2,3,8;\\cr G^{}_i,\\quad i=4,5,6,7},$$ \nsatisfy a current algebra with both commutators and anticommutators, \n\n\\[\n[\\,F^{}_i\\,,\\,F^{}_j\\,]~=~if^{}_{ijk} F^{}_k, \\]\n\\[ [\\,F^{}_i\\,,\\,G^{}_j\\,]~=~if^{}_{ijk} G^{}_k,\\]\n\\[ \\{\\,G^{}_i\\,,\\,G^{}_j\\,\\}~=~d^{}_{ijk} F^{}_k,\\]\na ``generalized Jordan algebra\" which he calls $V(3)$. This is the first example, albeit non-relativistic, of a superalgebra, today called $SU(2\/1)$ with even part  $SU(2)\\times U(1)$.\n\nIn 1967, he expanded his construction\\cite{Miyazawa2}, to  general superalgebras he calls $V(n,m)$ with the idea of including the decimet. Alas, the phenomenology was not as compelling as that of $SU(6)$; two of the quarks inside a nucleon do not seem live together in an antitriplet color state.\n\nIn 1969, F.A. Berezin and G. I. Kac\\cite{Berezin} show the mathematical consistency of graded Lie algebra which contains both commutators and anticommutators; they  give its simplest example generated by the three Pauli matrices $\\sigma_+,\\sigma_-,\\sigma_3$. Physical applications are not discussed, although Berezin's advocacy of Grassmann variables in path integrals was no doubt a motivation.  \n\n\\section{Dual Resonance Models}\n\\label{sec:3}\nIn the 1960's, physicists had all but given up on a Lagrangian description of the Strong Interactions, to be replaced by the S-matrix program:  amplitudes were  determined from general principles and symmetries,  locality, causality, and Lorentz invariance.  Further requirements on the amplitudes such as Regge behavior and its consequent bootstrap program were still not sufficient to determine the amplitudes.  \n\nIn 1967, Dolen, Horn and Schmid\\cite{Dolen} discovered a peculiar relation in $\\pi-N$ scattering.  At tree-level, its fermionic $s$-channel  ($\\pi\\,N\\rightarrow \\pi\\,N $) is dominated by resonances ($\\Delta^{++}$, ...), as shown by countless experiments. On the other hand, its bosonic $t$-channel ($\\pi\\,\\bar\\pi \\rightarrow N \\,\\overline N$) is dominated by the $\\rho$-meson. Using the tools of S-matrix theory in the form of ``finite energy sum rules\", they found that the Regge shadow of the bosonic $t$-channel's $\\rho$-meson {\\em averaged} the fermionic resonances in the $s$-channel! This was totally unexpected since these two contributions, described by different Feynman diagrams, should have been independent. Was this the additional piece of information needed to fully determine the amplitudes of Strong Interactions? This early example of fermion-boson kinship led, through an unlikely tortuous path, to modern Supersymmetry. \n\nAn intense theoretical search for amplitudes where the $s$- and $t$-channel contributions are automatically related to one another followed. Under the spherical cow principle, spin was set aside and the search for DHS-type amplitudes focused on the purely bosonic process $\\omega\\rightarrow \\pi\\pi\\pi$\\cite{Ademollo}. Soon thereafter, Veneziano\\cite{Veneziano} proposed a four-point amplitude with the desired crossing symmetry,\n\n$$A(s,t) \\sim \\frac{\\Gamma(-\\alpha(s))\\Gamma(-\\alpha(t))}{\\Gamma(-\\alpha(s)-\\alpha(t)},$$\nwhere $\\alpha(x)=\\alpha_0+\\alpha'x$  is the linear Regge trajectory. It displays an infinite number of poles in {\\em both}  s-channel $s>0,~ t<0$ and t-channel $s<0,~t>0$. \n \nVeneziano's construction was quickly generalized to n-point ``dual\" amplitudes. The infinite series of poles were recognized as the vibrations of a string\\cite{string}. \n\nThe amplitudes were linear combinations of tree chains which factorize into three-point vertices and propagators. A generalized coordinate emerged\\cite{Fubini} from this analysis,\n\n$$\\quad Q^{}_\\mu(\\tau)=x^{}_\\mu+\\tau\\,p^{}_\\mu+\\sum_{n=1}^\\infty\\frac{1}{\\sqrt{2n\\alpha'}}\\left(a^{}_{n\\mu}e^{in\\tau}_{}-a^{\\dagger }_{n\\mu}e^{-in\\tau}_{}\\right),$$\nwith an infinite set of oscillators,\n\n$$[a^{}_{n\\mu},a^{\\dagger}_{m\\nu}]=\\delta^{}_{nm}g^{}_{\\mu\\nu}$$\nThe vertex for emitting a particle of momentum $k_\\mu$ from the linear chain was simple,\n\n$$V(k,\\tau)~=:e^{ik\\cdot Q(\\tau)}_{}:.$$\nOut of its corresponding generalized momentum\n\n\\begin{equation} P^{}_\\mu(\\tau)~=~\\frac{dQ^{}_\\mu}{d\\tau}, \\end{equation}\none derived the operators,\n\n$$L^{}_n~=~\\frac{1}{2\\pi}\\int^\\pi_{-\\pi}d\\tau e^{in\\tau}_{}:P^\\mu_{}P_\\mu^{}:~\\equiv~<:P^\\mu_{}P_\\mu^{}:>^{}_n, $$\nwhich satisfy the Virasoro algebra\\footnote{c-number is added anachronostically},\n \n $$[\\,L^{}_m\\,,\\,L^{}_n\\,]~=~(m-n)L^{}_{n+m}+{{\\frac{D}{12}m(m^2-1)\\delta^{}_{m,-n}}}.$$\n Its finite subalgebra, $L_0,L_\\pm$, the Gliozzi algebra, generates conformal transformations in two dimensions. \nThe  propagator was given by\n\n$$\\frac{1}{(\\alpha'L^{}_0+1)}.$$\n\n\n\\section{Superstrings}\nThe Klein-Gordon equation for a point particle,\n\n$$0~=~p^2_{}+m^2_{}~=~ <P^\\mu_{}>^{}_0<P_\\mu^{}>^{}_0+m^2, $$\ncould then be interpreted as a special case of \n \n$$0~=~ <P^\\mu_{}P_\\mu^{}>^{}_0+ m^2$$\nsuggesting a correspondence\\cite{Ramond1} between point particles and dual amplitudes, \n\n$$<A><B>~\\rightarrow~<A\\,B>.\n$$ \nFermions should satisfy the Dirac equation, \n\n$$0~=~\\gamma^{}_\\mu\\,p^\\mu_{}+m~=~<\\Gamma_\\mu^{}>^{}_0<P^\\mu>^{}_0+m.$$\nThis requires a generalization of the Dirac matrices  as dynamical operators,\n\n$$\\gamma^{}_\\mu~~\\rightarrow~~ \\Gamma^{}_\\mu~=~\\gamma^{}_\\mu+i\\gamma^{}_5\\sum_{n=0}^\\infty\\left(b^{}_{n\\mu} e^{in\\tau}_{}+b^{\\dagger}_{n\\mu} e^{-in\\tau}_{}\\right)\n$$\nwhere the oscillators are {\\em Lorentz vectors}\\footnote{Later was it realized that this made sense only in ten space-time dimensions where the little group is the spinor-vector schizophrenic $SO(8)$}, which satisfy anticommuting relations,\n\n$$\\{b^{}_{n\\mu},b^{\\dagger}_{n\\mu}\\}~=~\\delta^{}_{nm}g^{}_{\\mu\\nu},$$\nthe sum running over the positive integers. \n\nThis led me to propose the string Dirac equation in \nthe winter of 1970\\cite{Ramond}, which readily followed from that correspondence,\n\n$$ 0~=~{{<\\Gamma^{}_\\mu\\,P^\\mu_{}>^{}_0+m}}.$$\nThe basic Dirac algebra, $\\{\\gamma\\cdot p,\\gamma\\cdot p\\}=p^2_{}$ \nis seen to be generalized to an algebra with both commutator and anticommutators,\n\n$$\\{\\,F_n^{}\\,,\\,F^{}_m\\,\\}~=~ 2L^{}_{n+m},\\quad [\\,L^{}_n\\,,\\,F^{}_m\\,]~=~(2m-n)F^{}_{m+n},$$\nwhere $F_n=<\\Gamma_\\mu P^\\mu>_n$, and these new $L_n$'s also satisfy the Virasoro algebra, but with a different $c$-number.\n\nAndr\\'e Neveu and John Schwarz then compute the amplitude for a dual fermion emitting three pseudoscalars with the Yukawa vertex, \n \n$$\\Gamma_5^{}:e^{ik\\cdot Q(\\tau)}:,\\quad \\Gamma_5~=~\\gamma_5(-1)^{\\sum b_{n}^\\dagger\\cdot b_{n}},$$ \nand find that the resulting amplitude contains an infinite number of poles in its fermion-antifermion channel, and even identify the residue of the first pole\\cite{Neveu2}!\n\n A new model with bosonic poles and vertices emerges, written in terms of  an infinite tower of anticommuting vector oscillators,\n\n\n$$\\{b^{}_{r\\mu},b^\\dagger_{s\\nu}\\}~=~\\delta^{}_{rs}g^{}_{\\mu\\nu},\\quad r,s={\\textstyle\\frac{1}{2},\\frac{3}{2},\\cdots}.$$\nThe triple boson vertex is given by\n\n$$V_{NS}^{}(k,\\tau)k_{}^\\mu~=~ H^{}_\\mu(\\tau):e^{ik\\cdot Q(\\tau)}_{}:,$$\nwhere\n\n$$\nH^{}_\\mu(\\tau)~=~\\sum_{ r=1\/2,3\/2,\\dots} [b^{}_{r\\mu}e^{-ir\\tau}+b^\\dagger_{r\\mu}e^{ir\\tau}].$$\nThese are the building blocks of the  ``Dual Pion model\"\\cite{Neveu}, published in April 1971. The algebraic structure found in the generalized Dirac equation remains the same, producing a super-Virasoro algebra which decouples unwanted modes\\cite{NST}, with $\\Gamma_\\mu$ replaced by $H_\\mu$, through the operators,\n\n$$G_r^{}~=~<H\\cdot P>^{}_r,\\quad r= {\\textstyle\\frac{1}{2},\\frac{3}{2},\\cdots}$$ \n \nThe close relation of the two sectors is soon after formalized by Jean-Loup Gervais and Bunji Sakita\\cite{Gervais} who write them in terms of  a world-sheet $\\sigma$-model, with different boundary conditions, symmetric for the fermions, antisymmetric for the bosons.  They call the transformations  generated by the anticommuting Virasoro  opertors, {\\em supergauge transformations}, the first time the name ``super\" appears in this context.   \n\nThe following years saw the formulation of the RNS (NSR to some) ``Dual Fermion Model\", generating  dual amplitudes  with bosons and fermions legs.\nIt lived in ten space-time dimensions, with states determined in terms of transverse fermionic and bosonic harmonic oscillator operators. \n\nIn the fermionic ``R-sector\", the spectrum of states is spanned by the fermionic ground state, $u|0>$ where $u$ is a fixed 32-dimensional spinor,  annihilated by both transverse  bosonic and fermionic oscillators, $a^{}_{ni}$ and $b^{}_{ni}$, $i=1,2\\dots ,8$, and integer $n$. The fermion masses are determined by\n\n$$\\alpha'm^2_R~=~\\sum_{n=1}^\\infty n\\Big[a^\\dagger_n\\cdot a^{}_n+b^\\dagger_n\\cdot b^{}_n\\Big] $$\nThe  bosonic ``NS-sector\"  spectrum starts with a tachyon,  $|0>$ annihilated by the same $a^{}_{ni}$, but also by the NS fermionic oscillators $b^{}_{ri}$, where $r$ runs over half-integers. The boson masses satisfy \n\n$$\\alpha'm^2_{NS}~=~\\sum_{n=1}^\\infty n a^\\dagger_n\\cdot a^{}_n+\\sum_{\\scriptstyle r=\\frac{1}{2}}rb^\\dagger_r\\cdot b^{}_r-\\frac{1}{2}.$$\n\nBut there were idiosyncrasies. The correspondence between Neveu-Schwarz and the dual fermion states differed for states with an even number ($G\\equiv(-1)^{\\sum b_r^\\dagger\\cdot b^{}_r}=-1$) of $b^\\dagger_r$, and states with an odd number, and there is a tachyon in the even number spectrum, at $\\alpha'm^2_{NS}=-1\/2$..  \n\nIn 1976, F. Gliozzi, Jo\\\"el Scherk, and David Olive\\cite{GSO} noticed that the NS tachyon can be eliminated by requiring an odd number of anticommuting operators in the bosonic spectrum, ($G=-1$). The NS ground state \n\n$$\\alpha'm^2_{NS}=0: ~~~b^\\dagger_{1i}|0\\rangle,$$\nnow consists of eight bosons, transforming as the vector(=spinor) $SO(8)$ representation.  The first excited states are \n\n$$\\alpha'm^2_{NS}=1:~~~b^\\dagger_{\\scriptstyle\\frac{1}{2}i}b^\\dagger_{\\scriptstyle\\frac{1}{2}j}b^\\dagger_{\\scriptstyle\\frac{1}{2}k}|0\\rangle,~~b^\\dagger_{\\scriptstyle\\frac{1}{2}i}a^\\dagger_{1j}|0\\rangle,~\nb^\\dagger_{\\scriptstyle\\frac{3}{2}i}|0\\rangle,$$\nthat is $128=56(8.7.6\/1.2.3)+64 (8.8)+8$ bosonic states, and so on.\n\nIn their next step, they  show that the R ground state solution could also be reduced to eight fermionic degrees of freedom. In ten dimensions, while a spinor has naturally thirty-two degrees of freedom, they showed that one can impose {\\em both} chiral and Majorana (reality) restrictions on it, and reduce the spinor to eight dimensions, the spinor(=vector) $SO(8)$ representation. \n\n$$\\alpha'm^2_{R}=0:~~~\\psi_{\\alpha}|0\\rangle,~~\\alpha=1,2\\cdots 8.$$\nThe first excited state of the R-sector consists of\n\n$$\\alpha'm^2_{R}=1:~~~b^\\dagger_{1i}\\psi_{\\alpha}|0\\rangle,~~a^\\dagger_{1j}\\psi_{\\alpha}|0\\rangle,$$\nwith $128=8.8+8.8$ fermionic states! This was no accident, and using one of Jacobi's most obtuse relations, they showed that this equality obtained at all levels. Indeed this was supersymmetry, with the same number of bosons and fermions, albeit in ten space-time dimensions.   \n\nFermion-boson symmetry, born in its world-sheet realization, reappears as supersymmetry in ten-dimensional space-time. \n\nMeanwhile, behind the iron curtain, ...\n\\section{Russians}\n\\label{sec:4}\nIn March 1971, there appears a remarkable and terse paper by Yu. Gol'fand and E. Likhtman\\cite{Golfand}  who extend the Poincar\\'e algebra \ngenerated by $P_\\mu$ and $M_{\\mu\\nu}$ to ``bispinor  generators\", $W_\\alpha$ and $\\overline W_\\beta$, which generate spinor translations.\n\nCognizant that spin-statistics  requires anticommutating spinors, they arrive at the parity-violating algebra,\n  \n\\begin{equation}\n\\{W,W\\}~=~[P_\\mu,P_\\nu]~=~0,\\quad \\{W,\\overline W\\}~=~\\frac{(1+\\gamma_5)}{2}\\gamma_\\mu P_\\mu.\\end{equation}\nassuming no other subalgebra of the Poincar\\'e group.  With  little stated motivation, they have written down the ${\\cal N}=1$ superPoincar\\'e algebra in four dimensions!\n \nThey identify its simplest representation: two ``scalar hermitean\" fields $\\phi(x)$ and $\\omega(x)$, and one left-handed spinor field $\\psi_1(x)$, of equal mass, the earliest mention of the Wess-Zumino supermultiplet. They do not consider auxiliary fields nor display the transformation properties of these fields. However, they show the spinor generators as bilinears in those fields, \n\n\\begin{equation}\nW~=~\\frac{(1+\\gamma_5)}{2}\\int d^3x\\Big[\\phi^*\\uplrarrow\\partial_0\\psi^{}_1(x)+\\omega(x)\\uplrarrow\\partial_0\\psi^{c}_1(x)\\Big].\\end{equation}\n\nThey also describe the  {\\em massive} vector multiplet follows with the vector field $A_\\mu(x)$, a scalar field $\\chi(x)$ and a spinor field $\\psi_2(x)$. They write down its  spinor current,\n\n\\begin{equation}\nW~=~\\frac{(1+\\gamma_5)}{2}\\int d^3x\\Big[\\chi\\uplrarrow\\partial_0\\psi^{}_2(x)+A_\\mu(x)\\uplrarrow\\partial_0\\gamma_\\mu\\psi^{}_2(x)\\Big].\\end{equation}\n\nThis ground-breaking paper ends with the difficult task of writing interactions. Self-interactions of the WZ multiplet are not presented, \n only  its interactions with a massive Abelian vector supermultiplet.  This,  the last formula in their paper, is a bit confusing since $\\phi$ and $\\omega$ now appear as complex fields (setting $\\omega=0$ and replacing the complex $\\phi$ by $\\phi+i\\omega$ is more what they need), but it contains now familiar features, such as the squared $D$-term. \n\nGol'fand and Likhtman had firmly planted the flag of supersymmetry in four-dimensions. \n\nInterestingly, physicists on both sides of the iron curtain seemed  oblivious to this epochal paper. \n\nE. Likhtman seems to be the only one who followed up on this paper. He notices\\cite{Lebedev} that the vacuum energy cancels out because of the equal number of  mass bosons and fermions with the same mass. He finds scalar masses  only logarithmically divergent, which he mentions in a later publication\\cite{Likhtman}.   \n\nIn December 1972, in an equally impressive paper,  D.V. Volkov, and V.P. Akulov\\cite{Volkov}, want to explain the masslessness of  neutrinos in terms of  an invariance principle. They note that the neutrino free Dirac equation is invariant under the transformations,\n\n$$\\psi\\rightarrow \\psi+ \\zeta,\\quad x_\\mu\\rightarrow x_\\mu-\\frac{a}{2i}(\\zeta^\\dagger\\sigma_\\mu\\psi-\\psi^\\dagger\\sigma_\\mu\\zeta),$$\nwhere $\\zeta$ is a global spinor. When added to the Poincar\\'e generators, they form a group, of the type   Berezin and G. I. Kac's had advocated\\cite{Berezin}  for algebras with commuting and anticommuting parameters. The translation of $\\psi$ makes the neutrino akin to a Nambu-Goldstone particle with only derivative couplings. \n\nThere follows a Lagrangian that describes its invariant interactions, which we can identify as a non-linear representation of supersymmetry.\n\nThe end of their paper contains this remarkable sentence ``We note that if one introduces gauge fields corresponding to the(se) transformations, then, as a consequence of the Higgs effect, a massive gauge field with spin $3\/2$ arises, and the Goldstone particles with spin $1\/2$ vanish\". This remark is followed in October 1973, when  D. V. Volkov and V. A. Soroka\\cite{Volkov2} generalize their transformations to local parameters and show explicitly that the fermionic Nambu-Goldstone particle indeed becomes a gauge artifact. Thus was born what became known as the ``Super Higgs Effect\".\n\n\\section{Wess-Zumino}\nIn October 1973, Julius Wess and Bruno Zumino\\cite{Wess}  generalize the world-sheet supergauge transformations of the RNS model to four dimensions. \n\nTheirs is the paper that launched the massive and systematic study of  supersymmetric field theories in four dimensions. \n\nThe scalar (now called chiral or Wess-Zumino) multiplet is introduced.  It consists of two real scalar bosons, $A$ and $B$, a Weyl (Majorana) fermion $\\psi$ and two auxiliary fields $F$ and $G$. Supergauge transformations generate the algebra,\n\n\\begin{eqnarray}\n\\delta A&=&i\\overline\\alpha\\psi,\\quad \\delta B=i\\overline\\alpha\\gamma_5\\psi,\\nonumber\\\\\n \\delta\\psi&=&\\partial_\\mu(A-\\gamma_5B)\\gamma^\\mu\\alpha+n(A-\\gamma_5B)\\gamma_\\mu\\partial_\\mu\\alpha\\nonumber\\\\\n&&~+~F\\alpha+G\\gamma_5\\alpha\\nonumber\\\\\n\\delta F&=&i\\overline\\alpha\\gamma^\\mu\\partial_\\mu\\psi+i(n-\\frac{1}{2})\\partial_\\mu\\overline\\alpha\\gamma^\\mu\\psi\\nonumber\\\\\n\\delta G&=&i\\overline\\alpha\\gamma_5\\gamma^\\mu\\partial_\\mu\\psi+i(n-\\frac{1}{2})\\partial_\\mu\\overline\\alpha\\gamma_5\\gamma^\\mu\\psi,\\nonumber\\\\\n&&\\nonumber\\end{eqnarray}\nwhere $\\alpha$ is an ``infinitesimal\" anticommuting spinor, and $n$ is an integer assigned to the multiplet. With impressive algebraic strength, they are shown to close on both conformal and  chiral transformations. In particular, two transformations with parameters $\\alpha_1$ and $\\alpha_2$ result in a shift of $x_\\mu$  by $i\\overline\\alpha_1\\gamma_\\mu\\alpha_2$.\n\nThe free Lagrangian for the scalar multiplet follows,\n\n$$\n {\\m L}_{WZ}~=~-\\frac{1}{2}\\partial_\\mu A\\partial^\\m A -\\frac{1}{2}\\partial_\\mu B\\partial^\\mu B-\\frac{i}{2}\\overline\\psi\\gamma_\\mu\\partial^\\mu\\psi+\\frac{1}{2}(F^2+G^2).\n $$\nIt is not invariant under supergauge transformations but since it transforms as a derivative, the action is invariant. In order to introduce invariant interactions, they derive the calculus necessary to produce covariant interactions, by assembling two scalar multiplets into a third, etc... .\n\nThey also introduce the vector supermultiplet, consisting of four scalar fields, $D$, $C$, $M$, $N$, a vector field $v_\\mu$, and two spinor fields $\\chi$ and $\\lambda$, on which they  derive the supergauge transformations. By identifying the vector field with the chiral current generated by a scalar multiplet,\n\n$$v_\\mu^{}~=~B\\partial_\\mu A-A\\partial_\\mu B-\\frac{1}{2}i\\overline\\psi\\gamma_5\\gamma_\\mu\\psi,$$\nand following it through the algebra, they express all the vector multiplet fields  as quadratic combinations of the scalar supermultiplet. In particular $D=2{\\m L}_{WZ}$. \n\nFinally, they notice that one can drop some of these fields, $C$, $N$, $M$, and $\\chi$ without affecting the algebra (soon to be called the Wess-Zumino gauge), and write the vector multiplet Lagrangian in a very simple form,\n\n$${\\m L}_V~=~-\\frac{1}{4}v^{}_{\\mu\\nu}v^{\\mu\\nu}_{}-\\frac{1}{2}i\\overline\\lambda \\gamma_\\mu\\partial^\\mu\\lambda+\\frac{1}{2}D^2.$$\nThis paper contains many of the techniques that were soon to be used in deriving many of the magical properties of supersymmetric theories in four dimensions. \n\nIn December 1973, Wess and Zumino present the one-loop analysis\\cite{Wess2} of an interacting Wess-Zumino multiplet, and find remarkable regularities: the SuSy tree-level relations are not altered by quantum effects, the vertex correction is finite (leaving only finally where they find that only wave function renormalization), and finally that the quadratic divergences of the scalar and pseudoscalar fields cancel. \nAs it was realized later, this addresses the ``gauge hierarchy problem\", and strongly suggests SuSy's application to the Standard Model.\n \\section{Representations }\nThe representations of the supersymmetry algebra were first systematically studied by Gell-Mann and Ne'eman (unpublished). They mapped the algebra in light-cone coordinates to one fermi oscillator, and found that in supersymmetry, the  massless representations of the Poincar\\'e group assemble into two states with helicities separated by one-half, \n\n$$ (\\lambda\\pm \\frac{1}{2}, \\lambda),$$\nand with the same light-like momentum, yielding an equal number of bosons and fermions.  The simplest is $\\lambda=0$, with a real scalar and half a left-handed Weyl fermion. However, CPT-symmetric local field theories require the other half of the Weyl fermion, $({\\textstyle \\frac{1}{2}, 0)+(0, - \\frac{1}{2}})$\nwhich describe one Weyl fermion and a complex scalar boson, the ingredients of the Gol'fand-Likhtman-Wess-Zumino multiplet. \n\nThe massless gauge supermultiplet,  ${\\textstyle(1 , \\frac{1}{2}) +(- \\frac{1}{2},-1)},$\ndescribes a gauge boson and its companion Weyl (Majorana) fermion, the gaugino.  \n\nThe supergravity supermultiplet, $ {\\textstyle (2, \\frac{3}{2})+(-\\frac{3}{2},-2)}$\ncontains the graviton and the gravitino, remarkably the ingredients of interacting supergravity\\cite{supergravity} \n\nThey extend their analysis to the case of $\\m N$ supersymmetries. Disregarding particles of spin higher than two, they find two cases with manifestly self-conjugate supermultiplets:  \n\n$\\m N=4$ supermultiplet, with helicities, \n\n$$\n {\\textstyle(1)+4( \\frac{1}{2})+6(0)+4(- \\frac{1}{2})+(-1)},$$\nand led in 1976 to the $\\m N=4$ superYang-Mills theory\\cite{N4}, with was found much later to have magical properties, such as an enhanced conformal symmetry, and ultraviolet finiteness! \n\n$\\m N=8$ supergravity with helicities,\n\n\\begin{eqnarray*}\n {\\textstyle(2)+8(\\frac{3}{2})+28(1)+56( \\frac{1}{2})+70(0)}+\\\\\n+ {\\textstyle56(- \\frac{1}{2})+28(-1)+8( -\\frac{3}{2})+(-2)},\n\\end{eqnarray*}\nwhich also led to a fully interacting theory, $\\m N=8$ Supergravity\\cite{N8}.\n \nMassive representations of supersymmetry can be assembled using a group-theoretical Higgs mechanism. The massive vector representation contains a Dirac spinor, a massive vector, and a scalar particle,\n\n$${\\textstyle(1,\\frac{1}{2})+(-1,-\\frac{1}{2})+(0,-\\frac{1}{2})+(0,\\frac{1}{2})},$$\nall of equal mass, as considered by Gol'fand and Likhtman.\n\n\n\\section{Towards the Supersymmetric Standard Model}\nWith the Wess-Zumino paper, the flood gates had been opened\\cite{Ferrara}. In short order,  a supersymmetric version\\cite{WZ3} of $QED$  is written down, with  Abelian gauge invariance, in which the Dirac electron spinor is accompanied by {\\em two} complex spin zero fields. In January 1974, Abdus Salam and J. A. Strathdee\\cite{Salam} assemble the fields within a supermultiplet into one superfield  with the help of anticommuting Grassmann variables. The same authors\\cite{Salam2} coin the word ``super-symmetry\" in a May 1974 paper which generalizes  supersymmetry to Non-Abelian gauge interactions. \n\nBefore applying supersymmetry to the real world, several conceptual steps must be resolved. The absence of fermion-boson symmetry at low energies, requires it to  be broken. Secondly, its application to the electroweak theory demands the extension of  the Higgs mechanism. Finally the known particles must be assigned to supermultiplets.   \n   \nIn 1974, Pierre Fayet and John Iliopoulos\\cite{ILIO} produce the first paper on spontaneous breaking of supersymmetry in theories with a gauged Abelian symmetry by giving its $D$ auxiliary field a constant value. Their proposal is remarkably simple, just add to the Lagrangian for a $U(1)$ vector multiplet a $D$-term\n\n$$\n{\\m L}^{FI}_V~=~{\\m L}^{}_V~+~\\xi D.$$\nThis extra term  violate neither Abelian gauge invariance, nor supergauge invariance, since its supergauge variation is a total derivative. The resulting field equation $<D>_0=\\xi$ yields a theory where both gauge and supergauge invariances are broken.\n\nA year later, Lochlainn O'Raifeartaigh\\cite{O'R} invents a different way to spontaneous breaking of supersymmetry, in theories with \nseveral interacting scalar supermultiplets. Its simplest model involves three scalar supermultiplets, with equations of motion\n\n$$\nF_1^{}=-m\\phi_2^*-2\\lambda\\phi_1^*\\phi_3^*, ~ F_2=-m\\phi_1^*,~ F_3=\\lambda(M^2-{\\phi_1^2}^*),\n$$\nwhere $m$, $M$ and $\\lambda$ are parameters. There are no solutions for which all three $F_i$ vanish, and supersymmetry is broken. From these two early examples, the auxiliary fields are the order parameters of SuSy breaking.\n \nBoth schemes yielded an embarassing massless Goldstone spinor, which may have impeded the application of supersymmetry\\footnote{In 1976, Weinberg and Gildener note that supersymmetry could explain a low mass scalar boson, but bemoan that it would produce a massless fermion!}. None of these authors  were aware of Volkov's papers. \n\nThe second hurdle is the generalization of the Higgs mechanism to supersymmetry. This is done in the context of an unusual model by Pierre Fayet\\cite{Fayet} in December 1974. Like Volkov and Akulov before, Fayet  builds models where the electron neutrino is the Goldstone spinor from the breakdown of supersymmetry\\footnote{In 1974,  the Standard Model was not yet ``standard\", and many authors were still presenting alternatives}, using the FI mechanism. \n\nAlthough the model building in this paper did not survive the test of time, two important and more permanent concepts  emerged. One is that the Higgs mechanism applies, but  {\\em two} scalar supermultiplets are needed to achieve $SU(2)\\times U(1)\\rightarrow U(1)$ electroweak breaking, in accord with the number of surviving scalars in the massive vector supermultiplets. Also the existence of  $R$-symmetry, a new kind of continuous symmetry acting on both the fields and the Grassmann parameters of the superfields.  \n\nIt was not until July 1976, that Pierre Fayet\\cite{Fayet2} generalizes the Weinberg-Salam (soon to be Weinberg-Salam-Glashow, and then Standard) model to SuSy. Its distinctive feature are:\n\\begin{itemize}\n\n\\item Two scalar superfields, $S$, $T$, (today's $H_{u,d}$) for EW breaking\n\n\\item Leptons and quarks are the fermions inside scalar supermultiplet. \n\n\\item  A continuous $R$-symmetry\n\n \\end{itemize}\nThe particle content is that the ``minimal supersymmetric model\" (MSSM). Some kinks still need to be ironed out. having to do with  SuSy breaking ( {\\em \\` a la} Fayet-Iliopoulos in this paper), which produces a massless Goldstone spinor. The continuous $R$-symmetry in this paper behaves like a ``leptonic\" number, but it prevents the spinor gluons from acquiring a mass. \n\nToday we know that SuSy breaking is an active area of theoretical research, even without the presence of a Goldstone fermion, eaten by the Super-Higgs mechanism.\n\n\n\\section{SuSy Today}\nBy stopping this history of fermion-boson symmetry in 1976, we rob the reader of the many wonderful concepts since discovered, but they are more than adequately covered in the \n articles in this volume.\n\nThe seeds of today's Susy research were planted in these early papers. \n\nAlmost forty years later, superstring theories have blossomed into a dazzling array of connected theories; the study of $\\m N=4$ superYang-Mills theories is an active field of research, as is the possible finiteness of $\\m N=8$ supergravity. \n\nThe Hamiltonian is no longer fundamental, but derived from translations along SuSy's fermionic dimensions.\n\n\nFew doubt of the existence of a deeper connection between bosons and fermions, but opinions differ at which scale it will be revealed: the breaking of Supersymmetry remains as mysterious as ever. \n\nYet, the recent discovery of a low mass Higgs suggests that the universe displays more symmetry at shorter distances. \n\nToday, SuSy is unfulfilled, beloved by theorists, but so far shunned by experiments. \n\nIn the words of the late Sergio Fubini, {\\em ``We  do not know if supersymmetry is just a beautiful painting to put on the wall, or something more\"}. \n\n\n\n\\section{Introduction}\n\\label{intro}\nBy the 1940's, physicists had identified two classes of ``elementary\" particles with widely different group behavior, bosons and fermions. The prototypic boson is the photon which generates electromagnetic forces;  electrons, the essential constituents of matter, are fermions which satisfy Pauli's exclusion principle. This distinction was quickly extended to Yukawa's particle (boson), the generator of  Strong Interactions, and to nucleons (fermions). A compelling characterization followed: matter is built out of fermions, while forces are generated by bosons.  \n\nEinstein's premature dream of unifying {\\em all} constituents of the physical world should have provided a clue for that of fermions and bosons; yet it took physicists a long time to relate them by symmetry. This fermion-boson symmetry is called ``{\\em Supersymmetry}\".  \n\nSupersymmetry, a necessary ingredient of string theory, turns out to have further remarkable formal properties when applied to local quantum field theory, by restricting its ultraviolet behavior, and  providing unexpected insights into its non-perturbative behavior. It  may also play a  pragmatic role as the glue that  explains the weakness of the elementary forces within the  Standard Model of Particle Physics at short distances.\n\n \n\n\\section{Early Hint}\nIn 1937, Eugene Wigner, with some help from his brother-in-law, publishes one of his many famous papers\\cite{Wigner} ``On Unitary Representations of the Inhomogeneous Lorentz Group\". He was then at the University of Wisconsin at Madison, a refugee from Princeton which had denied him tenure. It was not an easy paper to read, but its results were very simple: there were five types of representations labelled by the values of  $P^2\\equiv p^\\mu p_\\mu=m^2$, one of the Poincar\\'e group's Casimir operator. \n \n All but two representations describe familiar particles found in Nature. Massive particles come with momentum $\\bf p$, spin $j$, and $2j+1$ states of polarization, e.g.  electrons and nucleons with spin $1\/2$. There are also four types of massless representations  with spin replaced by helicity (spin projection along the momentum). The first two describe massless particles with a single helicity (photons with helicity $\\pm1$), or half-odd integer helicity, such as   ``massless\" neutrinos with helicity $+1\/2$.\n \nThe last two representations $O(\\Xi)$ and $O'(\\Xi)$ describe states which look like massless ``objects\", particle-like in the sense that they have four-momentum, but with bizarre helicities: each representation contains an infinite tower of helicities, one with  integer helicities, the other with half-odd integer helicities. These have no analogues in Nature\\footnote{``Infinite spin\" representations do not appear in the Poincar\\'e decomposition of the conformal group}.   \n \nPhysicists were slow in recognizing the importance of  group representations, even though Pauli's provided the first solution of the quantum-mechanical Hydrogen atom using  group-theory.  Wigner's paper does not seem to have moved any mountains, and infinite spin representations were simply ignored, except of course by Wigner. \n\nYet, $O(\\Xi)$ and $O'(\\Xi)$ contained important information: they are ``supersymmetric partners'' of one another!\n\n\\section{Hadrons \\& Mesons}\n\\label{sec:2}\nSymmetries were gaining credence among physicists, not as a simplifying device but as a guide to the organization of Nature. Wigner and St\\\"uckelberg's ``supermultiplet model\" unified $SU(2)$ isospin and spin. Once Gell-Mann and Ne'eman generalized isospin to $SU(3)$, it did not take long for \n Feza G\\\"ursey and Luigi Radicati\\cite{Gursey}, as well as Bunji Sakita\\cite{Sakita}, to propose its unification with spin into $SU(6)$.  Pseudoscalar and vector mesons (bosons)  were found in the $\\bf {35}$ of SU(6), while the hadrons (fermions) surprisingly lived in the $\\bf{56}$, not in the $\\bf{20}$\\cite{Sakita}, as expected by the statistics of the time. This non-relativistic unification\n proved very successful, both experimentally and conceptually, since it led to the hitherto unsuspected {\\em color} quantum number. \n \nIn 1966, Hironari Miyazawa\\cite{Miyazawa}  proposed further unification. His aim was to assemble the fermionic $\\bf{56}$ and the bosonic $\\bf{35}$ into one mathematical structure such as $SU(9)$, but at the cost of disregarding spin-statistics.\n\nIn order to explain the bounty of strange particle discovered in the 1950's, Sakata had proposed to explain mesons as $T\\overline T$  bound states of the spin one-half triplet \n \n$$T~=~(\\,p,\\,n,\\,\\Lambda\\,).$$ \nMiyazawa adds  a {\\em pseudoscalar} triplet  \n\n$$t~=~(\\,K^+_{},  K^0_{},\\,\\eta\\,),$$\nto the Sakata spinor triplet. The hadron octet would then be described by another bound state, $T\\bar t$, but he could not describe the spin three-half baryons decimet in the ${\\bf 56}$. \n \nHe introduces a toy model with two fundamental constituents, a spin one-half and a spin zero particle, $ {\\bf p}=(\\alpha_\\uparrow,\\alpha_\\downarrow,\\gamma)$. The nine currents\n \n$$ {\\bf p}^\\dagger\\lambda^{}_i{\\bf p}=\\cases{F^{}_i,\\quad i=0,1,2,3,8;\\cr G^{}_i,\\quad i=4,5,6,7},$$ \nsatisfy a current algebra with both commutators and anticommutators, \n\n\\[\n[\\,F^{}_i\\,,\\,F^{}_j\\,]~=~if^{}_{ijk} F^{}_k, \\]\n\\[ [\\,F^{}_i\\,,\\,G^{}_j\\,]~=~if^{}_{ijk} G^{}_k,\\]\n\\[ \\{\\,G^{}_i\\,,\\,G^{}_j\\,\\}~=~d^{}_{ijk} F^{}_k,\\]\na ``generalized Jordan algebra\" which he calls $V(3)$. This is the first example, albeit non-relativistic, of a superalgebra, today called $SU(2\/1)$ with even part  $SU(2)\\times U(1)$.\n\nIn 1967, he expanded his construction\\cite{Miyazawa2}, to  general superalgebras he calls $V(n,m)$ with the idea of including the decimet. Alas, the phenomenology was not as compelling as that of $SU(6)$; two of the quarks inside a nucleon do not seem live together in an antitriplet color state.\n\nIn 1969, F.A. Berezin and G. I. Kac\\cite{Berezin} show the mathematical consistency of graded Lie algebra which contains both commutators and anticommutators; they  give its simplest example generated by the three Pauli matrices $\\sigma_+,\\sigma_-,\\sigma_3$. Physical applications are not discussed, although Berezin's advocacy of Grassmann variables in path integrals was no doubt a motivation.  \n\n\\section{Dual Resonance Models}\n\\label{sec:3}\nIn the 1960's, physicists had all but given up on a Lagrangian description of the Strong Interactions, to be replaced by the S-matrix program:  amplitudes were  determined from general principles and symmetries,  locality, causality, and Lorentz invariance.  Further requirements on the amplitudes such as Regge behavior and its consequent bootstrap program were still not sufficient to determine the amplitudes.  \n\nIn 1967, Dolen, Horn and Schmid\\cite{Dolen} discovered a peculiar relation in $\\pi-N$ scattering.  At tree-level, its fermionic $s$-channel  ($\\pi\\,N\\rightarrow \\pi\\,N $) is dominated by resonances ($\\Delta^{++}$, ...), as shown by countless experiments. On the other hand, its bosonic $t$-channel ($\\pi\\,\\bar\\pi \\rightarrow N \\,\\overline N$) is dominated by the $\\rho$-meson. Using the tools of S-matrix theory in the form of ``finite energy sum rules\", they found that the Regge shadow of the bosonic $t$-channel's $\\rho$-meson {\\em averaged} the fermionic resonances in the $s$-channel! This was totally unexpected since these two contributions, described by different Feynman diagrams, should have been independent. Was this the additional piece of information needed to fully determine the amplitudes of Strong Interactions? This early example of fermion-boson kinship led, through an unlikely tortuous path, to modern Supersymmetry. \n\nAn intense theoretical search for amplitudes where the $s$- and $t$-channel contributions are automatically related to one another followed. Under the spherical cow principle, spin was set aside and the search for DHS-type amplitudes focused on the purely bosonic process $\\omega\\rightarrow \\pi\\pi\\pi$\\cite{Ademollo}. Soon thereafter, Veneziano\\cite{Veneziano} proposed a four-point amplitude with the desired crossing symmetry,\n\n$$A(s,t) \\sim \\frac{\\Gamma(-\\alpha(s))\\Gamma(-\\alpha(t))}{\\Gamma(-\\alpha(s)-\\alpha(t)},$$\nwhere $\\alpha(x)=\\alpha_0+\\alpha'x$  is the linear Regge trajectory. It displays an infinite number of poles in {\\em both}  s-channel $s>0,~ t<0$ and t-channel $s<0,~t>0$. \n \nVeneziano's construction was quickly generalized to n-point ``dual\" amplitudes. The infinite series of poles were recognized as the vibrations of a string\\cite{string}. \n\nThe amplitudes were linear combinations of tree chains which factorize into three-point vertices and propagators. A generalized coordinate emerged\\cite{Fubini} from this analysis,\n\n$$\\quad Q^{}_\\mu(\\tau)=x^{}_\\mu+\\tau\\,p^{}_\\mu+\\sum_{n=1}^\\infty\\frac{1}{\\sqrt{2n\\alpha'}}\\left(a^{}_{n\\mu}e^{in\\tau}_{}-a^{\\dagger }_{n\\mu}e^{-in\\tau}_{}\\right),$$\nwith an infinite set of oscillators,\n\n$$[a^{}_{n\\mu},a^{\\dagger}_{m\\nu}]=\\delta^{}_{nm}g^{}_{\\mu\\nu}$$\nThe vertex for emitting a particle of momentum $k_\\mu$ from the linear chain was simple,\n\n$$V(k,\\tau)~=:e^{ik\\cdot Q(\\tau)}_{}:.$$\nOut of its corresponding generalized momentum\n\n\\begin{equation} P^{}_\\mu(\\tau)~=~\\frac{dQ^{}_\\mu}{d\\tau}, \\end{equation}\none derived the operators,\n\n$$L^{}_n~=~\\frac{1}{2\\pi}\\int^\\pi_{-\\pi}d\\tau e^{in\\tau}_{}:P^\\mu_{}P_\\mu^{}:~\\equiv~<:P^\\mu_{}P_\\mu^{}:>^{}_n, $$\nwhich satisfy the Virasoro algebra\\footnote{c-number is added anachronostically},\n \n $$[\\,L^{}_m\\,,\\,L^{}_n\\,]~=~(m-n)L^{}_{n+m}+{{\\frac{D}{12}m(m^2-1)\\delta^{}_{m,-n}}}.$$\n Its finite subalgebra, $L_0,L_\\pm$, the Gliozzi algebra, generates conformal transformations in two dimensions. \nThe  propagator was given by\n\n$$\\frac{1}{(\\alpha'L^{}_0+1)}.$$\n\n\n\\section{Superstrings}\nThe Klein-Gordon equation for a point particle,\n\n$$0~=~p^2_{}+m^2_{}~=~ <P^\\mu_{}>^{}_0<P_\\mu^{}>^{}_0+m^2, $$\ncould then be interpreted as a special case of \n \n$$0~=~ <P^\\mu_{}P_\\mu^{}>^{}_0+ m^2$$\nsuggesting a correspondence\\cite{Ramond1} between point particles and dual amplitudes, \n\n$$<A><B>~\\rightarrow~<A\\,B>.\n$$ \nFermions should satisfy the Dirac equation, \n\n$$0~=~\\gamma^{}_\\mu\\,p^\\mu_{}+m~=~<\\Gamma_\\mu^{}>^{}_0<P^\\mu>^{}_0+m.$$\nThis requires a generalization of the Dirac matrices  as dynamical operators,\n\n$$\\gamma^{}_\\mu~~\\rightarrow~~ \\Gamma^{}_\\mu~=~\\gamma^{}_\\mu+i\\gamma^{}_5\\sum_{n=0}^\\infty\\left(b^{}_{n\\mu} e^{in\\tau}_{}+b^{\\dagger}_{n\\mu} e^{-in\\tau}_{}\\right)\n$$\nwhere the oscillators are {\\em Lorentz vectors}\\footnote{Later was it realized that this made sense only in ten space-time dimensions where the little group is the spinor-vector schizophrenic $SO(8)$}, which satisfy anticommuting relations,\n\n$$\\{b^{}_{n\\mu},b^{\\dagger}_{n\\mu}\\}~=~\\delta^{}_{nm}g^{}_{\\mu\\nu},$$\nthe sum running over the positive integers. \n\nThis led me to propose the string Dirac equation in \nthe winter of 1970\\cite{Ramond}, which readily followed from that correspondence,\n\n$$ 0~=~{{<\\Gamma^{}_\\mu\\,P^\\mu_{}>^{}_0+m}}.$$\nThe basic Dirac algebra, $\\{\\gamma\\cdot p,\\gamma\\cdot p\\}=p^2_{}$ \nis seen to be generalized to an algebra with both commutator and anticommutators,\n\n$$\\{\\,F_n^{}\\,,\\,F^{}_m\\,\\}~=~ 2L^{}_{n+m},\\quad [\\,L^{}_n\\,,\\,F^{}_m\\,]~=~(2m-n)F^{}_{m+n},$$\nwhere $F_n=<\\Gamma_\\mu P^\\mu>_n$, and these new $L_n$'s also satisfy the Virasoro algebra, but with a different $c$-number.\n\nAndr\\'e Neveu and John Schwarz then compute the amplitude for a dual fermion emitting three pseudoscalars with the Yukawa vertex, \n \n$$\\Gamma_5^{}:e^{ik\\cdot Q(\\tau)}:,\\quad \\Gamma_5~=~\\gamma_5(-1)^{\\sum b_{n}^\\dagger\\cdot b_{n}},$$ \nand find that the resulting amplitude contains an infinite number of poles in its fermion-antifermion channel, and even identify the residue of the first pole\\cite{Neveu2}!\n\n A new model with bosonic poles and vertices emerges, written in terms of  an infinite tower of anticommuting vector oscillators,\n\n\n$$\\{b^{}_{r\\mu},b^\\dagger_{s\\nu}\\}~=~\\delta^{}_{rs}g^{}_{\\mu\\nu},\\quad r,s={\\textstyle\\frac{1}{2},\\frac{3}{2},\\cdots}.$$\nThe triple boson vertex is given by\n\n$$V_{NS}^{}(k,\\tau)k_{}^\\mu~=~ H^{}_\\mu(\\tau):e^{ik\\cdot Q(\\tau)}_{}:,$$\nwhere\n\n$$\nH^{}_\\mu(\\tau)~=~\\sum_{ r=1\/2,3\/2,\\dots} [b^{}_{r\\mu}e^{-ir\\tau}+b^\\dagger_{r\\mu}e^{ir\\tau}].$$\nThese are the building blocks of the  ``Dual Pion model\"\\cite{Neveu}, published in April 1971. The algebraic structure found in the generalized Dirac equation remains the same, producing a super-Virasoro algebra which decouples unwanted modes\\cite{NST}, with $\\Gamma_\\mu$ replaced by $H_\\mu$, through the operators,\n\n$$G_r^{}~=~<H\\cdot P>^{}_r,\\quad r= {\\textstyle\\frac{1}{2},\\frac{3}{2},\\cdots}$$ \n \nThe close relation of the two sectors is soon after formalized by Jean-Loup Gervais and Bunji Sakita\\cite{Gervais} who write them in terms of  a world-sheet $\\sigma$-model, with different boundary conditions, symmetric for the fermions, antisymmetric for the bosons.  They call the transformations  generated by the anticommuting Virasoro  opertors, {\\em supergauge transformations}, the first time the name ``super\" appears in this context.   \n\nThe following years saw the formulation of the RNS (NSR to some) ``Dual Fermion Model\", generating  dual amplitudes  with bosons and fermions legs.\nIt lived in ten space-time dimensions, with states determined in terms of transverse fermionic and bosonic harmonic oscillator operators. \n\nIn the fermionic ``R-sector\", the spectrum of states is spanned by the fermionic ground state, $u|0>$ where $u$ is a fixed 32-dimensional spinor,  annihilated by both transverse  bosonic and fermionic oscillators, $a^{}_{ni}$ and $b^{}_{ni}$, $i=1,2\\dots ,8$, and integer $n$. The fermion masses are determined by\n\n$$\\alpha'm^2_R~=~\\sum_{n=1}^\\infty n\\Big[a^\\dagger_n\\cdot a^{}_n+b^\\dagger_n\\cdot b^{}_n\\Big] $$\nThe  bosonic ``NS-sector\"  spectrum starts with a tachyon,  $|0>$ annihilated by the same $a^{}_{ni}$, but also by the NS fermionic oscillators $b^{}_{ri}$, where $r$ runs over half-integers. The boson masses satisfy \n\n$$\\alpha'm^2_{NS}~=~\\sum_{n=1}^\\infty n a^\\dagger_n\\cdot a^{}_n+\\sum_{\\scriptstyle r=\\frac{1}{2}}rb^\\dagger_r\\cdot b^{}_r-\\frac{1}{2}.$$\n\nBut there were idiosyncrasies. The correspondence between Neveu-Schwarz and the dual fermion states differed for states with an even number ($G\\equiv(-1)^{\\sum b_r^\\dagger\\cdot b^{}_r}=-1$) of $b^\\dagger_r$, and states with an odd number, and there is a tachyon in the even number spectrum, at $\\alpha'm^2_{NS}=-1\/2$..  \n\nIn 1976, F. Gliozzi, Jo\\\"el Scherk, and David Olive\\cite{GSO} noticed that the NS tachyon can be eliminated by requiring an odd number of anticommuting operators in the bosonic spectrum, ($G=-1$). The NS ground state \n\n$$\\alpha'm^2_{NS}=0: ~~~b^\\dagger_{1i}|0\\rangle,$$\nnow consists of eight bosons, transforming as the vector(=spinor) $SO(8)$ representation.  The first excited states are \n\n$$\\alpha'm^2_{NS}=1:~~~b^\\dagger_{\\scriptstyle\\frac{1}{2}i}b^\\dagger_{\\scriptstyle\\frac{1}{2}j}b^\\dagger_{\\scriptstyle\\frac{1}{2}k}|0\\rangle,~~b^\\dagger_{\\scriptstyle\\frac{1}{2}i}a^\\dagger_{1j}|0\\rangle,~\nb^\\dagger_{\\scriptstyle\\frac{3}{2}i}|0\\rangle,$$\nthat is $128=56(8.7.6\/1.2.3)+64 (8.8)+8$ bosonic states, and so on.\n\nIn their next step, they  show that the R ground state solution could also be reduced to eight fermionic degrees of freedom. In ten dimensions, while a spinor has naturally thirty-two degrees of freedom, they showed that one can impose {\\em both} chiral and Majorana (reality) restrictions on it, and reduce the spinor to eight dimensions, the spinor(=vector) $SO(8)$ representation. \n\n$$\\alpha'm^2_{R}=0:~~~\\psi_{\\alpha}|0\\rangle,~~\\alpha=1,2\\cdots 8.$$\nThe first excited state of the R-sector consists of\n\n$$\\alpha'm^2_{R}=1:~~~b^\\dagger_{1i}\\psi_{\\alpha}|0\\rangle,~~a^\\dagger_{1j}\\psi_{\\alpha}|0\\rangle,$$\nwith $128=8.8+8.8$ fermionic states! This was no accident, and using one of Jacobi's most obtuse relations, they showed that this equality obtained at all levels. Indeed this was supersymmetry, with the same number of bosons and fermions, albeit in ten space-time dimensions.   \n\nFermion-boson symmetry, born in its world-sheet realization, reappears as supersymmetry in ten-dimensional space-time. \n\nMeanwhile, behind the iron curtain, ...\n\\section{Russians}\n\\label{sec:4}\nIn March 1971, there appears a remarkable and terse paper by Yu. Gol'fand and E. Likhtman\\cite{Golfand}  who extend the Poincar\\'e algebra \ngenerated by $P_\\mu$ and $M_{\\mu\\nu}$ to ``bispinor  generators\", $W_\\alpha$ and $\\overline W_\\beta$, which generate spinor translations.\n\nCognizant that spin-statistics  requires anticommutating spinors, they arrive at the parity-violating algebra,\n  \n\\begin{equation}\n\\{W,W\\}~=~[P_\\mu,P_\\nu]~=~0,\\quad \\{W,\\overline W\\}~=~\\frac{(1+\\gamma_5)}{2}\\gamma_\\mu P_\\mu.\\end{equation}\nassuming no other subalgebra of the Poincar\\'e group.  With  little stated motivation, they have written down the ${\\cal N}=1$ superPoincar\\'e algebra in four dimensions!\n \nThey identify its simplest representation: two ``scalar hermitean\" fields $\\phi(x)$ and $\\omega(x)$, and one left-handed spinor field $\\psi_1(x)$, of equal mass, the earliest mention of the Wess-Zumino supermultiplet. They do not consider auxiliary fields nor display the transformation properties of these fields. However, they show the spinor generators as bilinears in those fields, \n\n\\begin{equation}\nW~=~\\frac{(1+\\gamma_5)}{2}\\int d^3x\\Big[\\phi^*\\uplrarrow\\partial_0\\psi^{}_1(x)+\\omega(x)\\uplrarrow\\partial_0\\psi^{c}_1(x)\\Big].\\end{equation}\n\nThey also describe the  {\\em massive} vector multiplet follows with the vector field $A_\\mu(x)$, a scalar field $\\chi(x)$ and a spinor field $\\psi_2(x)$. They write down its  spinor current,\n\n\\begin{equation}\nW~=~\\frac{(1+\\gamma_5)}{2}\\int d^3x\\Big[\\chi\\uplrarrow\\partial_0\\psi^{}_2(x)+A_\\mu(x)\\uplrarrow\\partial_0\\gamma_\\mu\\psi^{}_2(x)\\Big].\\end{equation}\n\nThis ground-breaking paper ends with the difficult task of writing interactions. Self-interactions of the WZ multiplet are not presented, \n only  its interactions with a massive Abelian vector supermultiplet.  This,  the last formula in their paper, is a bit confusing since $\\phi$ and $\\omega$ now appear as complex fields (setting $\\omega=0$ and replacing the complex $\\phi$ by $\\phi+i\\omega$ is more what they need), but it contains now familiar features, such as the squared $D$-term. \n\nGol'fand and Likhtman had firmly planted the flag of supersymmetry in four-dimensions. \n\nInterestingly, physicists on both sides of the iron curtain seemed  oblivious to this epochal paper. \n\nE. Likhtman seems to be the only one who followed up on this paper. He notices\\cite{Lebedev} that the vacuum energy cancels out because of the equal number of  mass bosons and fermions with the same mass. He finds scalar masses  only logarithmically divergent, which he mentions in a later publication\\cite{Likhtman}.   \n\nIn December 1972, in an equally impressive paper,  D.V. Volkov, and V.P. Akulov\\cite{Volkov}, want to explain the masslessness of  neutrinos in terms of  an invariance principle. They note that the neutrino free Dirac equation is invariant under the transformations,\n\n$$\\psi\\rightarrow \\psi+ \\zeta,\\quad x_\\mu\\rightarrow x_\\mu-\\frac{a}{2i}(\\zeta^\\dagger\\sigma_\\mu\\psi-\\psi^\\dagger\\sigma_\\mu\\zeta),$$\nwhere $\\zeta$ is a global spinor. When added to the Poincar\\'e generators, they form a group, of the type   Berezin and G. I. Kac's had advocated\\cite{Berezin}  for algebras with commuting and anticommuting parameters. The translation of $\\psi$ makes the neutrino akin to a Nambu-Goldstone particle with only derivative couplings. \n\nThere follows a Lagrangian that describes its invariant interactions, which we can identify as a non-linear representation of supersymmetry.\n\nThe end of their paper contains this remarkable sentence ``We note that if one introduces gauge fields corresponding to the(se) transformations, then, as a consequence of the Higgs effect, a massive gauge field with spin $3\/2$ arises, and the Goldstone particles with spin $1\/2$ vanish\". This remark is followed in October 1973, when  D. V. Volkov and V. A. Soroka\\cite{Volkov2} generalize their transformations to local parameters and show explicitly that the fermionic Nambu-Goldstone particle indeed becomes a gauge artifact. Thus was born what became known as the ``Super Higgs Effect\".\n\n\\section{Wess-Zumino}\nIn October 1973, Julius Wess and Bruno Zumino\\cite{Wess}  generalize the world-sheet supergauge transformations of the RNS model to four dimensions. \n\nTheirs is the paper that launched the massive and systematic study of  supersymmetric field theories in four dimensions. \n\nThe scalar (now called chiral or Wess-Zumino) multiplet is introduced.  It consists of two real scalar bosons, $A$ and $B$, a Weyl (Majorana) fermion $\\psi$ and two auxiliary fields $F$ and $G$. Supergauge transformations generate the algebra,\n\n\\begin{eqnarray}\n\\delta A&=&i\\overline\\alpha\\psi,\\quad \\delta B=i\\overline\\alpha\\gamma_5\\psi,\\nonumber\\\\\n \\delta\\psi&=&\\partial_\\mu(A-\\gamma_5B)\\gamma^\\mu\\alpha+n(A-\\gamma_5B)\\gamma_\\mu\\partial_\\mu\\alpha\\nonumber\\\\\n&&~+~F\\alpha+G\\gamma_5\\alpha\\nonumber\\\\\n\\delta F&=&i\\overline\\alpha\\gamma^\\mu\\partial_\\mu\\psi+i(n-\\frac{1}{2})\\partial_\\mu\\overline\\alpha\\gamma^\\mu\\psi\\nonumber\\\\\n\\delta G&=&i\\overline\\alpha\\gamma_5\\gamma^\\mu\\partial_\\mu\\psi+i(n-\\frac{1}{2})\\partial_\\mu\\overline\\alpha\\gamma_5\\gamma^\\mu\\psi,\\nonumber\\\\\n&&\\nonumber\\end{eqnarray}\nwhere $\\alpha$ is an ``infinitesimal\" anticommuting spinor, and $n$ is an integer assigned to the multiplet. With impressive algebraic strength, they are shown to close on both conformal and  chiral transformations. In particular, two transformations with parameters $\\alpha_1$ and $\\alpha_2$ result in a shift of $x_\\mu$  by $i\\overline\\alpha_1\\gamma_\\mu\\alpha_2$.\n\nThe free Lagrangian for the scalar multiplet follows,\n\n$$\n {\\m L}_{WZ}~=~-\\frac{1}{2}\\partial_\\mu A\\partial^\\m A -\\frac{1}{2}\\partial_\\mu B\\partial^\\mu B-\\frac{i}{2}\\overline\\psi\\gamma_\\mu\\partial^\\mu\\psi+\\frac{1}{2}(F^2+G^2).\n $$\nIt is not invariant under supergauge transformations but since it transforms as a derivative, the action is invariant. In order to introduce invariant interactions, they derive the calculus necessary to produce covariant interactions, by assembling two scalar multiplets into a third, etc... .\n\nThey also introduce the vector supermultiplet, consisting of four scalar fields, $D$, $C$, $M$, $N$, a vector field $v_\\mu$, and two spinor fields $\\chi$ and $\\lambda$, on which they  derive the supergauge transformations. By identifying the vector field with the chiral current generated by a scalar multiplet,\n\n$$v_\\mu^{}~=~B\\partial_\\mu A-A\\partial_\\mu B-\\frac{1}{2}i\\overline\\psi\\gamma_5\\gamma_\\mu\\psi,$$\nand following it through the algebra, they express all the vector multiplet fields  as quadratic combinations of the scalar supermultiplet. In particular $D=2{\\m L}_{WZ}$. \n\nFinally, they notice that one can drop some of these fields, $C$, $N$, $M$, and $\\chi$ without affecting the algebra (soon to be called the Wess-Zumino gauge), and write the vector multiplet Lagrangian in a very simple form,\n\n$${\\m L}_V~=~-\\frac{1}{4}v^{}_{\\mu\\nu}v^{\\mu\\nu}_{}-\\frac{1}{2}i\\overline\\lambda \\gamma_\\mu\\partial^\\mu\\lambda+\\frac{1}{2}D^2.$$\nThis paper contains many of the techniques that were soon to be used in deriving many of the magical properties of supersymmetric theories in four dimensions. \n\nIn December 1973, Wess and Zumino present the one-loop analysis\\cite{Wess2} of an interacting Wess-Zumino multiplet, and find remarkable regularities: the SuSy tree-level relations are not altered by quantum effects, the vertex correction is finite (leaving only finally where they find that only wave function renormalization), and finally that the quadratic divergences of the scalar and pseudoscalar fields cancel. \nAs it was realized later, this addresses the ``gauge hierarchy problem\", and strongly suggests SuSy's application to the Standard Model.\n \\section{Representations }\nThe representations of the supersymmetry algebra were first systematically studied by Gell-Mann and Ne'eman (unpublished). They mapped the algebra in light-cone coordinates to one fermi oscillator, and found that in supersymmetry, the  massless representations of the Poincar\\'e group assemble into two states with helicities separated by one-half, \n\n$$ (\\lambda\\pm \\frac{1}{2}, \\lambda),$$\nand with the same light-like momentum, yielding an equal number of bosons and fermions.  The simplest is $\\lambda=0$, with a real scalar and half a left-handed Weyl fermion. However, CPT-symmetric local field theories require the other half of the Weyl fermion, $({\\textstyle \\frac{1}{2}, 0)+(0, - \\frac{1}{2}})$\nwhich describe one Weyl fermion and a complex scalar boson, the ingredients of the Gol'fand-Likhtman-Wess-Zumino multiplet. \n\nThe massless gauge supermultiplet,  ${\\textstyle(1 , \\frac{1}{2}) +(- \\frac{1}{2},-1)},$\ndescribes a gauge boson and its companion Weyl (Majorana) fermion, the gaugino.  \n\nThe supergravity supermultiplet, $ {\\textstyle (2, \\frac{3}{2})+(-\\frac{3}{2},-2)}$\ncontains the graviton and the gravitino, remarkably the ingredients of interacting supergravity\\cite{supergravity} \n\nThey extend their analysis to the case of $\\m N$ supersymmetries. Disregarding particles of spin higher than two, they find two cases with manifestly self-conjugate supermultiplets:  \n\n$\\m N=4$ supermultiplet, with helicities, \n\n$$\n {\\textstyle(1)+4( \\frac{1}{2})+6(0)+4(- \\frac{1}{2})+(-1)},$$\nand led in 1976 to the $\\m N=4$ superYang-Mills theory\\cite{N4}, with was found much later to have magical properties, such as an enhanced conformal symmetry, and ultraviolet finiteness! \n\n$\\m N=8$ supergravity with helicities,\n\n\\begin{eqnarray*}\n {\\textstyle(2)+8(\\frac{3}{2})+28(1)+56( \\frac{1}{2})+70(0)}+\\\\\n+ {\\textstyle56(- \\frac{1}{2})+28(-1)+8( -\\frac{3}{2})+(-2)},\n\\end{eqnarray*}\nwhich also led to a fully interacting theory, $\\m N=8$ Supergravity\\cite{N8}.\n \nMassive representations of supersymmetry can be assembled using a group-theoretical Higgs mechanism. The massive vector representation contains a Dirac spinor, a massive vector, and a scalar particle,\n\n$${\\textstyle(1,\\frac{1}{2})+(-1,-\\frac{1}{2})+(0,-\\frac{1}{2})+(0,\\frac{1}{2})},$$\nall of equal mass, as considered by Gol'fand and Likhtman.\n\n\n\\section{Towards the Supersymmetric Standard Model}\nWith the Wess-Zumino paper, the flood gates had been opened\\cite{Ferrara}. In short order,  a supersymmetric version\\cite{WZ3} of $QED$  is written down, with  Abelian gauge invariance, in which the Dirac electron spinor is accompanied by {\\em two} complex spin zero fields. In January 1974, Abdus Salam and J. A. Strathdee\\cite{Salam} assemble the fields within a supermultiplet into one superfield  with the help of anticommuting Grassmann variables. The same authors\\cite{Salam2} coin the word ``super-symmetry\" in a May 1974 paper which generalizes  supersymmetry to Non-Abelian gauge interactions. \n\nBefore applying supersymmetry to the real world, several conceptual steps must be resolved. The absence of fermion-boson symmetry at low energies, requires it to  be broken. Secondly, its application to the electroweak theory demands the extension of  the Higgs mechanism. Finally the known particles must be assigned to supermultiplets.   \n   \nIn 1974, Pierre Fayet and John Iliopoulos\\cite{ILIO} produce the first paper on spontaneous breaking of supersymmetry in theories with a gauged Abelian symmetry by giving its $D$ auxiliary field a constant value. Their proposal is remarkably simple, just add to the Lagrangian for a $U(1)$ vector multiplet a $D$-term\n\n$$\n{\\m L}^{FI}_V~=~{\\m L}^{}_V~+~\\xi D.$$\nThis extra term  violate neither Abelian gauge invariance, nor supergauge invariance, since its supergauge variation is a total derivative. The resulting field equation $<D>_0=\\xi$ yields a theory where both gauge and supergauge invariances are broken.\n\nA year later, Lochlainn O'Raifeartaigh\\cite{O'R} invents a different way to spontaneous breaking of supersymmetry, in theories with \nseveral interacting scalar supermultiplets. Its simplest model involves three scalar supermultiplets, with equations of motion\n\n$$\nF_1^{}=-m\\phi_2^*-2\\lambda\\phi_1^*\\phi_3^*, ~ F_2=-m\\phi_1^*,~ F_3=\\lambda(M^2-{\\phi_1^2}^*),\n$$\nwhere $m$, $M$ and $\\lambda$ are parameters. There are no solutions for which all three $F_i$ vanish, and supersymmetry is broken. From these two early examples, the auxiliary fields are the order parameters of SuSy breaking.\n \nBoth schemes yielded an embarassing massless Goldstone spinor, which may have impeded the application of supersymmetry\\footnote{In 1976, Weinberg and Gildener note that supersymmetry could explain a low mass scalar boson, but bemoan that it would produce a massless fermion!}. None of these authors  were aware of Volkov's papers. \n\nThe second hurdle is the generalization of the Higgs mechanism to supersymmetry. This is done in the context of an unusual model by Pierre Fayet\\cite{Fayet} in December 1974. Like Volkov and Akulov before, Fayet  builds models where the electron neutrino is the Goldstone spinor from the breakdown of supersymmetry\\footnote{In 1974,  the Standard Model was not yet ``standard\", and many authors were still presenting alternatives}, using the FI mechanism. \n\nAlthough the model building in this paper did not survive the test of time, two important and more permanent concepts  emerged. One is that the Higgs mechanism applies, but  {\\em two} scalar supermultiplets are needed to achieve $SU(2)\\times U(1)\\rightarrow U(1)$ electroweak breaking, in accord with the number of surviving scalars in the massive vector supermultiplets. Also the existence of  $R$-symmetry, a new kind of continuous symmetry acting on both the fields and the Grassmann parameters of the superfields.  \n\nIt was not until July 1976, that Pierre Fayet\\cite{Fayet2} generalizes the Weinberg-Salam (soon to be Weinberg-Salam-Glashow, and then Standard) model to SuSy. Its distinctive feature are:\n\\begin{itemize}\n\n\\item Two scalar superfields, $S$, $T$, (today's $H_{u,d}$) for EW breaking\n\n\\item Leptons and quarks are the fermions inside scalar supermultiplet. \n\n\\item  A continuous $R$-symmetry\n\n \\end{itemize}\nThe particle content is that the ``minimal supersymmetric model\" (MSSM). Some kinks still need to be ironed out. having to do with  SuSy breaking ( {\\em \\` a la} Fayet-Iliopoulos in this paper), which produces a massless Goldstone spinor. The continuous $R$-symmetry in this paper behaves like a ``leptonic\" number, but it prevents the spinor gluons from acquiring a mass. \n\nToday we know that SuSy breaking is an active area of theoretical research, even without the presence of a Goldstone fermion, eaten by the Super-Higgs mechanism.\n\n\n\\section{SuSy Today}\nBy stopping this history of fermion-boson symmetry in 1976, we rob the reader of the many wonderful concepts since discovered, but they are more than adequately covered in the \n articles in this volume.\n\nThe seeds of today's Susy research were planted in these early papers. \n\nAlmost forty years later, superstring theories have blossomed into a dazzling array of connected theories; the study of $\\m N=4$ superYang-Mills theories is an active field of research, as is the possible finiteness of $\\m N=8$ supergravity. \n\nThe Hamiltonian is no longer fundamental, but derived from translations along SuSy's fermionic dimensions.\n\n\nFew doubt of the existence of a deeper connection between bosons and fermions, but opinions differ at which scale it will be revealed: the breaking of Supersymmetry remains as mysterious as ever. \n\nYet, the recent discovery of a low mass Higgs suggests that the universe displays more symmetry at shorter distances. \n\nToday, SuSy is unfulfilled, beloved by theorists, but so far shunned by experiments. \n\nIn the words of the late Sergio Fubini, {\\em ``We  do not know if supersymmetry is just a beautiful painting to put on the wall, or something more\"}. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is well known that the rate of dilepton production from a thermal\nsystem is proportional to the two-point correlation function of vector\ncurrents.\nHence the spectral function of vector meson, the $\\rho$ meson particular,\nplays such an important role in the analysis of the late stages of\nheavy ion collisions~\\cite{Annals,RappAdv}. The NA60 experiment at the CERN SPS measured dimuon pairs in\nIn-In collisions in which an excess was observed over the contribution \nfrom hadronic decays at freeze-out in the mass region below the $\\rho$ \npeak~\\cite{na60}. This was attributed to the broadening of the $\\rho$ \nin hot and dense medium~\\cite{RappAdv}. More recently, the PHENIX experiment \nreported a substantial excess of electron pairs in the same region of\ninvariant mass~\\cite{Phenixdil}. This has been investigated by several\ngroups but the yield in all these cases have remained\ninsufficient to explain the data. Thus the issue of low mass \nlepton pair yield in heavy ion collisions is far from closed\nand is one of the key issues to be addressed in the forthcoming\n Compressed Baryonic Matter(CBM)\nexperiment to be performed at the FAIR facility in GSI~\\cite{CBM}.\n\n\nA substantial volume of work has been devoted to the study\nof $\\rho$ meson properties in hot and dense medium.\nWe do not attempt to review the existing literature but mention a few of them\nto put our work in perspective.\nWe find that it is only for the $\\pi-\\pi$ loop~\\cite{Gale,Friman}\nthat one calculates the thermal \nloop directly. In the case of other loops\ntypically involving one heavy and one light particle or both heavy particles \none uses in general either the virial formula~\\cite{Smilga,Post1,Eletsky}\nor the Lindhard function~\\cite{Fetter,RappNPA,Peters,Dani,Post2}. \nMost of the calculations involving baryonic effects mentioned above\nwere performed at zero temperature. Finite temperature \neffects on the $\\rho$ spectral function in dense matter have been evaluated by \nRapp et al~\\cite{RappNPA} in terms of resonant interactions of the $\\rho$ with\nsurrounding mesons and baryons in addition to modifying the pion cloud.\nEletsky~\\cite{Eletsky} and collaborators have also evaluated \nthe spectral function of vector mesons at finite temperature and density\nin terms of forward scattering amplitudes \nconstructed  using experimental inputs assuming resonance dominance at low\nenergies and a Regge-type approach at higher energies. \n\n\nThe sources modifying the free propagation\nof a particle find a unified description in terms of contributions from the \nbranch cuts of the self energy function as shown \nby Weldon~\\cite{Weldon}. In addition to the unitary cut present \nalready in vacuum, the thermal amplitude generates a new\ncut, the so called the Landau cut which provides the effect of collisions \nwith the surrounding particles in the medium. This formalism was applied \nto obtain the $\\rho$ self-energy in hot mesonic matter~\\cite{Ghosh1} by evaluating the\none loop self-energies involving the $\\pi$, $\\omega$, $h_1$  and  $a_1$ \nmesons. A significant broadening of the spectral function was obtained without\nappreciable shift in the mass as expected from chiral interactions. \n\n\nIn this work, we extend this analysis to the case of baryonic matter at\nfinite temperature considering an exhaustive set of 4-star resonances in the \nbaryonic loops making up the $\\rho$ self-energy. \nThe framework of real time \nthermal field theory~\\cite{Umezawa,Niemi,Kobes,Mallik_RT} that we use, enables us to evaluate \nthe imaginary part \nof the self-energy from the branch cuts for real and positive values of\nenergy and momentum without having to resort to analytic continuation\nas in the imaginary time approach~\\cite{Matsubara}.\nHere we work with the full relativistic baryon propagator\nin which baryons and anti-baryons manifestly appear on an equal footing. Thus\nthe contributions from all the \nsingularities in the self-energy function including the distant ones coming from \nthe unitary cut\nof the loops involving heavy baryons are also included. \nThese are usually not considered but can contribute appreciably to the real part\nof the $\\rho$ meson self-energy as shown~\\cite{Ghosh3} in the case of a $N\\Delta$\nloop. In addition\nwe have used\nthe covariant form of the momentum dependent vertex functions in the loop \nintegrals in which\nadditional terms~\\cite{Peccei} required to\ndescribe the coupling of\noff-shell spin 3\/2 fields have been introduced.\n\n\nIn the following section we define the correlation function of vector currents\nand its relation to the transverse propagator of the $\\rho$. We also provide the\nvarious Lagrangian densities which will be used at the vertices of the loop\ngraphs. Next, in section 3 we specify the kinematic decomposition of the\nthermal propagators. In section 4 we evaluate the baryonic self-energy graphs \nas well as the discontinuities across the branch cuts. Section 5 contains the results\nof the numerical evaluation of the real and imaginary parts as well as the\n$\\rho$ spectral function followed by a summary and discussions in section 6. In\nthe appendix  we provide the details of various factors appearing in the expression\nfor self-energy for the different loops and provide some details of\nevaluation of the imaginary part in addition to a brief discussion on propagators\nand self-energies in the real time formalism.    \n\n\\section{The two point function in the medium}\nWe begin our discussion with the two point function of vector currents in vacuum,\n\\begin{equation}\nT^{ij}_{{\\mu\\nu}}(E,\\vec q)=i\\int d^3xd\\tau\\,e^{iq\\cdot x}\\langle 0|\nT V^i_\\mu(x) V^j_\\nu(0)|0\\rangle\n\\end{equation}\nwhere $V^i_\\mu(x)$ are the vector currents of two flavour QCD, given by\n\\begin{equation}\nV_\\mu^i(x)=\\bar q(x)\\gamma_\\mu\\frac{\\tau^i}{2}q(x),~~~~~~ q=\\left(\\begin{array}{c}\nu \\\\ d \\end{array}\\right) \n\\end{equation}\n$\\tau^i$ being the Pauli matrices.\nIn the real time formulation of thermal field theory, \nthe in medium two point function assumes a $2\\times2$ matrix structure~\\cite{Kobes}.\nThe thermal two point function is given by\n\\begin{equation}\nT^{ij,ab}_{{\\mu\\nu}}(E,\\vec q)=i\\int d^3xd\\tau\\,e^{iq\\cdot x}\\langle\n T_c V^i_\\mu(x) V^j_\\nu(0)\\rangle^{ab}\n\\label{Tmed}\n\\end{equation}\nwhere $\\langle{\\cal{O}}\\rangle$ denotes the ensemble average of an operator ${\\cal{O}}$,\n\\begin{equation}\n\\langle{\\cal{O}}\\rangle=Tr(e^{-\\beta H}{\\cal O})\/Tr e^{-\\beta H}\n\\end{equation}\nand $Tr$ indicating trace over a complete set of states.\nThe superscripts \n$a,b\\,(=1,2)$ are thermal indices and $T_c$ denotes time ordering with respect\nto a contour in the plane of the complex time variable~\\cite{Mallik_RT}.\nThe two point function of vector currents can be related to the $\\rho$ meson\npropagator using the method of external fields~\\cite{Gasser} \nwhere one introduces a classical vector field $v^i_\\mu(x)$\ncoupled to the vector current $V^i_\\mu(x)$. The free propagator of the rho \nmeson can be obtained by coupling the external field to the \n$\\rho$ meson field operator using the Lagrangian~\\cite{Mallik_VA} \n\\begin{equation}\n{\\cal L}_{\\rho v}=\\frac{F_\\rho}{m_\\rho}\n\\partial^\\mu\\vec v^\\nu\\cdot(\\partial_\\mu\\vec\\rho_\\nu-\\partial_\\nu\\vec\\rho_\\mu)\\nonumber\n\\end{equation}\nwhere F$_\\rho = 154$ MeV is obtained from the decay \n$\\rho^0\\to e^+\\,e^-$.\n\nThe transverse $\\rho$ meson propagator $G^{ab}_{{\\mu\\nu}}$ \nis then obtained from the relation $T^{ab}_{{\\mu\\nu}}=K_\\rho G^{ab}_{{\\mu\\nu}}$ where the factor \n$K_\\rho=({F_\\rho q^2}\/{m_\\rho})^2$ \ncomes from the coupling of the current with the $\\rho$ field~\\cite{Ghosh1}.\n The isospin structure is given by \n$\\delta^{ij}$ which we omit from now on.  \n\nThe free propagation of the $\\rho$ meson is modified by interactions in the medium which is \npopulated by mesons and baryons. \nHere we consider one loop graphs shown in Fig.~\\ref{loop_BB}\nconsisting of the nucleon $N$ and another  baryon $R$ denoted by the\ndouble lines. \nWe have included all spin one-half and three-half $4-$star\nresonances listed by the PDG~\\cite{PDG} so that $R$ stands for the \n$N^*(1520)$,  $N^*(1650)$, $N^*(1700)$, $\\Delta(1230)$, \n$\\Delta^*(1620),$ $\\Delta^*(1720)$ as well as the $N(940)$ itself.\nOmitting isospin factors,\nthe $\\rho N$ couplings with the resonances \nare described by the gauge invariant interactions~\\cite{Post2}    \n\\begin{eqnarray}\n{\\cal L}&=&\\frac{f}{m_\\rho}[\\overline{\\psi}_R\\sigma^{{\\mu\\nu}}{\\rho}_{{\\mu\\nu}}\\psi_N+ h.c.]\n~~~~~~~~~~J^{P}_R=\\frac{1}{2}^+ \\nonumber\\\\\n{\\cal L}&=&\\frac{f}{m_\\rho}[\\overline{\\psi}_R\\sigma^{{\\mu\\nu}}\\gamma^5{\\rho}_{{\\mu\\nu}}\\psi_N+ h.c.]\n~~~~~~~J^{P}_R=\\frac{1}{2}^- \\nonumber\\\\\n{\\cal L}&=&\\frac{f}{m_\\rho}[\\overline{\\psi}^{\\mu}_R\\gamma^{\\nu}\\gamma^5\\rho_{{\\mu\\nu}}\\psi_N  + h.c.]\n~~~~~~~~J^{P}_R=\\frac{3}{2}^+ \\nonumber\\\\\n{\\cal L}&=&\\frac{f}{m_\\rho}[\\overline{\\psi}^{\\mu}_R\\gamma^{\\nu}\\rho_{{\\mu\\nu}}\\psi_N  + h.c.]\n~~~~~~~~~~~J^{P}_R=\\frac{3}{2}^-\n\\label{lag1}\n\\end{eqnarray}\nwhere $\\rho_{{\\mu\\nu}}=\\partial_\\mu\\rho_\\nu-\\partial_\\nu\\rho_\\mu$ and\n$\\sigma^{{\\mu\\nu}}=\\frac{i}{2}[\\gamma^\\mu\\gamma^\\nu-\\gamma^\\nu\\gamma^\\mu]$.\nThe isospin part of the $RN\\rho$ interaction is given by\n\\begin{eqnarray}\n&&\\overline\\psi_{R\\, a}(\\vec\\tau\\cdot\\rho)^a_b\\psi_N^b~~~~~~~~~~~~~~~~~I=1\/2\\nonumber\\\\\n&&\\frac{1}{\\sqrt{2}}\\overline\\psi_{R\\, abc}(\\vec\\tau\\cdot\\rho)^b_d\\psi_N^a\\epsilon^{cd}~~~~~~~I=3\/2\n\\end{eqnarray}\nwhere the indices $a,b,c,d$ take values 1 and 2 and $\\epsilon^{12}=-\\epsilon^{21}=1$.\nFor the self-energy diagrams shown in Fig 1, the isospin factor  \n$I_F$ comes out to be 2 for $I=\\frac{1}{2}$ and $\\frac{4}{3}$ for \n$I=\\frac{3}{2}$.\n\nIt is essential to point out that for the spin ${3}\/{2}$ resonances \nthis coupling is not quite correct\n owing to the fact that the free Lagrangian\n for the Rarita-Schwinger field $\\psi^{\\mu}_R$ has a free parameter~\\cite{Rarita}. \n A symmetry is associated with a point transformation under\nwhich the free Lagrangian remains invariant up to a change in the value of\n the parameter~\\cite{Peccei}. The standard practice \nis to make a choice of the value of this parameter so that the spin-3\/2 \n propagator has a simple form.\nIn order that the interaction also remains invariant under this transformation\n an additional term is added to it.\nThus the Lagrangians involving spin-3\/2 fields take the form\n\\begin{eqnarray}\n{\\cal L}&=&\\frac{f}{m_\\rho}[\\overline{\\psi}^{\\alpha}_R {\\cal O}_{\\alpha\\beta}\n\\gamma_{\\nu}\\gamma^5\\rho^{\\beta\\nu}\\psi_N  + h.c.]\n~~~~~~~J^{P}_R=\\frac{3}{2}^+ \\nonumber\\\\\n{\\cal L}&=&\\frac{f}{m_\\rho}[\\overline{\\psi}^{\\alpha}_R {\\cal O}_{\\alpha\\beta}\n\\gamma_{\\nu}\\rho^{\\beta\\nu}\\psi_N  + h.c.]\n~~~~~~~~~~J^{P}_R=\\frac{3}{2}^- \n\\label{lag2}\n\\end{eqnarray}\nwith ${\\cal O}_{\\mu\\alpha}=g_{\\mu\\alpha}-\\frac{1}{4}\\gamma_\\mu\\gamma_\\alpha$, the\nsecond term contributing only when the spin $3\/2$ field is off the mass shell.\nThe value of the coupling strength $f$ thus remains unaffected by this exercise.\n\n\n\\begin{figure}\n\\includegraphics[scale=0.8]{rholoop_BB.eps}\n\\caption{One-loop Feynman diagrams for the two-point function contributing to\nthe $\\rho$ self-energy in baryonic matter. The solid and double\nlines stand for nucleons and resonances respectively.}\n\\label{loop_BB}\n\\end{figure} \n\n\\section{Kinematics of the $\\rho$ propagator}\nThe complete propagator of the $\\rho$ is obtained from the Dyson\nequation~\\cite{Kobes,Bellac}\n\\begin{equation}\nG_{{\\mu\\nu}}^{ab}(q)=G_{{\\mu\\nu}}^{(0)ab}(q)- G_{\\mu\\lambda}^{(0)ac}(q)\\Pi^{\\lambda\\sigma, cd}_{\\rm tot}(q)\n G_{\\sigma\\nu}^{db}(q)\n\\label{dyson_G}\n\\end{equation}\nwhere $\\Pi_{\\rm tot}^{{\\mu\\nu},ab}$ denotes the thermal self-energy matrix and  \n$G_{{\\mu\\nu}}^{(0)ab}(q)$ stands for the free thermal propagator. \n\nAs described briefly in the appendix, one can get rid of the thermal \nindices by diagonalisation. In terms of the diagonal \nelements (denoted by bar) which are analytic functions, \nthe Dyson equation for the $\\rho$ propagator reads\n\\begin{equation}\n\\og_{{\\mu\\nu}}(q)=\\og_{{\\mu\\nu}}^{(0)}(q)-\\og_{\\mu\\lambda}^{(0)}(q)\\overline\\Pi^{\\lambda\\sigma}_{\\rm tot}(q)\n\\og_{\\sigma\\nu}(q)~,\n\\label{dyson_G_2}\n\\end{equation}\nwhere\n\\begin{equation}\n\\og^{(0)}_{{\\mu\\nu}}(q)=\\left(-g_{{\\mu\\nu}}+\n\\frac{q_\\mu q_\\nu}{q^2}\\right)\\frac{-1}{q^2-m_\\rho^2+i\\epsilon}~.\n\\end{equation}\nThe one loop self energy  with baryons is obtained from the \ntwo diagrams shown in Fig.~\\ref{loop_BB} so that\n\\begin{equation}\n\\overline\\Pi^{\\lambda\\sigma}_B(q)=\\overline\\Pi^{\\lambda\\sigma}(q)+\\overline\\Pi^{\\sigma\\lambda}(-q)~.\n\\label{diag1a-b}\n\\end{equation}\nOn addition of the contribution from the meson loops\nthe total $\\rho$ self-energy is given by\n\\begin{equation}\n\\overline\\Pi^{\\lambda\\sigma}_{\\rm tot}(q)=\\overline\\Pi^{\\lambda\\sigma}_B(q)+\\overline\\Pi^{\\lambda\\sigma}_M(q)~.\n\\end{equation}\nIn the medium, the presence of the four velocity $u_\\mu$ introduces an additional \nscalar variable \n$u\\cdot q$ in addition to $q^2$\nleading to two independent tensors $P_{{\\mu\\nu}}$ and $Q_{{\\mu\\nu}}$ in terms of which \nthe propagator and self-energy can be\n written as \n\\begin{eqnarray}\n\\og_{{\\mu\\nu}}&=&P_{{\\mu\\nu}}\\og_t + Q_{{\\mu\\nu}}\\og_l\\nonumber\\\\\n\\overline\\Pi_{{\\mu\\nu}}&=&P_{{\\mu\\nu}}\\overline\\Pi_t + Q_{{\\mu\\nu}}\\overline\\Pi_l\n\\label{T_L}\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\nP_{{\\mu\\nu}}&=&-g_{{\\mu\\nu}}+\\frac{q_\\mu q_\\nu}{q^2}-\\frac{q^2}{\\overline{q}^{\\,2}}\\widetilde u_\\mu \\widetilde\nu_\\nu,~\\widetilde u_\\mu=u_\\mu-(u\\cdot q)q_{\\mu}\/q^2~;\\nonumber\\\\\nQ_{{\\mu\\nu}}&=&\\frac{(q^2)^2}{\\overline{q}^{\\,2}}\\widetilde u_\\mu \\widetilde\nu_\\nu,~\\overline{q}^{\\,2}=(u\\cdot q)^2-q^2~.\n\\label{defP+Q}\n\\end{eqnarray}\nUsing (\\ref{T_L}), the Dyson equation (\\ref{dyson_G_2}) can be solved to get,\n\\begin{equation}\n\\og_t(q)=\\frac{-1}{q^2-m_\\rho^2-\\overline\\Pi_t(q)},~~~~~\n\\og_l(q)=\\frac{1}{q^2}\\frac{-1}{q^2-m_\\rho^2-q^2\\overline\\Pi_l(q)}\n\\end{equation}\nwhere \n\\begin{equation}\n\\overline\\Pi_t=-\\frac{1}{2}(\\overline\\Pi_\\mu^\\mu +\\frac{q^2}{\\bar q^2}\\overline\\Pi_{00}),~~~~\n\\overline\\Pi_l=\\frac{1}{\\bar q^2}\\overline\\Pi_{00} , ~~~\\overline\\Pi_{00}\\equiv u^\\mu u^\\nu \\overline\\Pi_{{\\mu\\nu}}~.\n\\label{pitpil}\n\\end{equation}\nThe self-energy function $\\overline\\Pi_{{\\mu\\nu}}$  can be obtained from \nthe 11-component of the in-medium self-energy matrix using (see appendix)\n\\begin{eqnarray}\n{\\rm Re}\\,\\overline\\Pi_{{\\mu\\nu}}&=&{\\rm Re}\\,\\Pi_{{\\mu\\nu}}^{11}\\nonumber\\\\\n{\\rm Im}\\,\\overline\\Pi_{{\\mu\\nu}}&=&\\epsilon(q_0)\\tanh(\\beta q_0\/2){\\rm Im}\\,\\Pi_{{\\mu\\nu}}^{11}\n\\label{def_diag}\n\\end{eqnarray}\nin terms of which the retarded self-energy is given by~\\cite{Bellac}\n\\begin{eqnarray}\n{\\rm Re}\\,\\Pi_{{\\mu\\nu}}&=&{\\rm Re}\\,\\overline\\Pi_{{\\mu\\nu}}\\nonumber\\\\\n{\\rm Im}\\,\\Pi_{{\\mu\\nu}}&=&\\epsilon(q_0){\\rm Im}\\,\\overline\\Pi_{{\\mu\\nu}}~.\n\\label{def_ret}\n\\end{eqnarray}\nWe now proceed to evaluate the 11-component of the rho self-energy in the\nfollowing section.\n\n\\section{The self energy and its analytic structure}\n\n\nLet us begin by writing the expression for the $\\rho$ self-energy in vacuum\ncorresponding to the first diagram in Fig.~1.\nFor spin 1\/2 resonances in the loop, this is given by\n\\begin{equation}\n\\Pi^{{\\mu\\nu}}(q)=i I_F\\left(\\frac{fF(q)}{m_\\rho}\\right)^2 \n\\int \\frac{d^4p}{(2\\pi)^4} Tr[\\Gamma^{\\mu}S(p,m_N)\\Gamma^{\\nu}S(p-q, m_R)] \n\\label{spin1\/2}\n\\end{equation}\nwhere $S(p,m)=(p \\!\\!\\! \/+m)\\Delta(p,m)$ is the fermion propagator, \n$\\Delta(p,m)$ being the free propagator for a scalar field of mass $m$ and is given by\n\\begin{equation}\n\\Delta(p,m)=\\frac{-1}{p^2-m^2+i\\epsilon}~.\n\\end{equation}\nAlso included is a monopole form factor $F(q)=\\Lambda^2\/\\Lambda^2+\\vec q^2$ with $\\Lambda=2$\nGeV~\\cite{RappNPA} to take into account the finite size of the $\\rho N R$ vertex.\n\nThe corresponding expression for the case of loop graphs with spin 3\/2 resonances \nis given by\n\\begin{equation}\n\\Pi^{{\\mu\\nu}}(q)=i I_F\\left(\\frac{fF(q)}{m_\\rho}\\right)^2 \\int \\frac{d^4p}{(2\\pi)^4} \nTr[\\Gamma^{\\mu\\alpha}S(p,m_N)\\Gamma^{\\nu\\beta}S_{\\beta\\alpha}(p-q, m_R)]\n\\label{spin3\/2}\n\\end{equation} \nwhere the spin-3\/2 propagator is \n$S_{{\\mu\\nu}}(k,m)=(k \\!\\!\\! \/ + m)K_{{\\mu\\nu}}(k)( \\Delta(k,m)$ with\n$K_{{\\mu\\nu}}(k)= -g_{{\\mu\\nu}}+ \\frac{2}{3m^2}k_\\mu k_\\nu + \\frac{1}{3}\\gamma_\\mu\\gamma_\\nu \n+ \\frac{1}{3m}(\\gamma_\\mu k_\\nu - \\gamma_\\nu k_\\mu )$.\nObtaining the vertex factors $\\Gamma^{\\mu}$ and $\\Gamma^{\\mu\\alpha}$ \nfrom the interaction Lagrangians (\\ref{lag1}) and (\\ref{lag2}) \nboth the expressions (\\ref{spin1\/2}) and (\\ref{spin3\/2}) can\nbe expressed in the general form\n\\begin{equation}\n\\Pi_{{\\mu\\nu}}(q)=i\\int\\frac{d^4p}{(2\\pi)^4}L_{{\\mu\\nu}}(p,q)\\Delta (p,m_N)\\Delta(p-q,m_R) \n\\end{equation}\nwhere the factor $L_{{\\mu\\nu}}(p,q)$ consists of the trace over Dirac matrices appearing \nin the two fermion propagators along with\ntheir associated tensor structures, isospin and form factors coming\nfrom the $\\rho NR$ vertex. Since the self-energy is transverse,\n$L^{\\mu\\nu}$ can be expressed as\n\\begin{equation}\nL^{{\\mu\\nu}}(p,q)=I_F \\left(\\frac{fF(q)}{m_\\rho}\\right)^2 [\\alpha(p,q) A^{{\\mu\\nu}}+\\beta(p,q)\nB^{{\\mu\\nu}}+\\gamma(p,q) C^{{\\mu\\nu}}]\n\\end{equation}\nwhere the three gauge-invariant tensors $A^{{\\mu\\nu}}$, $B^{{\\mu\\nu}}$ and $C^{{\\mu\\nu}}$ are \ngiven by\n\\begin{eqnarray}\nA_{{\\mu\\nu}}(q)&=&-g_{{\\mu\\nu}}+{q_\\mu q_\\nu}\/{q^2} ,\\nonumber\\\\\nB_{{\\mu\\nu}}(q,p)&=&q^2 p_\\mu p_\\nu-q\\cdot p(q_\\mu p_\\nu+p_\\mu q_\\nu)\n+(q\\cdot p)^2g_{{\\mu\\nu}} ,\\nonumber\\\\\nC_{{\\mu\\nu}}(q,p)&=&q^4 p_\\mu p_\\nu-q^2(q\\cdot p)(q_\\mu p_\\nu+p_\\mu\nq_\\nu) +(q\\cdot p)^2q_\\mu q_\\nu~.\n\\label{ABC}\n\\end{eqnarray}\nThe coefficient functions $\\alpha(p,q)$, $\\beta(p,q)$ and $\\gamma(p,q)$ for the different loops are tabulated in\nthe appendix.\n\nWe now extend the vacuum self-energy to the nuclear medium.\nIn the real-time version of thermal field theory \nthat we are using \nthe propagators assume the form of matrices. \nThe spin and isospin structure of the self-energy graph remaining the same, \nit is only the scalar part $\\Delta(p,m)$ of the propagators that assumes \na matrix structure. \nThe required $11-$component of the fermion propagator is given by \n\\begin{equation}\nE^{11}(p)=\\Delta(p)+2\\pi i\nN(p_0)\\delta(p^2-m^2);~~N(p_0)=n_+(\\omega)\\theta(p_0)+n_-(\\omega)\\theta(-p_0)~.\n\\label{de11}\n\\end{equation}\nThe function $n_\\pm(\\omega)=\\displaystyle\\frac{1}{e^{\\beta(\\omega \\mp \\mu)}+1}$ is\nthe Fermi distribution where the $\\pm$ sign in the subscript refers to baryons and \nanti-baryons respectively, $\\omega=\\sqrt{\\vec p^2+m^2}$ \nand $\\mu$ is the baryonic chemical potential which is \ntaken to be equal for all the baryons considered here. Expressed as\n\\begin{equation}\nE^{11}(p)=-\\frac{1}{2\\omega}\\left(\\frac{1-n_+}{p_0-\\omega+i\\epsilon}+\n\\frac{n_+}{p_0-\\omega-i\\epsilon}-\\frac{1-n_-}{p_0+\\omega-i\\epsilon}\n-\\frac{n_-}{p_0+\\omega+i\\epsilon}\\right)\n\\label{fullprop}\n\\end{equation}\nthe first and the second terms can be identified with the propagation of baryons\nabove the Fermi sea and holes in the Fermi sea respectively~\\cite{Vol_16} while\nthe third and\nfourth terms correspond to anti-baryons. \n\nAs noted in the previous section, the in-medium self-energy function of the\n$\\rho$\ncan be obtained from the 11-component of the thermal self-energy \nmatrix. For the one-loop graphs shown in Fig.~1, the latter is given by\n\\begin{equation}\n\\Pi^{11}_{{\\mu\\nu}}(q)=i\\int\\frac{d^4p}{(2\\pi)^4}L_{{\\mu\\nu}}(p,q)\nE^{11}(p,m_N)E^{11}(p-q,m_R) ~.\n\\label{eq_b}\n\\end{equation}\nUpon inserting the form of $E^{11}$ from (\\ref{de11}) we get three types of \nterms. One is the vacuum contribution involving \nthe vacuum parts of the two propagators, the other two being medium dependent, \none linear and the other quadratic in the\nthermal distribution function. \nPerforming the $p_0-$integration and using the relations (\\ref{def_diag})\nconnecting the real and imaginary parts of the $11$-component of the \nself-energy matrix with those of the diagonal \nelement (defined using a bar), the self-energy function is written as\n\\begin{eqnarray}\n\\overline\\Pi^{\\mu\\nu}(q_0,\\vec q)=\\int\\frac{d^3\\vec p}{(2\\pi)^3 4\\omega_N\\omega_R}&\\times &\n\\left[\\frac{L^{\\mu\\nu}_1 n^N_+ -L^{\\mu\\nu}_3 n^R_+}{q_0 -\\omega_N+\\omega_R+i\\epsilon(q_0)\\eta}-\n\\frac{L^{\\mu\\nu}_2 n^N_- -L^{\\mu\\nu}_4 n^R_-}{q_0 +\\omega_N-\\omega_R+i\\epsilon(q_0)\\eta}\n\\right. \\nonumber\\\\\n&&\\left.+\\frac{L^{\\mu\\nu}_1 (1-n^N_+) -L^{\\mu\\nu}_4 n^R_-}{q_0 -\\omega_N-\\omega_R+i\\epsilon(q_0)\\eta}\n-\\frac{L^{\\mu\\nu}_2 (1-n^N_-) -L^{\\mu\\nu}_3 n^R_+}{q_0\n+\\omega_N+\\omega_R+i\\epsilon(q_0)\\eta}\\right] \n\\label{Pi_a}\n\\end{eqnarray} \nwhere $n^N\\equiv n(\\omega_N)$ with $\\omega_N=\\sqrt{\\vec p^2+m_{N}^2}$, $n^R\\equiv \nn(\\omega_R)$ with $\\omega_R=\\sqrt{(\\vec p-\\vec q)^2+m_{R}^2}$\nand $L^{\\mu\\nu}_i,i=1,..4$ denote the values of $L^{\\mu\\nu}(p_0)$ for\n$p_0=\\omega_N,-\\omega_N,q_0+\\omega_R,q_0-\\omega_R$ respectively. \n\n\nLet us first consider the imaginary part of the self-energy. \nThe retarded self-energy defined by (\\ref{def_ret}) can be\neasily read off from the self-energy function (\\ref{Pi_a}) to get\n\\begin{eqnarray}\n&& {\\rm Im} \\Pi^{\\mu\\nu}(q_0,\\vec q)=-\\pi\\coth(\\beta q_0\/2) \\int\\frac{d^3\\vec p}{(2\\pi)^3 4\\omega_N\\omega_R}\\times  \\nonumber\\\\\n&&[L^{\\mu\\nu}_1\\{(1-n^N_+-n^R_-)\\delta(q_0-\\omega_N-\\omega_R)\n+(n_+^N-n_+^R)\\delta(q_0-\\omega_N+\\omega_R)\\}\\nonumber\\\\\n&& + L^{\\mu\\nu}_2\\{(n_-^R-n_-^N)\\delta(q_0+\\omega_N-\\omega_R)\n-(1-n_-^N-n_+^R)\\delta(q_0+\\omega_N+\\omega_R)\\}]\n\\label{ImPi_a}\n\\end{eqnarray} \nin which the factors $L^{\\mu\\nu}_{3,4}$ have converted to $L^{\\mu\\nu}_{1,2}$ respectively in \nassociation with the $\\delta$-functions.\nFollowing~\\cite{Weldon,Ghosh1} it is interesting to relate the\nterms appearing in the above expression with scattering and decay processes \ninvolving the $\\rho$, nucleon $(N)$ and the heavy resonances $(R)$ \n in the medium. The delta functions in each of the terms in (\\ref{ImPi_a}) \n precisely define the kinematic domains where these processes can occur. \n The regions which are non-vanishing \n  give rise to cuts in the self-energy function. Thus, the first and \n the fourth terms are non-vanishing for $q^2>(m_R+m_N)^2$\n giving rise to the unitary cut and second and third terms are non-vanishing for\n  $q^2<(m_R-m_N)^2$ giving rise to the Landau cut. Note that\nthe unitary cut is present in vacuum but the Landau cut appears\n only in the medium. \n\nConsider, for example the first term with $1-n^N_+-n^R_-$ in (\\ref{ImPi_a}). Written as \n$(1-n^N_+)(1-n^R_-)-n^N_+n^R_-$ this indicates a process in which\n a  (virtual) $\\rho$ decays into a $NR^{-1}$ pair with the \n Pauli blocked probability  $(1-n^N_+)(1-n^R_-)$ minus the process in\n which  $NR^{-1}$ pair gets absorbed in the medium with a statistical weight\n  factor $n^N_+n^R_-$. This process can obviously \ntake place for $\\rho$'s with invariant mass $(\\sqrt{q^2}) > (m_R+m_N)$,\n a requirement that is in conformity with the kinematic threshold of the\nunitary cut coming from the associated $\\delta-$function. \n\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.6]{landau_cut.eps}}\n\\caption{The positions of the Landau and unitary cuts in the complex $q_0$\nplane for $q^2>0$.}\n\\label{cutfig}\n\\end{figure}  \n\nThe kinematic domains where the four terms contribute can be summarised \nas follows. For\n$q_0>0$, the first term in (\\ref{Pi_a}) contributes for time-like values of\n$q^2$, the second at space-like $q^2$ and the third at all $q^2$. Likewise, \nfor negative values of the variable $q_0$, the second term is non-zero\nat all $q^2$, the third at space-like $q^2$ and the fourth only for time-like\n$q^2$. In view of the fact that the spectral function of the $\\rho$ will be \nmeasured in the invariant mass spectra of lepton pairs we will henceforth\nconfine ourself to the kinematic region $q_0>0$ and  $q^2>0$.\nThe position of the relevant cuts in the complex energy plane \nfor this region are shown in Fig.~\\ref{cutfig} where we have ignored the portion\nof the Landau cut for $q_0<|\\vec q|$. (For a discussion on \nthe branch cuts on the entire $q_0$ axis, see~\\cite{Ghosh1}).\nFurthermore, we will not include the unitary cut contribution\nin the imaginary part since the \nthreshold of this cut begins at $q_0>m_R+m_N$\nwhich being far away from the $\\rho$ pole is not expected to contribute \nto the $\\rho$ spectral \nfunction in a substantial way. Thus only the Landau cut as given by the third\nterm in eq.~(\\ref{ImPi_a}) will be considered.\n\n\nCollecting the Landau contributions from both the diagrams using (\\ref{diag1a-b})\nwe finally write down the imaginary part of the $\\rho$ self-energy due to baryonic \nloops\n\\begin{eqnarray}\n&&{\\rm Im}\\Pi_B^{\\mu\\nu}(q_0,\\vec q)=\\pi\\int\\frac{d^3\\vec p}{(2\\pi)^3 2\\omega_N}\n\\left[\\frac{L_1^{\\mu\\nu} (-q)}{2\\omega_{R'}}\\{(n^N_+ -n^{R'}_+)\\delta(q_0+\\omega_N-\\omega_{R'})\\}\\right.\\nonumber\\\\\n&& + \\left.\\frac{L_2^{\\mu\\nu} (q)}{2\\omega_R}\\{(n^N_- - n^R_-)\\delta(q_0+\\omega_N-\\omega_R)\\}\\right]\n\\end{eqnarray}\nwhere $\\omega_{R'}=\\sqrt{(\\vec p+\\vec q)^2+m_{R}^2}$. The two terms in this\nexpression describe the contributions from scattering processes. The factor \n$(n_+^N-n_+^{R'})$\nexpressed as $(1-n^{R'}_+)n^N_+ - (1-n^N_+)n^{R'}_+$ can be interpreted as the \nprobability of a $\\rho$ meson scattering on a nucleon from the \nmedium producing a resonance minus the process in which it\nscatters from the resonance to produce a nucleon, \nthe final states in both cases being Pauli-blocked. The corresponding processes\ninvolving anti-baryons are included in the second term.\n\n\nLet us now proceed to evaluate the integral over the momenta in the \n$NR^{-1}$ loop. The integral over $\\cos\\theta$ in \n$d^3p=-2\\pi\\sqrt{\\omega^2_N-m^2_N}\\omega_N d\\omega_Nd(\\cos\\theta)$ is done \nusing the $\\delta-$function. Also the condition \n$|\\cos\\theta|\\leq 1$ puts restriction on the range of \nintegration over $\\omega_N$.\nA substantial simplification is obtained by changing the integration variable \nto $x$ using $\\omega_N=\\frac{S^2}{2q^2}(-q_0+|\\vec q|x)$ where \n$S^2=q^2-m^2_R+m^2_N$ to get\n\\begin{equation}\n{\\rm Im}\\Pi_B^{\\mu\\nu}(q_0,\\vec q)=-\\frac{S^2}{32\\pi q^2}\\int_{-W}^W dx\\ L^{\\mu\\nu}(x)\n\\{n_+(q_0+\\omega_N)-n_+(\\omega_N)+n_-(q_0+\\omega_N)-n_-(\\omega_N) \\}\n\\label{piB_final}\n\\end{equation}\nwhere $W=\\sqrt{1-4q^2m_N^2\/S^4}$ and the factor $L^{\\mu\\nu}$ in terms of the variable $x$ has the same form for\nboth diagrams in Fig.~1. This is shown in the appendix. \n\nThe real part consists of principal value integrals which remain after removing the\nimaginary part from (\\ref{Pi_a}). Note that unlike the imaginary part, \nthe real part of the self-energy at a given value of $q$\nreceives contribution from all the four terms.\n\nUp to now we have been treating the baryon resonances $R$ in the narrow width \napproximation. It is indeed necessary to consider the width of the\nunstable baryons in a realistic evaluation of the spectral function. \nFor this, we follow the procedure (see\ne.g.~\\cite{SarkarNPA,Vijande})\nof convoluting the self energy calculated in the narrow width \napproximation with the spectral function of the baryons. \nThis approach has the advantage that the analytic structure of the self energy \ndiscussed above remains undisturbed.\n\n\\begin{equation}\n\\Pi_B^{\\mu\\nu}(q;m_R)= \\frac{1}{N_R}\\int^{m_R+2\\Gamma_R}_{m_R-2\\Gamma_R}dM\\frac{1}{\\pi} \n{\\rm Im} \\left[\\frac{1}{M-m_R + \\frac{i}{2}\\Gamma_R(M)} \\right] \\Pi_B^{\\mu\\nu}(q;M) \n\\end{equation}\nwith $N_R=\\displaystyle\\int^{m_R+2\\Gamma_R}_{m_R-2\\Gamma_R}dM\\frac{1}{\\pi} {\\rm Im} \n\\left[\\frac{1}{M-m_R + \\frac{i}{2}\\Gamma_R(M)} \\right]$ and \n$\\Gamma_R(M)=\\Gamma_{R\\rightarrow N\\pi} (M) + \\Gamma_{R\\rightarrow N\\rho} (M)$.\nAs a consequence of this convolution, the sharp ends of the regions of\nnon-zero imaginary part smoothly go to zero at a higher value of $M$\ndepending upon the width of the resonance. This is shown in\nFig.~\\ref{impi_B} for the $N^*(1520)$ resonance. \n\nNuclear medium at finite temperature is also substantially populated by mesons \nwhich modify \nthe $\\rho$ propagation in the medium \nin a non-trivial way. This has been studied~\\cite{Ghosh1}\nfollowing the same procedure as described here for mesonic loop graphs with\none internal pion line and another meson line $h$ where $h=\\pi,\\omega,h_1,a_1$\nusing interactions from chiral perturbation theory. \nCollecting the real and\nimaginary parts, the self-energy from \nmesonic loops can be written as\n\\begin{eqnarray}\n&&\\Pi^{{\\mu\\nu}}_M(q_0,\\vec q)= \\int \\frac{d^3 \\vec k}{(2\\pi)^3}\n\\frac{1}{4\\omega_\\pi\\omega_h}\\left[-\\frac{N^{{\\mu\\nu}}_1 n_\\pi - N^{{\\mu\\nu}}_3 n_h}\n{q_0 - \\omega_\\pi + \\omega_h+i\\epsilon(q_0)\\eta}+\\frac{N^{{\\mu\\nu}}_2 n_\\pi - N^{{\\mu\\nu}}_4 n_h}\n{q_0 + \\omega_{\\pi} - \\omega_h+i\\epsilon(q_0)\\eta}\n\\right.\\nonumber\\\\\n&& \\left.+ \\frac{N^{{\\mu\\nu}}_1 (1+n_\\pi)  + N^{{\\mu\\nu}}_4 n_h}{q_0 + \\omega_\\pi - \n\\omega_h +i\\epsilon(q_0)\\eta}- \\frac{N^{{\\mu\\nu}}_2(1+ n_\\pi) + N^{{\\mu\\nu}}_3 n_h}\n{q_0 + \\omega_\\pi + \\omega_h +i\\epsilon(q_0)\\eta}\\right]\n\\label{pi_meson}\n\\end{eqnarray}\nwhere the  Bose distribution functions $n_\\pi\\equiv n(\\omega_\\pi)$ with\n$\\omega_\\pi=\\sqrt{\\vec k^2+m_\\pi^2}$ and\n$n_h\\equiv n(\\omega_h)$ with $\\omega_h=\\sqrt{(\\vec q-\\vec k)^2+m_h^2}$.\n $N^{{\\mu\\nu}}_i(i=1,4)$ are the values of $N^{{\\mu\\nu}}(k_0)$ for\n$k_0=\\omega_\\pi,-\\omega_\\pi,q_0+\\omega_h,q_0-\\omega_h$ respectively and can \nbe expressed in terms of the gauge-invariant tensors (\\ref{ABC}).\nThe complete expressions are provided in the appendix. \nUsing the procedure described above \nwe have improved upon the\ncalculations in~\\cite{Ghosh1} by including  the\nwidth of the heavy mesons $a_1$ and $h_1$.\nIn this case we use a slightly different formula~\\cite{Vijande}, \n\\begin{equation}\n\\Pi_M^{\\mu\\nu}(q;m_h)= \\frac{1}{N_h}\\int^{(m_h+2\\Gamma_h)^2}_{(m_h-2\\Gamma_h)^2}dM^2\\frac{1}{\\pi} \n{\\rm Im}\\left[\\frac{1}{M^2-m_h^2 + iM\\Gamma_h(M) } \\right] \\Pi_M^{\\mu\\nu}(q;M) \n\\end{equation}\nwith $N_h=\\displaystyle\\int^{(m_h+2\\Gamma_h)^2}_{(m_h-2\\Gamma_h)^2}\ndM^2\\frac{1}{\\pi} {\\rm Im}\\left[\\frac{1}{M^2-m_h^2 + iM\\Gamma_h(M)} \\right]$\nand $\\Gamma_h(M)=\\Gamma_{h\\rightarrow \\rho\\pi} (M)$.\n\nThe transverse and longitudinal components can then be obtained from the\nself-energy tensors using the relations (\\ref{pitpil}).\n\n\\section{Numerical Results}\n\n\\begin{figure}\n\\includegraphics[scale=0.35]{im_qv_00.eps}\n\\includegraphics[scale=0.35]{im_qv_300.eps}\n\\caption{Imaginary part of $\\rho$ meson self-energy  \nshowing the individual contributions for different $NR$ loops.\nLeft panel shows results for $\\vec q=0$ and the right panel shows the\ntransverse (solid) and longitudinal (dotted) parts for $\\vec q=300$ MeV. The \nlong dashed line in the upper panel on the left shows\nthe imaginary part of the $NN^*(1520)$ loop evaluated in the narrow width approximation.}\n\\label{impi_B}\n\\end{figure}  \n\n\n\\begin{figure}\n\\includegraphics[scale=0.35]{re_qv_00.eps}\n\\includegraphics[scale=0.35]{re_qv_300.eps}\n\\caption{Same as Fig.~\\ref{impi_B} for the real part}\n\\label{repi_B}\n\\end{figure}  \n\n\nIn this section we present the results of numerical evaluation beginning with the\nimaginary part of the $\\rho$ self-energy\nas a function of the invariant mass $\\sqrt{q^2}\\equiv M$ for two values of the\nthree momentum. \nThe Landau cut\ncontribution starts from $M=0$ for all values of the three momentum.\nShown in Fig.~\\ref{impi_B} left panel are the contributions from the individual\n$NR$ loops for a $\\rho$ meson at rest.\nThe $NN^*(1520)$ loop makes the most significant contribution followed by\nthe $N^*(1720)$ and $\\Delta(1700)$. The right panel shows the \ncorresponding results for $\\vec q=300$ MeV where the transverse and \nlongitudinal components $\\Pi_t$ and $q^2\\Pi_l$ have been shown separately\nby solid and dotted lines respectively.\n(Note that for a $\\rho$ meson at rest $\\Pi_t=q_0^2\\Pi_l$.)\nThe corresponding results for the thermal contribution to the real part are shown in Fig.~\\ref{repi_B}.\nThe divergent vacuum contribution in this case\nis assumed to renormalize the $\\rho$ mass to its physical value. \nAlso shown by the long dashed lines in the left upper panels of Figs.~\\ref{impi_B} and\n\\ref{repi_B} are the corresponding contributions to the real and\nimaginary parts coming from the $NN^*(1520)$ loop computed in the narrow width\napproximation.\n\\begin{figure}\n\\includegraphics[scale=0.35]{im_re_M.eps}~~\n\\includegraphics[scale=0.35]{im_re_MBt.eps}\n\\caption{(Left) Imaginary and real parts of $\\rho$ meson self-energy  \nshowing the individual contributions for different $\\pi-h$ loops.\n(Right) The total contribution from meson and baryon loops.}\n\\label{imrepi_MBt}\n\\end{figure}  \n\nNext we show the results of the spin-averaged $\\rho$ self-energy \ndefined by \n\\begin{equation}\n\\Pi=\\frac{1}{3}(2\\Pi_t+q^2\\Pi_l)~.\n\\end{equation}\nwhere the transverse and longitudinal components are obtained\nusing (\\ref{pitpil}). In Fig.~\\ref{imrepi_MBt} left panel the imaginary and real parts of $\\pi-h$ loop\ngraphs are shown in the upper and lower panels respectively. \nThe Landau and unitary cut contributions for the $\\pi-\\omega$ loop are clearly\ndiscernible though the contribution at the $\\rho$ pole is dominated by the $a_1$.\nOn the right panel we plot the total contribution from the baryon and meson \nloops for two values of the baryonic chemical potential. The small positive\ncontribution from the baryon loops to the real part is partly compensated\nby the negative contributions from the meson loops. The\nsubstantial baryon contribution at vanishing baryonic chemical potential \nreflects the importance of anti-baryons.\n\\begin{figure}\n\\includegraphics[scale=0.35]{spec_T.eps}~~~\n\\includegraphics[scale=0.35]{spec_qv.eps}\n\\caption{ The spectral function of the $\\rho$ meson  \nfor (left) different values of the temperature $T$ and\n(right) different values of the three-momentum $\\vec q$.}\n\\label{spec_T_qv}\n\\end{figure} \n\n \nWe now turn to the spin averaged spectral function given by\n\\begin{equation}\n\\mathrm{Im}\\,\\og(q)=\\frac{1}{3}(2\\mathrm{Im}\\,\\og_t+q^2\\mathrm{Im}\\,\\og_l)\n\\label{spavgG}\n\\end{equation}\nwhere\n\\begin{equation}\n\\mathrm{Im}\\,\\og_{t,l}(q)=\\frac{-\\sum{\\rm Im}\\,\\overline\\Pi_{t,l}^{\\rm tot}}{(M^2-m_\\rho^2-\n(1,q^2)\\sum\\mathrm{Re}\\,\\overline\\Pi_{t,l}^{\\rm\ntot})^2+\\{(1,q^2)\\sum{\\rm Im}\\,\\overline\\Pi_{t,l}^{\\rm tot}\\}^2}~.\n\\end{equation}\nFirst, in Fig.\\ref{spec_T_qv} left panel we plot the spectral function at fixed\nvalues of the baryonic chemical potential and three-momentum for various\nrepresentative values of the temperature. \nWe observe an increase of spectral strength at lower \ninvariant masses resulting in broadening of the spectral function \nwith increase in temperature. This is purely a Landau cut\ncontribution from the baryonic loops arising from the scattering of the $\\rho$ from\nbaryons in the medium.\nHowever, we do not observe much\nvariation with the three-momentum of the $\\rho$ as seen from the figure on the\nright panel.\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.35]{spec_mu.eps}}\n\\caption{ The spectral function of the $\\rho$ meson \nfor different values of the baryonic chemical potential $\\mu$}\n\\label{spec_mu}\n\\end{figure} \nWe then plot in Fig.~\\ref{spec_mu}, the spectral function for \nvarious values of the baryonic chemical potential for a fixed temperature.\nFor high values of $\\mu$\nwe observe an almost flattened spectral density of the $\\rho$.  \n\nOwing to differences in the various approaches to the evaluation of the $\\rho$\nspectral function followed in the literature, a direct numerical comparison  \nwith earlier results does not appear to be meaningful. These differences \nare at the level of the basic formulae arising from the type of couplings \nof the nucleon and the rho fields with the various baryon resonances considered\nas well as in the form of the propagators used in the calculations. There\nalso exist differences at the level of formalism employed in the\nevaluation of the $\\rho$ self-energy.\nWe thus end this section by showing how the in-medium spectral function\nof the $\\rho$ is \nmanifested in the dilepton emission rate.\nThis rate from thermalised \nhadronic matter is given by~\\cite{Toimela}\n\\begin{equation}\n\\frac{dR}{d^4q}=-\\frac{\\alpha^2}{3\\pi^3 q^2}H(M^2)n_{BE}(q_0)\\,g^{\\mu\\nu}{\\rm Im}\nT_{\\mu\\nu}(q_0,\\vec q)\n\\end{equation}\nwhere ${\\rm Im}T_{\\mu\\nu}$ is the imaginary part of the (retarded) two-point \nfunction of vector currents which can be obtained from (\\ref{Tmed}) using\nrelations analogous to (\\ref{def_diag}). The quantity $H(M^2)=(1+{2m_l^2}\/{M^2})~\n(1-4m_l^2\/M^2)^{1\/2}~$ is of the order of unity for electrons and will be\nomitted henceforth.\nIn the low invariant mass ($M$) region, ${\\rm Im}T_{\\mu\\nu}$ is usually \nexpressed as a sum over\nthe spectral densities of the vector mesons $\\rho$, $\\omega$ and\n$\\phi$~\\cite{Annals}. This is however justified only in vacuum. \nThe vector mesons $\\rho$ and $\\omega$ can in general undergo mixing in the presence \nof matter which can lead to non-trivial modifications, \nfor example, of the electromagnetic form factor\nof the pion~\\cite{Poda_JPG} and consequently the invariant mass distribution\nof lepton pairs. In the following, we will consider only the\n$\\rho$ pole contribution which is known to play the most dominant role.\nThe dilepton rate in this case can be \nexpressed in terms of the spin-averaged spectral function of the $\\rho$ \n(\\ref{spavgG}) getting,\n\\begin{equation}\n\\frac{dR}{dM^2q_Tdq_Tdy}=\\frac{\\alpha^2}{\\pi^2 M^2}n_{BE}(q_0)K_\\rho{\\rm Im}\\overline G(q_0,\\vec q)\n\\end{equation}\nwhere we have used $T_{\\mu\\nu}=K_\\rho G_{\\mu\\nu}$ as defined earlier. \nIntegrating over the transverse momentum $q_T$ and rapidity $y$ of the \nelectron\npairs we plot $dR\/dM^2$ vs $M$ in Fig.~\\ref{dilfig} for $T$=175 MeV. \nBecause of the\nkinematical factors multiplying the $\\rho$ spectral function \nthe broadening appears magnified in the dilepton\nemission rate.\nA significant enhancement is seen in the low mass lepton production rate\ndue to baryonic\nloops over and above the mesonic ones shown by the dot-dashed line.\nThe substantial contribution from baryonic loops even for vanishing\nchemical potential points to the important role played by antibaryons in thermal\nequilibrium in systems created at RHIC and LHC energies. \n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.35]{dRdM2_inp.eps}}\n\\caption{ The lepton pair emission rate at $T=175$ MeV with and without\nbaryon (B) loops in addition to the meson (M) loops}\n\\label{dilfig}\n\\end{figure} \n\n\n\\section{Summary and Discussion}\n\nWe evaluate the $\\rho$ self-energy to one loop in nuclear matter at finite\ntemperature and baryon density. Loop graphs involving the nucleon and 4-star\n$N^*$ and $\\Delta$ resonances up to spin 3\/2 were calculated using gauge invariant\ninteractions in the framework of real time thermal field theory to obtain the\ncorrect relativistic expressions for the $\\rho$ self-energy. The singularities in\nthe complex energy plane were analysed and the imaginary part obtained from the\nLandau cut contribution. Results for the real and imaginary parts at non-zero\nthree-momenta for various\nvalues of temperature and baryonic chemical potential were shown for the\nindividual loop graphs. Adding the contributions from mesons obtained  \nin the same formalism, the spectral function of the \n$\\rho$ was observed to undergo a significant modification at and below \nthe nominal rho mass which was seen to bring about a large enhancement of\nlepton pair yield in this region.\n\nIt may be emphasised that the determination of the $\\rho$ spectral\ndensity at finite\ntemperature and baryon density by an explicit evaluation of loop\ngraphs using thermal field theoretic techniques such as performed here\nis of relevance in view of precision data from experiments at\nRHIC and LHC as well as from the\nFAIR facility at GSI in future. But an actual comparison with data will \ninvolve a\nspace-time evolution of the static rates using a framework like\nrelativistic hydrodynamics.   \nEfforts in this direction are in progress and\nwill be reported in due coarse.   \n\n\\section{Acknowledgement}\n\nThe authors gratefully acknowledge  discussions with S. Mallik\nduring the coarse of this work and to J. Alam for useful suggestions.\n\n\\section{Appendix}\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\subsection{The real-time propagators and the self-energy matrix}\n\n\nIn the real time formulation of thermal field\ntheory a two-point function of local operators assumes a $2\\times 2$ matrix\nstructure on account of the shape of the contour in the complex time plane.\nThus the current correlator $T^{\\mu\\nu}$~\\cite{SarkarIJP} as well as propagators for various\nfields\\cite{Kobes}\nassume a matrix structure. For the contour shown in Fig.~\\ref{contourfig} the\ncomponents of a free scalar propagator are as\n\\begin{eqnarray}\n&&D^{11}=-(D^{22})^*=\\Delta(q)+2\\pi in\\delta(q^2-m^2)\\nonumber\\\\\n&&D^{12}=D^{21}=2\\pi i\\sqrt{n(1+n)}\\delta(q^2-m^2)\n\\label{Dlam}\n\\end{eqnarray}\nwhere $\\Delta(q)$ is the Feynman propagator in vacuum,\n\\begin{equation}\n\\Delta (q^2)=\\frac{-1}{q^2 -m^2 +i\\epsilon}~.\n\\end{equation}\n\\begin{figure}\n\\centerline{\\includegraphics{real_time.eps}}\n\\caption{Contour in the complex time plane for real time formalism}.\n\\label{contourfig}\n\\end{figure}\nThe thermal propagator may be diagonalised in the form \n\\begin{equation}\nD^{ab}(q_0,\\vec q)=U^{ac}(q_0)[{\\rm diag}\\{\\Delta(q_0,\\vec q),-\\Delta^*(q_0,\\vec q)\\}]^{cd}\nU^{db}(q_0)\n\\label{diag_D}\n\\end{equation}\nwith the elements of the diagonalising matrix as\n\\[U^{11}=U^{22}=\\sqrt{1+n},~~~ U^{12}=U^{21}=\\sqrt{n}~.\\]\nUsing the (transverse) vector propagator given by\n\\begin{equation}\nG^{(0)ab }_{{\\mu\\nu}}(q)=\\left(-g_{{\\mu\\nu}}+\\frac{q_\\mu q_\\nu}{q^2}\\right)D^{ab}(q)\n\\end{equation}\nin the Dyson equation and the fact that $U$ diagonalises not\nonly the free propagator, but also the complete one~\\cite{Kobes,Mallik_RT} it\nturns out that the self-energy matrix $\\Pi^{ab}_{{\\mu\\nu}}$ is also diagonalisable by\n$(U^{-1})^{ab}$, \n\\begin{equation}\n\\Pi^{ab}_{{\\mu\\nu}}(q)=[U^{-1}(q_0)]^{ac}[{\\rm\ndiag}\\{\\overline\\Pi_{{\\mu\\nu}}(q),-\\overline\\Pi_{{\\mu\\nu}}^*(q)\\}]^{cd}[U^{-1}(q_0)]^{db}~.\n\\end{equation}\nThe relations \n\\begin{eqnarray}\n{\\rm Re}\\,\\overline\\Pi_{{\\mu\\nu}}&=&{\\rm Re}\\,\\Pi_{{\\mu\\nu}}^{11}\\nonumber\\\\\n{\\rm Im}\\,\\overline\\Pi_{{\\mu\\nu}}&=&\\epsilon(q_0)\\tanh(\\beta q_0\/2){\\rm Im}\\,\\Pi_{{\\mu\\nu}}^{11}\n\\end{eqnarray}\ntrivially follow showing that the thermal matrices are actually given by a {\\em single} \nanalytic function which essentially coincides with the corresponding \nresult from the imaginary time formulation. \n\n\n\\subsection{The factor $L^{{\\mu\\nu}}$ in baryonic loops}\n\nThe factor $L^{{\\mu\\nu}}$ which appears in the loop integrals is given by\n\\begin{equation}\nL^{{\\mu\\nu}}(p,q)=I_F \\left(\\frac{fF(q)}{m_\\rho}\\right)^2 [\\alpha A^{{\\mu\\nu}}+\\beta\nB^{{\\mu\\nu}}+\\gamma C^{{\\mu\\nu}}]\n\\end{equation}\nwhere $A^{{\\mu\\nu}},\\,B^{{\\mu\\nu}}$ and $C^{{\\mu\\nu}}$ are the transverse tensors defined in\n(\\ref{ABC}). The coefficient factors $\\alpha,\\,\\beta,\\,\\gamma$ are given in Table~1 where\n\\begin{eqnarray}\n\\alpha_{1\/2+}&=&4q^2(p^2-m_N m_R+p\\cdotp q)\\nonumber\\\\\n\\alpha_{1\/2-}&=&\\alpha_{1\/2+}(m_N\\to -m_N)\\nonumber\\\\\n\\beta_{1\/2}&=&8\\nonumber\\\\\n\\gamma_{1\/2}&=&0\\nonumber\\\\\n\\alpha_{3\/2+}&=&\\frac{2q^2}{3m_R^2}[p^2(p^2-3q^2)+p\\cdot q(3p^2+q^2)\\nonumber\\\\\n&&+3m_R^2(p^2+2m_Nm_R+p\\cdot q)\\nonumber\\\\\n&&-2m_Nm_R(p^2+q^2-2p\\cdot q)]\\nonumber\\\\\n\\alpha_{3\/2-}&=&\\alpha_{3\/2+}(m_N\\to -m_N)\\nonumber\\\\\n\\beta_{3\/2}&=&4(1 +{p^2}\/{3m_R^2})\\nonumber\\\\\n\\gamma_{3\/2}&=&-{4}\/{3m_R^2}~.\n\\end{eqnarray}\n\n\n\\begin{table\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n& & & & &\\\\\n$R$ & $J^P$ & $f$ & $\\alpha$ & $\\beta$ & $\\gamma$ \\\\\n& & & & &\\\\\n\\hline\n& & & & &\\\\\n$N(940)$ & $\\frac{1}{2}^+$ & 7.7 & $\\frac{1}{2}\\alpha_{1\/2+}$ & $\\frac{1}{2}\\beta_{1\/2}$ &\n$\\frac{1}{2}\\gamma_{1\/2}$ \\\\\n& & & & &\\\\\n$N^*(1520)$ & $\\frac{3}{2}^-$ & 7.0 & $\\alpha_{3\/2-}$ & $\\beta_{3\/2}$ & $\\gamma_{3\/2}$ \\\\\n& & & & &\\\\\n$N^*(1650)$ & $\\frac{1}{2}^-$ & 0.9 & $\\alpha_{1\/2-}$ & $\\beta_{1\/2}$ & $\\gamma_{1\/2}$ \\\\\n& & & & &\\\\\n$N^*(1720)$ & $\\frac{3}{2}^+$ & 7.0 & $\\alpha_{3\/2+}$ & $\\beta_{3\/2}$ & $\\gamma_{3\/2}$ \\\\\n& & & & &\\\\\n$\\Delta(1232)$ & $\\frac{3}{2}^+$ & 10.5 & $\\alpha_{3\/2+}$ & $\\beta_{3\/2}$ & $\\gamma_{3\/2}$ \\\\\n& & & & &\\\\\n$\\Delta(1620)$ & $\\frac{1}{2}^-$ & 2.7 & $\\alpha_{1\/2-}$ & $\\beta_{1\/2}$ & $\\gamma_{1\/2}$ \\\\\n& & & & &\\\\\n$\\Delta(1700)$ & $\\frac{3}{2}^-$ & 5.0 & $\\alpha_{3\/2-}$ & $\\beta_{3\/2}$ & $\\gamma_{3\/2}$ \\\\\n& & & & &\\\\\n\\hline\n\\end{tabular}\n\\caption{Table showing the coefficients $\\alpha$, $\\beta$ and $\\gamma$ for loops \ncontaining the  various resonances considered.}\n\\end{center}\n\\end{table}\n\nThe corresponding expressions of $\\alpha,\\beta$ and $\\gamma$ for the second \ngraph of Fig.~1 can be obtained by replacing $q\\to-q$ as indicated in \n(\\ref{diag1a-b}).\nThe factor 1\/2 in front of the coefficients for the $NN$ loop is put\nto prevent a double counting of this contribution. \nThe coupling constants $f$ have been\nobtained as in~\\cite{Peters,Post2}. The values are also in reasonable agreement\nas seen in Table~1.\n\n\nA considerable simplification in the imaginary part\ncan be achieved in the sum of the two diagrams\nin Fig.~1 by a change of variables. \nNote that only the factor $p\\cdot q$ which appears with various powers in the tensors $A^{\\mu\\nu},\\,\nB^{\\mu\\nu}$ and $C^{\\mu\\nu}$ and in the factors $\\alpha$\ntakes on different values in the various terms in the expression for the imaginary part of \nself energy. We recall for convenience the imaginary part coming from \nfirst diagram in Fig.~1,\n\\begin{eqnarray}\n&& {\\rm Im} \\overline\\Pi^{(1)}(q_0,\\vec q)=-\\pi \\int\\frac{d^3\\vec p}{(2\\pi)^3 4\\omega_N\\omega_R}\\times  \\nonumber\\\\\n&&[L_1(q)\\{(1-n^N_+-n^R_-)\\delta(q_0-\\omega_N-\\omega_R)\n+(n_+^N-n_+^R)\\delta(q_0-\\omega_N+\\omega_R)\\}\\nonumber\\\\\n&& + L_2(q)\\{(n_-^R-n_-^N)\\delta(q_0+\\omega_N-\\omega_R)\n-(1-n_-^N-n_+^R)\\delta(q_0+\\omega_N+\\omega_R)\\}]\n\\label{impi1}\n\\end{eqnarray} \nwhere we have dropped the Lorentz indices for brevity. For the first two \nterms $p_0=\\omega_N$ and on integration over the angle\nusing either of the two delta functions one has \n$\\vp\\cdot\\vq=-\\frac{1}{2}(S^2-2q_0\\omega_N)$ with $S^2=q^2-m^2_R+m^2_N$. One thus gets\n$p\\cdot q=p_0q_0-\\vp\\cdot\\vq=\\frac{S^2}{2}$. Changing variable to $x$ \nusing $\\omega_N=\\frac{S^2}{2q^2}(q_0+|\\vec q|x) $ one has\n\\begin{eqnarray}\n&& A_\\mu^\\mu =-3,~~~~ A_{00}=\\frac{|\\vec q|^2}{q^2}\\\\\n&& B_\\mu^\\mu =m_\\pi^2 q^2 +\\frac{S^4}{2},~~~~ B_{00}=-\\frac{|\\vec q|^2S^4}{4q^2}(1-x^2)\\\\\n&&  C_\\mu^\\mu =q^2(m_\\pi^2 q^2 -\\frac{S^4}{4}),~~~~\nC_{00}=\\frac{|\\vec q|^2S^4}{4}x^2~.\n\\label{abc00}\n\\end{eqnarray}\nFor the last two terms, $\\vp\\cdot\\vq=-\\frac{1}{2}(S^2+2q_0\\omega_N)$ and \n$p_0=-\\omega_N$ so that here too $p\\cdot q=\\frac{S^2}{2} $.\nDefining $x$ in this case through $\\omega_N=\n\\frac{S^2}{2q^2}(-q_0+|\\vec q|x)$ one recovers the\nsame expression for the tensor component as in the first two terms.\n\n\nThe corresponding expression for the imaginary part from the second graph of Fig.~1 \nis obtained by changing $q\\rightarrow -q$ in the expression for \n${\\rm Im}\\Pi$  given by (\\ref{impi1}) getting\n\\begin{eqnarray}\n&& {\\rm Im} \\overline\\Pi^{(2)}(q_0,\\vec q)=-\\pi \\int\\frac{d^3\\vec p}{(2\\pi)^3 4\\omega_N\\omega_{R'}}\\times  \\nonumber\\\\\n&&[L_2(-q)\\{(1-n^N_--n^{R'}_+)\\delta(q_0-\\omega_N-\\omega_{R'})\n+(n_-^N-n_-^{R'})\\delta(q_0-\\omega_N+\\omega_{R'})\\}\\nonumber\\\\\n&& + L_1(-q)\\{(n_+^{R'}-n_+^N)\\delta(q_0+\\omega_N-\\omega_{R'})\n-(1-n_+^N-n_-^{R'})\\delta(q_0+\\omega_N+\\omega_{R'})\\}]\n\\end{eqnarray} \nwhere $\\omega_{R'}=\\sqrt{(\\vec p+\\vec q)^2+m_{R'}^2}$.\nIn this case, for the first two terms $\\vp\\cdot\\vq=\\frac{1}{2}(S^2-2q_0\\omega_N)$ \nwhich along with \n$p_0=-\\omega_N$ gives $p\\cdot q=-\\frac{S^2}{2} $.\nThe same is obtained for the last two terms for which \n$\\vp\\cdot\\vq=\\frac{1}{2}(S^2+2q_0\\omega_N)$ and $p_0=\\omega_N$. \nMaking a change of variables\nas before, identical values of the tensor components are obtained as in\n(\\ref{abc00}) \nas a consequence of the fact that the gauge invariant tensors\n$A_{{\\mu\\nu}},\\, B_{{\\mu\\nu}}$ and $ C_{{\\mu\\nu}}$ are even under $q\\rightarrow -q$.\nThe coefficients $\\alpha,\\, \\beta$ and $\\gamma$ also remain unchanged in the two\ndiagrams \nthe sign of $p\\cdot q$ in the expressions remaining the same under the \ncombined effect of $q\\rightarrow -q$ and a reversal in its magnitude\n($\\frac{S^2}{2}\\rightarrow -\\frac{S^2}{2}$). The value of $L(x)$ thus comes out \nto be the same for all cases and we end up with the final expression \ngiven by (\\ref{piB_final}).\n\n\n\\subsection{Expressions for $N_{\\mu\\nu}$ for mesonic loops}\n\nThe expressions for $N_{\\mu\\nu}$ appearing in the $\\rho$ self-energy (\\ref{pi_meson}) \nfor $\\pi-h$ loops have been obtained in~\\cite{Ghosh1} and are given below,\n\\begin{eqnarray}\nN_{{\\mu\\nu}}^{(\\pi)}(q,k)&=&\\left(\\frac{2G_\\rho}{m_\\rho F_\\pi^2}\\right)^2\nC_{{\\mu\\nu}}\\nonumber\\\\\nN_{{\\mu\\nu}}^{(\\omega)}(q,k)&=&-4\\left(\\frac{g_1}{F_\\pi}\\right)^2(B_{{\\mu\\nu}}+q^2k^2\nA_{{\\mu\\nu}})\\nonumber\\\\\nN_{{\\mu\\nu}}^{(h_1)}(q,k)&=&-\\left(\\frac{g_2}{F_\\pi}\\right)^2(B_{{\\mu\\nu}}-\\frac{1}{m_{h_1}^2}\nC_{{\\mu\\nu}})\\nonumber\\\\\nN_{{\\mu\\nu}}^{(a_1)}(q,k)&=&-2\\left(\\frac{g_3}{F_\\pi}\\right)^2(B_{{\\mu\\nu}}-\\frac{1}{m_{a_1}^2}\nC_{{\\mu\\nu}})\n\\end{eqnarray}\nwhere the constants $G_\\rho=69$ MeV, $F_\\pi=93$ MeV, $g_1=0.87,\\, g_2=1.0$ and $g_3=1.1$. \nThese can be simplified as shown above and finally expressed in terms of\nthe tensor components defined in (\\ref{abc00}). As in the case of baryon loops\nthe factors $N_{{\\mu\\nu}}$ have also been multiplied by the square of\nthe monopole form factor $F(q)$ defined earlier.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \n\nThe study of complex manifolds admitting Hermitian metrics with special properties is currently one of the central topics in complex geometry. While the development of K\\\"ahler geometry is the premier example, the literature abounds in K\\\"ahler-like metric properties of complex manifolds for which a rich theory is being developed -- see, for example, [AB95], [FPS04], [Gau77a], [Gau77b], [MP21], [Pop13] and the references therein. Arguably, the two most heavily studied ones are the {\\bf SKT} and the {\\bf balanced} metrics\/manifolds. (We refer to the next section for the relevant definitions.) \n\nIn this paper we study these two classes of complex manifolds and, partially inspired by [DP20], we take up the question of characterising K\\\"ahlerianity within them. Indeed, it is well known that there are many compact complex manifolds which admit SKT but no K\\\"ahler metrics. For example, any compact complex surface is SKT, whereas K\\\"ahlerianity of a surface is a topological property, equivalent to the first Betti number being even. Meanwhile, starting from dimension 3, there are numerous examples of non-K\\\"ahler balanced manifolds. Nevertheless, the question of whether a compact complex manifold that is both SKT and balanced must be K\\\"ahler (see [FV15]) is still open.\n \nOur approach is variational: we propose two new functionals defined on the spaces of SKT, respectively balanced, metrics and prove that their critical points are the K\\\"ahler metrics, if any. As these functionals are non-negative by definition, a family of SKT metrics (respectively a family of balanced metrics) asymptotically minimising the corresponding functional {\\bf either} converges to a K\\\"ahler metric {\\bf or} degenerates. After a suitable normalisation of the metrics, this should produce in the limit an SKT {\\bf $(1,1)$-current}, resp. a balanced {\\bf $(n-1,n-1)$-current}. Such a current ought to constitute an {\\bf obstruction} to the existence of K\\\"ahler metrics. We expect that a suitable pairing of two such currents, one SKT, the other balanced, on an SKT balanced manifold will help with the efforts of settling the K\\\"ahlerianity conjecture for the SKT balanced manifolds.\n\nThe main idea behind the functionals is as follows: in the SKT case the pluriclosedness of the metric $\\omega$ can be rewritten as $\\partial\\omega\\in ker d$ i.e. $\\partial\\omega$ is {\\bf orthogonal} to $Im\\ d^\\star$. Exploiting the three-space decomposition at the level of complex-valued $3$-forms, this means that $\\partial\\omega$ can be decomposed into a harmonic part and a $d$-exact part. The harmonic subspace being always finite-dimensional (by standard elliptic theory on a compact manifold), we consider the orthogonal projection onto the image of $d$ and, generalising a construction from [DP20], we define a {\\bf torsion form} $\\rho_\\omega$ as the minimal $L^2$-norm primitive (for the operator $d$) of the projection of $\\partial\\omega$. The functional $F$ is then defined as the squared $L^2$-norm of this torsion form. The major difference from [DP20], where we dealt with Hermitian-symplectic metrics, is that $\\rho_\\omega$ is now merely a (complex-valued) $2$-form and its decomposition into pure types plays a major role in the computations. \n\nSimilarly, in the balanced case we exploit the three-space decomposition associated with the $\\bar\\partial$-Laplacian on $(n-1,n-1)$-forms and the orthogonal projection onto $Im\\ \\bar\\partial$ and again we define a suitable torsion form which is now of pure type $(n-1,n-2)$.\n\nIn this paper we deal with non-normalised functionals. Our main results consist in the computations of the critical point equations and the variation of our functionals in the scaling direction. To make this possible, we first compute the first variation of various differential geometric operators that are frequently used in this context, such as the Hodge star operator $\\star_\\omega$, the adjoint $\\Lambda_\\omega$ of the Lefschetz operator $L_\\omega$ of multiplication by a metric $\\omega$ and the orthogonal projectors onto the kernels of the Laplacians $\\Delta'$, $\\Delta''$, $\\Delta$. We hope that these preliminary computations are of sufficient generality to be of independent interest. \n\nAs our functionals are defined on open cones, it is natural to expect that if minimisers thereof exist, the variation in the scaling direction has to vanish and the normalised functional has to be constant along the rays. This is confirmed by our computations which, moreover, show that vanishing of any of the functionals yields a K\\\"ahler metric.\n\nThe paper is organised as follows: in Section \\ref{section:two-functionals}, after fixing the notation and recalling a few basic definitions, we define the functional $F$ acting on the space of SKT metrics on a fixed compact complex manifold and then the functional $G$ acting on $(n-1)$-st wedge powers of balanced metrics. Section \\ref{section:1st-variation_operators} is devoted to the computations of the first variations of various operators that are involved in the definitions of $F$ and $G$. The next two sections contain the computations of the critical points of both functionals (Section \\ref{section:critical-points-computation_functional_F} for $F$ and Section \\ref{section:critical-points-computation_functional_G} for $G$) and the variations in the scaling direction. We utilise here the computations of Section \\ref{section:1st-variation_operators} together with a trick that we already used in [DP20]. In the last section \\ref{section:simultaneous_SKT-bal}, we briefly discuss a simultaneous approach to the variations of SKT and balanced metrics, when both are supposed to exist, on a compact complex manifold by proposing a third functional $H$ depending on two arguments. The starting point is an observation that generalises the quick proof given in [Pop15] to the fact that if a same Hermitian metric is both SKT and balanced then it must be K\\\"ahler.\n\n\\vspace{2ex}\n\n\\noindent {\\it Acknowledgements}.  The first named author is partially supported by grant no.\n2021\/41\/B\/ST1\/01632 from the National Science Center, Poland.\n\n\\section{Two functionals}\\label{section:two-functionals}  \n\nThroughout the paper $X$ will be a fixed compact complex manifold with $\\mbox{dim}_\\C X=n$. We tacitly assume $n\\geq 2$, although some of the computations also make sense for compact Riemann surfaces. As is customary, we identify Hermitian metrics on $X$ with the corresponding $C^\\infty$ positive definite $(1,1)$-forms. For any $(1,\\,1)$-form $\\gamma$ on $X$ and any $p\\in\\{1,\\dots , n\\}$, we use the notation $\\gamma_p:=\\gamma^p\/p!$. \n\nRecall that a metric $\\omega$ is said to be {\\bf SKT} if $\\partial\\bar\\partial\\omega=0$. A manifold admitting such a metric is called an SKT manifold. These metrics are also known as {\\it pluriclosed} in the literature (see [ST10]). A metric $\\omega$ is said to be {\\bf balanced} if $d(\\omega_{n-1})=0$ (see [Gau77b] for the notion, [Mic83] for the name) and again any manifold admitting such metrics is called a balanced manifold. Both notions impose topological and analytical restrictions on the underlying manifold -- see [Pop15] for more details. It is by now a standard fact (see [IP12] for a proof or [Pop15] for a short proof) that a metric which is both SKT and balanced is K\\\"ahler. \n\nWe will study conditions under which  these two special types of Hermitian metrics have the stronger K\\\"ahler property. We introduce a functional in each of these two cases and prove that its critical points are precisely the K\\\"ahler metrics on $X$.\n\n\\subsection{The SKT functional}\\label{subsection:functional_skt-Delta}\n\nSuppose there exists an {\\bf SKT metric} $\\omega$ on $X$, namely a $C^\\infty$ positive definite $(1,\\,1)$-form $\\omega$ on $X$ such that $\\partial\\bar\\partial\\omega=0$. The last condition is equivalent to $\\partial\\omega\\in\\ker d$, hence also to $$\\star(\\partial\\omega)\\in\\ker d^\\star,$$ where $\\star=\\star_\\omega$ is the Hodge star operator induced by $\\omega$ and $d^\\star = d^\\star_\\omega$ is the adjoint of $d$ w.r.t. the $L^2$-inner product $\\langle\\langle\\,\\,,\\,\\,\\rangle\\rangle_\\omega$ defined by $\\omega$. We used the facts that $d^\\star = -\\star d\\star$, $\\star$ is an isomorphism and $\\star\\star = -\\mbox{Id}$ on odd-degreed forms. (Note that $\\partial\\omega$ is of degree $3$.)\n\nNow, let $P^{(3)}_\\omega:C^\\infty_3(X,\\,\\C)\\longrightarrow\\mbox{Im}\\,d$ be the orthogonal projection (w.r.t. the $L^2$-inner product defined by $\\omega$) onto the subspace $\\mbox{Im}\\,d$ of smooth $3$-forms, induced by the classical orthogonal $3$-space decomposition: \\begin{equation}\\label{eqn:3-space-decomp_d-3}C^\\infty_3(X,\\,\\C) = {\\cal H}^3_\\Delta(X,\\,\\C)\\oplus\\mbox{Im}\\,d\\oplus\\mbox{Im}\\,d^\\star\\end{equation} in which ${\\cal H}^3_\\Delta(X,\\,\\C)$ is the kernel of the $d$-Laplacian $\\Delta=\\Delta_\\omega=dd^\\star_\\omega + d^\\star_\\omega d:C^\\infty_3(X,\\,\\C)\\longrightarrow C^\\infty_3(X,\\,\\C)$ induced by $\\omega$, $\\ker d = {\\cal H}^3_\\Delta(X,\\,\\C)\\oplus\\mbox{Im}\\,d$ and $\\ker d^\\star = {\\cal H}^3_\\Delta(X,\\,\\C)\\oplus\\mbox{Im}\\,d^\\star$.\n\nSince $\\partial\\omega\\in\\ker d$, the projection of $\\partial\\omega\\in C^\\infty_3(X,\\,\\C)$ onto $ \\mbox{Im}\\,d^\\star$ vanishes.\n\n\\begin{Def}\\label{Def:torsion-form} Given an {\\bf SKT} metric $\\omega$ on a compact complex manifold $X$, the unique smooth $2$-form $\\rho=\\rho_\\omega\\in C^\\infty_2(X,\\,\\C)$ that solves the system of equations: \\begin{eqnarray}\\label{eqn:system_torsion-form}(a)\\hspace{1ex} d\\rho_\\omega = P^{(3)}_\\omega(\\partial\\omega) \\hspace{3ex} \\mbox{and} \\hspace{3ex} (b)\\hspace{1ex}\\rho_\\omega\\in\\mbox{Im}\\,d^\\star_\\omega\\end{eqnarray} is called the {\\bf torsion form} of $\\omega$.\n\n\\end{Def}  \n\nNote that equation (a) of (\\ref{eqn:system_torsion-form}) is solvable since $P^{(3)}_\\omega(\\partial\\omega)\\in\\mbox{Im}\\,d$. However, the solution of equation (a) is unique only up to $\\ker d$, so the extra condition (b) makes it unique in the absolute sense since $\\mbox{Im}\\,d^\\star$ is the orthogonal complement w.r.t. $\\langle\\langle\\,\\,,\\,\\,\\rangle\\rangle_\\omega$ of $\\ker d$ inside the space $C^\\infty_2(X,\\,\\C)$. For the same reason, the solution $\\rho_\\omega$ of system (\\ref{eqn:system_torsion-form}) is the {\\it minimal $L^2_\\omega$-norm solution} of equation (a). It is given by the classical {\\it Neumann formula}: \\begin{eqnarray}\\label{eqn:Neumann_system_torsion-form}\\rho_\\omega = \\Delta_\\omega^{-1}d^\\star_\\omega(P^{(3)}_\\omega(\\partial\\omega)),\\end{eqnarray} where $\\Delta_\\omega^{-1}$ is the {\\it Green operator} of the (elliptic) $d$-Laplacian $\\Delta_\\omega=dd^\\star_\\omega + d^\\star_\\omega d:C^\\infty_2(X,\\,\\C)\\longrightarrow C^\\infty_2(X,\\,\\C)$ induced by $\\omega$.\n\n\n\\begin{Lem}\\label{Lem:Kaehler_torsion-vanishing} Let $\\omega$ be an {\\bf SKT} metric on a compact complex manifold $X$. The following equivalences hold: \\begin{eqnarray*}\\label{eqn:Kaehler_torsion-vanishing}(i)\\hspace{1ex} \\omega \\hspace{1ex}\\mbox{is K\\\"ahler}\\hspace{1ex} \\iff \\hspace{1ex} (ii)\\hspace{1ex}\\rho_\\omega =0 \\hspace{1ex} \\iff \\hspace{1ex} (iii) \\hspace{1ex} P^{(3)}_\\omega(\\partial\\omega)=0.\\end{eqnarray*}\n\n\\end{Lem}  \n\n\\noindent {\\it Proof.} We will prove the implications: $(i)\\implies (ii) \\implies (iii) \\implies (i)$.\n\n\\hspace{1ex}\n\n$(i)\\implies (ii).$ If $\\omega$ is K\\\"ahler, $\\partial\\omega=0$, hence $P^{(3)}_\\omega(\\partial\\omega)=0$, so the minimal $L^2$-norm solution of equation $d\\rho_\\omega = 0$ is $\\rho_\\omega=0$. \n\n\\hspace{1ex}\n\n$(ii)\\implies (iii).$ If $\\rho_\\omega=0$, then $P^{(3)}_\\omega(\\partial\\omega)=d\\rho_\\omega=0$.\n\n\\hspace{1ex}\n\n$(iii)\\implies (i).$ Suppose that $P^{(3)}_\\omega(\\partial\\omega)=0$. Since $\\partial\\omega\\in\\ker d = {\\cal H}_\\Delta(X,\\,\\C)\\oplus\\mbox{Im}\\,d$, this is equivalent to $\\partial\\omega\\in{\\cal H}_\\Delta(X,\\,\\C)$.\n\nOn the other hand, for any pure-type form $\\alpha$ we have: \\begin{eqnarray}\\label{eqn:Laplacians_pure-type}\\langle\\langle\\Delta\\alpha,\\,\\alpha\\rangle\\rangle = \\langle\\langle\\Delta'\\alpha,\\,\\alpha\\rangle\\rangle + \\langle\\langle\\Delta''\\alpha,\\,\\alpha\\rangle\\rangle\\end{eqnarray} Indeed, we have: \\begin{eqnarray*}\\langle\\langle\\Delta\\alpha,\\,\\alpha\\rangle\\rangle = ||d\\alpha||^2 + ||d^\\star\\alpha||^2 \\stackrel{(a)}{=} ||\\partial\\alpha||^2 + ||\\partial^\\star\\alpha||^2 + ||\\bar\\partial\\alpha||^2 + ||\\bar\\partial^\\star\\alpha||^2 = \\langle\\langle\\Delta'\\alpha,\\,\\alpha\\rangle\\rangle + \\langle\\langle\\Delta''\\alpha,\\,\\alpha\\rangle\\rangle,\\end{eqnarray*} where the crucial equality (a) follows from $\\partial\\alpha$ and $\\bar\\partial\\alpha$ being of {\\it different} pure types, hence orthogonal to each other, hence $||d\\alpha||^2 = ||\\partial\\alpha + \\bar\\partial\\alpha||^2 = ||\\partial\\alpha||^2 + ||\\bar\\partial\\alpha||^2$, while $\\partial^\\star\\alpha$ and $\\bar\\partial^\\star\\alpha$ too are of {\\it different} pure types, hence orthogonal to each other, hence $||d^\\star\\alpha||^2 = ||\\partial^\\star\\alpha + \\bar\\partial^\\star\\alpha||^2 = ||\\partial^\\star\\alpha||^2 + ||\\bar\\partial^\\star\\alpha||^2$.\n\n  Now, taking $\\alpha=\\partial\\omega$ (which is of pure type $(2,\\,1)$) in (\\ref{eqn:Laplacians_pure-type}), the property $\\Delta(\\partial\\omega)=0$ implies that $\\Delta'(\\partial\\omega)=0$ and $\\Delta''(\\partial\\omega)=0$. In particular, we get $$\\partial\\omega\\in\\ker\\Delta'\\cap\\mbox{Im}\\,\\partial =\\{0\\},$$ where the last equality of spaces follows from $\\ker\\Delta'\\perp\\mbox{Im}\\,\\partial$.\n\nThus, $\\partial\\omega=0$, which means that $\\omega$ is K\\\"ahler.  \\hfill $\\Box$\n\n\\vspace{2ex}\n\n\nNow, we consider the set \\begin{equation*}\\label{eqn:def-Somega}{\\cal S}:=\\bigg\\{\\omega\\,\\mid\\,\\omega\\in C^\\infty_{1,\\,1}(X,\\,\\R) \\hspace{1ex}\\mbox{such that}\\hspace{1ex} \\omega>0 \\hspace{1ex}\\mbox{and}\\hspace{1ex} \\partial\\bar\\partial\\omega= 0\\bigg\\}\\subset \\ker(\\partial\\bar\\partial)\\cap C^\\infty_{1,\\,1}(X,\\,\\R)\\end{equation*} of all the SKT metrics on $X$. The set ${\\cal S}$ is an {\\it open convex cone} in the real vector space $\\ker(\\partial\\bar\\partial)\\cap C^\\infty_{1,\\,1}(X,\\,\\R)$.\n\n\n\\begin{Def}\\label{Def:F_energy-functional_SKT} Let $X$ be a compact complex SKT manifold with $\\mbox{dim}_{\\C}X=n$. We define the following {\\bf energy functional}: \\begin{equation}\\label{eqn:F_energy-functional_SKT}F : {\\cal S} \\to [0,\\,+\\infty), \\hspace{3ex} F(\\omega) = \\int\\limits_X|\\rho_\\omega|^2_\\omega\\,dV_\\omega = ||\\rho_\\omega||^2_\\omega,  \\end{equation}\n\n\\noindent where $\\rho_{\\omega}$ is the torsion form of the SKT metric $\\omega\\in{\\cal S}$ defined in Definition \\ref{Def:torsion-form}, while $|\\,\\,\\,|_\\omega$ is the pointwise norm and $||\\,\\,\\,||_\\omega$ is the $L^2$-norm induced by $\\omega$.\n\n\n\\end{Def}\n\n\nThe first trivial observation that justifies the introduction of the functional $F$ is the following.\n\n\n\\begin{Lem}\\label{Lem:vanishing-F-Kaehler} Let $\\omega$ be an SKT metric on $X$. The following equivalence holds:\n\n\n\\begin{equation}\\label{eqn:vanishing-F-Kaehler}\\omega \\hspace{1ex} \\mbox{is K\\\"ahler} \\iff F(\\omega)=0.\\end{equation}\n\n\n\\end{Lem} \n\n\\noindent {\\it Proof.} The condition $F(\\omega)=0$ is equivalent to $\\rho_{\\omega}=0$. This, in turn, is equivalent to $\\omega$ being K\\\"ahler by Lemma \\ref{Lem:Kaehler_torsion-vanishing}. \\hfill $\\Box$\n\n\\vspace{3ex}\n\nThe next observation goes further after specifying the behaviour of $F(\\omega)$ under rescalings of $\\omega$. \n\n\\begin{Lem}\\label{Lem:rescaling_F-critical} Let $\\omega$ be an SKT metric on $X$. For every real $\\lambda>0$, the following equality holds: \\begin{equation}\\label{eqn:rescaling_F-critical}F(\\lambda\\,\\omega) = \\lambda^n\\,F(\\omega).\\end{equation}\n\nIn particular, \\begin{eqnarray}\\label{eqn:rescaling_F-critical_1+t}\\frac{d}{dt}_{|t=0}\\bigg(F((1+t)\\omega)\\bigg) = nF(\\omega)\\end{eqnarray} and the following equivalence holds:\n\n\n\\begin{equation}\\label{eqn:critical-points_F-Kaehler}\\omega \\hspace{1ex} \\mbox{is K\\\"ahler} \\iff \\omega \\hspace{1ex} \\mbox{is a critical point for} \\hspace{1ex} F.\\end{equation}\n\n\n\\end{Lem} \n\n\\noindent {\\it Proof.} The equivalence (\\ref{eqn:critical-points_F-Kaehler}) follows from Lemma \\ref{Lem:vanishing-F-Kaehler} and from (\\ref{eqn:rescaling_F-critical_1+t}) by considering the vanishing of $(d\/dt)_{|t=0}F(\\omega+t\\gamma)$ with the tangent direction provided by the choice $\\gamma=\\omega$. So, only (\\ref{eqn:rescaling_F-critical}) needs a proof.\n\nLemma 3.6. in [BP18] shows that, for any Hermitian metric $\\omega$ and any positive real $\\lambda$, one has $\\bar\\partial^\\star_{\\lambda\\omega} = (1\/\\lambda)\\,\\bar\\partial^\\star_\\omega$ and $\\Delta''_{\\lambda\\omega} = (1\/\\lambda)\\,\\Delta''_\\omega$. The same argument yields: \\begin{eqnarray}\\label{eqn:rescaling_d-star_Delta}d^\\star_{\\lambda\\omega} = \\frac{1}{\\lambda}\\,d^\\star_\\omega \\hspace{3ex} \\mbox{and} \\hspace{3ex} \\Delta_{\\lambda\\omega} = \\frac{1}{\\lambda}\\,\\Delta_\\omega\\end{eqnarray} on differential forms of any degree. In particular, $\\Delta_\\omega$ and $\\Delta_{\\lambda\\omega}$ have the same kernels, while $d^\\star_{\\lambda\\omega}$ and $d^\\star_\\omega$ have the same image.\n\nNow, let $P_{\\Delta,\\,\\omega}$, resp. $P_{\\Delta,\\,\\lambda\\,\\omega}$, be the $L^2_\\omega$-orthogonal projection, resp. the $L^2_{\\lambda\\omega}$-orthogonal projection, from the space of $C^\\infty$ $3$-forms onto the kernel of $\\Delta_\\omega$, resp. the kernel of $\\Delta_{\\lambda\\omega}$. The decomposition $\\partial\\omega = P_{\\Delta,\\,\\omega}(\\partial\\omega) + d\\rho_\\omega$, with $\\rho_\\omega\\in\\mbox{Im}\\,d^\\star_\\omega$, leads to $\\partial(\\lambda\\,\\omega) = \\lambda\\,P_{\\Delta,\\,\\omega}(\\partial\\omega) + d(\\lambda\\rho_\\omega)$ and, thanks to (\\ref{eqn:rescaling_d-star_Delta}), the latter is the decomposition of $\\partial(\\lambda\\,\\omega)\\in\\ker d$ according to the $L^2_{\\lambda\\omega}$-orthogonal splitting $\\ker d = \\ker\\Delta_{\\lambda\\omega}\\oplus\\mbox{Im}\\,d$. Moreover, $\\lambda\\rho_\\omega\\in\\mbox{Im}\\,d^\\star_\\omega = \\mbox{Im}\\,d^\\star_{\\lambda\\omega}$, so we get \\begin{eqnarray*}\\rho_{\\lambda\\omega} = \\lambda\\rho_\\omega.\\end{eqnarray*}\n\nFrom this, we infer: \\begin{eqnarray*}F(\\lambda\\omega) = \\int\\limits_X|\\rho_{\\lambda\\omega}|^2_{\\lambda\\omega}\\,dV_{\\lambda\\omega} = \\lambda^2\\int\\limits_X\\frac{1}{\\lambda^2}\\,|\\rho_\\omega|^2_\\omega\\,(\\lambda^n\\,dV_\\omega) = \\lambda^nF(\\omega), \\hspace{5ex} \\lambda>0,\\end{eqnarray*} which proves (\\ref{eqn:rescaling_F-critical}).  \\hfill $\\Box$\n\n\n\\vspace{2ex}\n\nIn view of further use, we note that the above proof also gives $P^{(3)}_{\\lambda\\omega}(\\partial(\\lambda\\omega)) = \\lambda\\,P^{(3)}_\\omega(\\partial\\omega)$, hence \\begin{eqnarray}\\label{eqn:P^3_del-omega_lambda}P^{(3)}_{\\lambda\\omega}(\\partial\\omega) = P^{(3)}_\\omega(\\partial\\omega),  \\hspace{5ex} \\lambda>0.\\end{eqnarray}\n\n \n\n\n\\subsection{The balanced functional}\\label{subsection:functional_balanced}\n\nSuppose there exists a balanced metric $\\omega$ on $X$, namely a $C^\\infty$ positive definite $(1,\\,1)$-form $\\omega$ such that $\\bar\\partial\\omega_{n-1}=0$. Since every $C^\\infty$ positive definite $(n-1,\\,n-1)$-form $\\Omega$ on $X$ admits a unique $(n-1)$-st root, namely a unique $C^\\infty$ positive definite $(1,\\,1)$-form $\\omega$ such that $\\omega_{n-1} = \\Omega$ (see [Mic83]), $\\omega$ and $\\omega_{n-1}$ determine each other. Thus, we can refer to either $\\omega$ or $\\omega_{n-1}$ as a balanced metric when $\\omega_{n-1}$ lies in $\\ker\\bar\\partial$.\n\nFor any balanced metric $\\omega$ on $X$, let  $\\Delta''=\\Delta''_\\omega=\\bar\\partial\\bar\\partial^\\star_\\omega + \\bar\\partial^\\star_\\omega\\bar\\partial:C^\\infty_{n-1,\\,n-1}(X,\\,\\C)\\longrightarrow C^\\infty_{n-1,\\,n-1}(X,\\,\\C)$ be the induced $\\bar\\partial$-Laplacian acting in bidegree $(n-1,\\,n-1)$ and let \\begin{equation*}C^\\infty_{n-1,\\,n-1}(X,\\,\\C) = \\ker\\Delta_\\omega''\\oplus\\mbox{Im}\\,\\bar\\partial\\oplus\\mbox{Im}\\,\\bar\\partial^\\star\\end{equation*} be the classical $L^2_\\omega$-orthogonal $3$-space decomposition. We denote by $P^{(n-1,\\,n-1)}_{\\omega_{n-1}}:C^\\infty_{n-1,\\,n-1}(X,\\,\\C)\\longrightarrow\\mbox{Im}\\,\\bar\\partial$ the $L^2_\\omega$-orthogonal projection onto the image of $\\bar\\partial$. It is standard (for any Hermitian metric $\\omega$) that $\\ker\\bar\\partial = \\ker\\Delta_\\omega''\\oplus\\mbox{Im}\\,\\bar\\partial$, so if $\\omega$ is balanced, $\\omega_{n-1}\\in\\ker\\bar\\partial$ and therefore $\\omega_{n-1}$ splits uniquely as an $L^2_\\omega$-orthogonal sum of a $\\Delta_\\omega''$-harmonic form and $P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})$.\n\n\\begin{Def}\\label{Def:balanced_torsion-form} Given a {\\bf balanced} metric $\\omega$ on a compact complex $n$-dimensional manifold $X$, the unique smooth $(n-1,\\,n-2)$-form $\\Gamma=\\Gamma_{\\omega_{n-1}}\\in C^\\infty_{n-1,\\,n-2}(X,\\,\\C)$ that solves the system of equations: \\begin{eqnarray}\\label{eqn:system_balanced_torsion-form}(a)\\hspace{1ex} \\bar\\partial\\Gamma_{\\omega_{n-1}} = P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1}) \\hspace{3ex} \\mbox{and} \\hspace{3ex} (b)\\hspace{1ex}\\Gamma_{\\omega_{n-1}}\\in\\mbox{Im}\\,\\bar\\partial^\\star_\\omega\\end{eqnarray} is called the {\\bf torsion $(n-1,\\,n-2)$-form} of $\\omega_{n-1}$.\n\n\\end{Def}  \n\n\nNote that equation (a) of (\\ref{eqn:system_balanced_torsion-form}) is solvable since $P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})\\in\\mbox{Im}\\,\\bar\\partial$. However, the solution of equation (a) is unique only up to $\\ker\\bar\\partial$, so the extra condition (b) makes it unique in the absolute sense since $\\mbox{Im}\\,\\bar\\partial^\\star_\\omega$ is the orthogonal complement of $\\ker\\bar\\partial$ w.r.t. the $L^2_\\omega$-inner product $\\langle\\langle\\,\\,,\\,\\,\\rangle\\rangle_\\omega$ inside the space $C^\\infty_{n-1,\\,n-1}(X,\\,\\C)$. For the same reason, the solution $\\Gamma_{\\omega_{n-1}}$ of system (\\ref{eqn:system_balanced_torsion-form}) is the {\\it minimal $L^2_\\omega$-norm solution} of equation (a). It is given by the classical {\\it Neumann formula}: \\begin{eqnarray}\\label{eqn:Neumann_system_balanced_torsion-form}\\Gamma_{\\omega_{n-1}} = \\Delta_\\omega^{''-1}\\bar\\partial^\\star_\\omega(P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})),\\end{eqnarray} where $\\Delta_\\omega^{''-1}$ is the {\\it Green operator} of the (elliptic) $\\bar\\partial$-Laplacian $\\Delta_\\omega''$.\n\n\n\\begin{Lem}\\label{Lem:Kaehler_balanced_torsion-vanishing} Let $\\omega$ be a {\\bf balanced} metric on a compact complex $n$-dimensional manifold $X$. The following equivalences hold: \\begin{eqnarray*}\\label{eqn:Kaehler_balanced_torsion-vanishing}(i)\\hspace{1ex} \\omega \\hspace{1ex}\\mbox{is K\\\"ahler}\\hspace{1ex} \\iff \\hspace{1ex} (ii)\\hspace{1ex} P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})=0 \\hspace{1ex} \\iff \\hspace{1ex} (iii) \\hspace{1ex} \\Gamma_{\\omega_{n-1}}=0.\\end{eqnarray*}\n\n\n\\end{Lem}  \n\n\\noindent {\\it Proof.} We will prove the implications: $(i)\\implies (ii) \\implies (iii) \\implies (i)$.\n\n\\hspace{1ex}\n\n$(i)\\implies (ii).$ If $\\omega$ is K\\\"ahler, $\\omega_{n-1}\\in\\ker\\Delta_\\omega''$, hence $P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})=0$. Indeed, since $\\ker\\Delta_\\omega'' = \\ker\\bar\\partial\\cap\\ker\\bar\\partial^\\star$, to see that $\\omega_{n-1}$ lies in $\\ker\\Delta_\\omega''$, we notice that:\n\n\\vspace{1ex}\n\n$\\bar\\partial\\omega_{n-1} = \\omega_{n-2}\\wedge\\bar\\partial\\omega = 0$ since $\\bar\\partial\\omega = 0$ by the K\\\"ahler assumption on $\\omega$;\n\n\\vspace{1ex}\n\n$\\bar\\partial^\\star\\omega_{n-1} = -\\star\\partial(\\star\\omega_{n-1}) = -\\star\\partial\\omega = 0$ since $\\partial\\omega = 0$ by the K\\\"ahler assumption on $\\omega$.\n\n\n\\hspace{1ex}\n\n$(ii)\\implies (iii).$ If $P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})=0$, the zero form is a solution of equation (a) of (\\ref{eqn:system_balanced_torsion-form}). Then, the zero form is necessarily the minimal $L^2_\\omega$-norm solution $\\Gamma_{\\omega_{n-1}}$ of that equation, hence $\\Gamma_{\\omega_{n-1}}=0$.\n\n\\hspace{1ex}\n\n$(iii)\\implies (i).$ Suppose that $\\Gamma_{\\omega_{n-1}}=0$. Then $\\bar\\partial\\Gamma_{\\omega_{n-1}}=0$, hence $P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})=0$. Since $\\omega_{n-1}\\in\\ker\\bar\\partial = \\ker\\Delta_\\omega''\\oplus\\mbox{Im}\\,\\bar\\partial$, by the balanced hypothesis on $\\omega$, the property $P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})=0$ is equivalent to $\\omega_{n-1}\\in\\ker\\Delta_\\omega'' = \\ker\\bar\\partial\\cap\\ker\\bar\\partial^\\star$. In particular, $\\bar\\partial^\\star\\omega_{n-1}=0$, which means that $0=-\\star\\partial(\\star\\omega_{n-1}) = -\\star\\partial\\omega$. Since $\\star$ is an isomorphism, this amounts to $\\partial\\omega=0$, which means that $\\omega$ is K\\\"ahler.  \\hfill $\\Box$\n\n\n\\vspace{2ex}\n\n\nNow, we consider the set \\begin{equation*}\\label{eqn:def-Bomega}{\\cal B}:=\\bigg\\{\\omega_{n-1}\\,\\mid\\,\\omega\\in C^\\infty_{1,\\,1}(X,\\,\\R) \\hspace{1ex}\\mbox{such that}\\hspace{1ex} \\omega>0 \\hspace{1ex}\\mbox{and}\\hspace{1ex} \\bar\\partial\\omega_{n-1}= 0\\bigg\\}\\subset \\ker(\\bar\\partial)\\cap C^\\infty_{n-1,\\,n-1}(X,\\,\\R)\\end{equation*} of all the balanced metrics on $X$. The set ${\\cal B}$ is an {\\it open convex cone} in the real vector space $\\ker(\\bar\\partial)\\cap C^\\infty_{n-1,\\,n-1}(X,\\,\\R)$.\n\n\\begin{Def}\\label{Def:F_energy-functional_balanced} Let $X$ be a compact complex balanced manifold with $\\mbox{dim}_{\\C}X=n$. We define the following {\\bf energy functional}: \\begin{equation}\\label{eqn:G_energy-functional_balanced}G :{\\cal B} \\to [0,\\,+\\infty), \\hspace{3ex} G(\\omega_{n-1}) = \\int\\limits_X|\\Gamma_{\\omega_{n-1}}|^2_\\omega\\,dV_\\omega = ||\\Gamma_{\\omega_{n-1}}||^2_\\omega,  \\end{equation}\n\n\\noindent where $\\Gamma_{\\omega_{n-1}}$ is the torsion $(n-1,\\,n-2)$-form of the balanced metric $\\omega_{n-1}\\in{\\cal B}$ defined in Definition \\ref{Def:balanced_torsion-form}, while $|\\,\\,\\,|_\\omega$ is the pointwise norm and $||\\,\\,\\,||_\\omega$ is the $L^2$-norm induced by $\\omega$.\n\n\n\\end{Def}\n\n\nThe first trivial observation that justifies the introduction of the functional $G$ is the following.\n\n\n\\begin{Lem}\\label{Lem:vanishing-G-Kaehler} Let $\\omega$ be a balanced metric on $X$. The following equivalence holds:\n\n\n\\begin{equation}\\label{eqn:vanishing-G-Kaehler}\\omega \\hspace{1ex} \\mbox{is K\\\"ahler} \\iff G(\\omega_{n-1})=0.\\end{equation}\n\n\n\\end{Lem} \n\n\\noindent {\\it Proof.} The condition $G(\\omega_{n-1})=0$ is equivalent to $\\Gamma_{\\omega_{n-1}}=0$. This, in turn, is equivalent to $\\omega$ being K\\\"ahler by Lemma \\ref{Lem:Kaehler_balanced_torsion-vanishing}. \\hfill $\\Box$\n\n\n\n\\vspace{3ex}\n\nThe next observation is the analogue in the balanced case of Lemma \\ref{Lem:rescaling_F-critical}.\n\n\\begin{Lem}\\label{Lem:rescaling_G-critical} Let $\\omega$ be a balanced metric on $X$. For every real $\\lambda>0$, the following equality holds: \\begin{equation}\\label{eqn:rescaling_G-critical}G(\\lambda\\,\\omega_{n-1}) = \\lambda^{\\frac{n+1}{n-1}}\\,G(\\omega_{n-1}).\\end{equation}\n\nIn particular, \\begin{eqnarray}\\label{eqn:rescaling_G-critical_1+t}\\frac{d}{dt}_{|t=0}\\bigg(G((1+t)\\omega_{n-1})\\bigg) = \\frac{n+1}{n-1}\\,G(\\omega_{n-1})\\end{eqnarray} and the following equivalence holds:\n\n\n\\begin{equation}\\label{eqn:critical-points_G-Kaehler}\\omega \\hspace{1ex} \\mbox{is K\\\"ahler} \\iff \\omega_{n-1} \\hspace{1ex} \\mbox{is a critical point for} \\hspace{1ex} G.\\end{equation}\n\n\n\\end{Lem} \n\n\\noindent {\\it Proof.} The equivalence (\\ref{eqn:critical-points_G-Kaehler}) follows from Lemma \\ref{Lem:vanishing-G-Kaehler} and from (\\ref{eqn:rescaling_G-critical_1+t}) by considering the vanishing of $(d\/dt)_{|t=0}G(\\omega_{n-1}+t\\Omega)$ with the tangent direction provided by the choice $\\Omega=\\omega_{n-1}$. So, only (\\ref{eqn:rescaling_G-critical}) needs a proof.\n\nAs previously said, Lemma 3.6. in [BP18] shows that, for any Hermitian metric $\\omega$ and any positive real $\\lambda$, one has: \\begin{eqnarray}\\label{eqn:rescaling_d-bar-star_Delta''}\\bar\\partial^\\star_{\\lambda\\omega} = \\frac{1}{\\lambda}\\,\\bar\\partial^\\star_\\omega \\hspace{3ex} \\mbox{and} \\hspace{3ex} \\Delta''_{\\lambda\\omega} = \\frac{1}{\\lambda}\\,\\Delta''_\\omega\\end{eqnarray} on differential forms of any bidegree. In particular, $\\Delta''_\\omega$ and $\\Delta''_{\\lambda\\omega}$ have the same kernels, while $\\bar\\partial^\\star_{\\lambda\\omega}$ and $\\bar\\partial^\\star_\\omega$ have the same image.\n\nNow, let $P_{\\Delta'',\\,\\omega}$, resp. $P_{\\Delta'',\\,\\lambda\\,\\omega}$, be the $L^2_\\omega$-orthogonal projection, resp. the $L^2_{\\lambda\\omega}$-orthogonal projection, from the space of $C^\\infty$ $(n-1,\\,n-1)$-forms onto the kernel of $\\Delta''_\\omega$, resp. the kernel of $\\Delta''_{\\lambda\\omega}$. The decomposition $\\omega_{n-1} = P_{\\Delta'',\\,\\omega}(\\omega_{n-1}) + \\bar\\partial\\Gamma_{\\omega_{n-1}}$, with $\\Gamma_{\\omega_{n-1}}\\in\\mbox{Im}\\,\\bar\\partial^\\star_\\omega$, leads to $(\\lambda^{\\frac{1}{n-1}}\\,\\omega)_{n-1} = \\lambda\\,P_{\\Delta'',\\,\\omega}(\\omega_{n-1}) + \\bar\\partial(\\lambda\\Gamma_{\\omega_{n-1}})$ and, thanks to (\\ref{eqn:rescaling_d-bar-star_Delta''}), the latter is the decomposition of $\\lambda\\,\\omega_{n-1}\\in\\ker\\bar\\partial$ according to the $L^2_{\\lambda\\omega}$-orthogonal splitting $\\ker\\bar\\partial = \\ker\\Delta''_{\\lambda\\omega}\\oplus\\mbox{Im}\\,\\bar\\partial$. Moreover, $\\lambda\\Gamma_{\\omega_{n-1}}\\in\\mbox{Im}\\,\\bar\\partial^\\star_\\omega = \\mbox{Im}\\,\\bar\\partial^\\star_{\\lambda^{1\/(n-1)}\\omega}$, so we get \\begin{eqnarray*}\\Gamma_{\\lambda\\omega_{n-1}} = \\lambda\\Gamma_{\\omega_{n-1}}.\\end{eqnarray*}\n\nFrom this, we infer: \\begin{eqnarray*}G(\\lambda\\omega_{n-1}) = \\int\\limits_X|\\Gamma_{\\lambda\\omega_{n-1}}|^2_{\\lambda^{1\/(n-1)}\\omega}\\,dV_{\\lambda^{1\/(n-1)}\\omega} & = & \\lambda^2\\int\\limits_X\\bigg(\\frac{1}{\\lambda^{\\frac{1}{n-1}}}\\bigg)^{2n-3}\\,|\\Gamma_{\\omega_{n-1}}|^2_\\omega\\,(\\lambda^{\\frac{n}{n-1}}\\,dV_\\omega) \\\\\n & = & \\lambda^{\\frac{n+1}{n-1}}G(\\omega_{n-1}), \\hspace{5ex} \\lambda>0,\\end{eqnarray*} which proves (\\ref{eqn:rescaling_G-critical}).  \\hfill $\\Box$ \n\n \n\\vspace{2ex}\n\nIn view of further use, we note that the above proof also gives $P^{(n-1,\\,n-1)}_{\\lambda\\omega_{n-1}}(\\lambda\\omega_{n-1}) = \\lambda\\,P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})$, hence \\begin{eqnarray}\\label{eqn:P^n-1n-1_omega_n-1_lambda}P^{(n-1,\\,n-1)}_{\\lambda\\omega_{n-1}}(\\omega_{n-1}) = P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1}),  \\hspace{5ex} \\lambda>0.\\end{eqnarray}\n\n \n\n\n\n\n\\vspace{2ex}\n\n We will need to compute the critical points of the energy functionals $F$ and $G$. Before proceeding, we pause to compute the differentials of several of the classical operators involved in our construction that will come in handy later on and have an interest of their own.\n\n\n \\section{The first variation of various operators }\\label{section:1st-variation_operators} Let $X$ be a complex manifold with $\\mbox{dim}_\\C X=n$. We fix a Hermitian metric $\\omega$ on $X$. This means that $\\omega$ is a $C^\\infty$ positive definite $(1,\\,1)$-form on $X$. Meanwhile, let $\\gamma$ be a {\\it real} (in general signless) $C^\\infty$ $(1,\\,1)$-form on $X$. We will be considering reals $t$ sufficiently close to $0$ such that $\\omega + t\\gamma$ is positive definite.\n\n Fix a point $x\\in X$ and choose local holomorphic coordinates $z_1,\\dots , z_n$ about $x$ such that \\begin{equation}\\label{eqn:omega-gamma_diagonalisation}\\omega(x)=\\sum\\limits_{j=1}^n idz_j\\wedge d\\bar{z}_j  \\hspace{3ex} \\mbox{and} \\hspace{3ex} \\gamma(x)=\\sum\\limits_{j=1}^n \\gamma_j\\,idz_j\\wedge d\\bar{z}_j,\\end{equation} where $\\gamma_1,\\dots , \\gamma_n\\in\\R$ are the eigenvalues of $\\gamma$ w.r.t. $\\omega$ at $x$. Then, obviously, \\begin{equation}\\label{eqn:omega-plus-t-gamma_x}(\\omega + t\\gamma)(x)=\\sum\\limits_{j=1}^n (1+ t\\gamma_j)\\, idz_j\\wedge d\\bar{z}_j,  \\hspace{6ex} t\\in\\R.\\end{equation}\n\n\n \\subsection{Computation of the first variation of the trace of $(1,\\,1)$-forms}\\label{subsection:1st-variation_trace_11}\n\n In the setting described above, let $(\\alpha_t)_t$ be a $C^\\infty$ family of smooth $(1,\\,1)$-forms on $X$. In the local coordinates about a given point $x\\in X$ chosen above, we have \\begin{equation*}\\alpha_t =\\sum\\limits_{l,\\,r=1}^n\\alpha^{(t)}_{l\\bar{r}}\\,idz_l\\wedge d\\bar{z}_r,\\end{equation*} where the $\\alpha^{(t)}_{l\\bar{r}}$'s are $C^\\infty$ functions on a neighbourhood of $x$. For any Hermitian metric $\\rho$ on $X$, $\\Lambda_\\rho$ will denote the adjoint of the multiplication operator $\\rho\\wedge\\cdot$ w.r.t. the pointwise inner product $\\langle\\,\\,,\\,\\,\\rangle_\\rho$ defined by $\\rho$.\n\n\\begin{Lem}\\label{Lem:1st-variation_trace_11} For every $C^\\infty$ family $(\\alpha_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $\\alpha_t\\in C^\\infty_{1,\\,1}(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formula holds: \\begin{eqnarray}\\label{eqn:1st-variation_trace_11}\\frac{d}{dt}\\,(\\Lambda_{\\omega+t\\gamma}\\alpha_t) = \\Lambda_{\\omega+t\\gamma}\\bigg(\\frac{d\\alpha_t}{dt} \\bigg) -\\langle\\alpha_t,\\,\\gamma\\rangle_{\\omega+t\\gamma},  \\hspace{10ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray}\n     \n\\end{Lem}\n\n \\noindent {\\it Proof.} At an arbitrary point $x\\in X$ about which the local coordinates have been chosen such that (\\ref{eqn:omega-gamma_diagonalisation}) holds, (\\ref{eqn:omega-plus-t-gamma_x}) implies the following equality at $x$: \\begin{equation*}\\Lambda_{\\omega+t\\gamma}\\alpha_t = \\sum\\limits_{l=1}^n\\frac{\\alpha^{(t)}_{l\\bar{l}}}{1+t\\gamma_l},  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{equation*} Deriving with respect to $t$, we get the first equality below at $x$: \\begin{equation*}\\frac{d}{dt}\\,(\\Lambda_{\\omega+t\\gamma}\\alpha_t) = -\\sum\\limits_{l=1}^n\\frac{\\alpha^{(t)}_{l\\bar{l}}\\,\\gamma_l}{(1+t\\gamma_l)^2} + \\sum\\limits_{l=1}^n\\frac{\\frac{d\\alpha^{(t)}_{l\\bar{l}}}{dt}}{1+t\\gamma_l} = -\\langle\\alpha_t,\\,\\gamma\\rangle_{\\omega+t\\gamma} + \\Lambda_{\\omega+t\\gamma}\\bigg(\\frac{d\\alpha_t}{dt} \\bigg),  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon),\\end{equation*} where the second equality follows again from (\\ref{eqn:omega-plus-t-gamma_x}).  \\hfill $\\Box$\n\n\n \\subsection{Computation of the first variation of the Hodge star operator}\\label{subsection:1st-variation_Hodge-star}\n\n We will prove the following formula for the differential of the Hodge star operator $\\star_\\omega$ induced by the metric $\\omega$ in the direction of the given real $(1,\\,1)$-form $\\gamma$.\n\n\\begin{Lem}\\label{Lem:1st-variation_Hodge-star} For every bidegree $(p,\\,q)$ and every $C^\\infty$ family $(v_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $v_t\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formula holds: \\begin{eqnarray}\\label{eqn:1st-variation_Hodge-star}\\frac{d}{dt}\\bigg|_{t=0}(\\star_{\\omega+t\\gamma}v_t) = \\star_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg),\\end{eqnarray} where $\\gamma\\wedge\\cdot$ is the operator of multiplication by $\\gamma$ and $\\Lambda_\\omega$ is the adjoint of the multiplication operator $\\omega\\wedge\\cdot$ w.r.t. the pointwise inner product $\\langle\\,\\,,\\,\\,\\rangle_\\omega$ defined by $\\omega$. \n\n\n\\end{Lem}\n\n\\noindent {\\it Proof.} The formula to prove being pointwise, it suffices to prove it at a pregiven point $x$ with a choice of local coordinates such that (\\ref{eqn:omega-gamma_diagonalisation}) holds at $x$. \n\nFor every $t\\in(-\\varepsilon,\\,\\varepsilon)$, (\\ref{eqn:omega-plus-t-gamma_x}) yields the following equalities at $x$: \\begin{equation*}\\langle dz_j,\\,dz_k\\rangle_{\\omega+t\\gamma} = \\langle d\\bar{z}_j,\\,d\\bar{z}_k\\rangle_{\\omega+t\\gamma} = \\delta_{jk}\\,\\frac{1}{1+t\\gamma_j},  \\hspace{6ex} j,k.\\end{equation*}\n\nNow, let $u,v_t\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ be arbitrary forms. In a neighbourhood of $x$, they are of the shape \\begin{equation*}u = \\sum\\limits_{|I|=p \\atop |J|=q} u_{I\\bar{J}}\\,dz_I\\wedge d\\bar{z}_J  \\hspace{3ex} \\mbox{and} \\hspace{3ex} v_t = \\sum\\limits_{ |I|=p \\atop |J|=q} v_{I\\bar{J}}(t)\\,dz_I\\wedge d\\bar{z}_J, \\hspace{6ex} t\\in(-\\varepsilon,\\,\\varepsilon),\\end{equation*} where $I=(i_1<\\dots <i_p)$ and $J=(j_1<\\dots <j_q)$ are multi-indices of lengths $p$, resp. $q$.\n\nThe definition of the Hodge star operator $\\omega_{\\omega+t\\gamma}$ induced by the metric $\\omega+t\\gamma$ (when the real $t$ is so close to $0$ that $\\omega+t\\gamma>0$) gives the first equality below at every point, while the second equality holds at $x$ and follows from the above preparations: \\begin{eqnarray*}u\\wedge\\star_{\\omega+t\\gamma}\\bar{v_t} = \\langle u,\\,v_t\\rangle_{\\omega+t\\gamma}\\,\\frac{(\\omega+t\\gamma)^n}{n!} = \\sum\\limits_{ |I|=p \\atop |J|=q} u_{I\\bar{J}}\\,\\overline{v_{I\\bar{J}}(t)}\\,\\bigg(\\prod\\limits_{l\\in I}\\frac{1}{1+t\\gamma_l}\\bigg)\\,\\bigg(\\prod\\limits_{r\\in J}\\frac{1}{1+t\\gamma_r} \\bigg)\\,\\frac{(\\omega+t\\gamma)^n}{n!}.\\end{eqnarray*}\n\n\nDeriving this equality w.r.t. $t$, we get: \\begin{eqnarray*}u\\wedge\\frac{d}{dt}(\\star_{\\omega+t\\gamma}\\bar{v_t}) & = & \\sum\\limits_{ |I|=p \\atop |J|=q} u_{I\\bar{J}}\\,\\overline{\\frac{dv_{I\\bar{J}}(t)}{dt}}\\,\\bigg(\\prod\\limits_{l\\in I}\\frac{1}{1+t\\gamma_l}\\bigg)\\,\\bigg(\\prod\\limits_{r\\in J}\\frac{1}{1+t\\gamma_r} \\bigg)\\,\\frac{(\\omega+t\\gamma)^n}{n!} \\\\\n  & - & \\sum\\limits_{ |I|=p \\atop |J|=q} u_{I\\bar{J}}\\,\\overline{v_{I\\bar{J}}(t)}\\bigg(\\sum\\limits_{l\\in I}\\frac{\\gamma_l}{(1+t\\gamma_l)^2}\\prod\\limits_{s\\in I\\setminus\\{l\\}}\\frac{1}{1+t\\gamma_s}\\bigg)\\,\\bigg(\\prod\\limits_{r\\in J}\\frac{1}{1+t\\gamma_r} \\bigg)\\,\\frac{(\\omega+t\\gamma)^n}{n!}  \\\\\n  & - & \\sum\\limits_{|I|=p \\atop |J|=q} u_{I\\bar{J}}\\,\\overline{v_{I\\bar{J}}(t)}\\,\\bigg(\\prod\\limits_{l\\in I}\\frac{1}{1+t\\gamma_l}\\bigg)\\,\\bigg(\\sum\\limits_{r\\in J}\\frac{\\gamma_r}{(1+t\\gamma_r)^2}\\prod\\limits_{s\\in J\\setminus\\{r\\}}\\frac{1}{1+t\\gamma_s}\\bigg)\\,\\frac{(\\omega+t\\gamma)^n}{n!}  \\\\\n  & + & \\sum\\limits_{ |I|=p \\atop |J|=q} u_{I\\bar{J}}\\,\\overline{v_{I\\bar{J}}(t)}\\,\\bigg(\\prod\\limits_{l\\in I}\\frac{1}{1+t\\gamma_l}\\bigg)\\,\\bigg(\\prod\\limits_{r\\in J}\\frac{1}{1+t\\gamma_r} \\bigg)\\,\\frac{(\\omega+t\\gamma)^{n-1}}{(n-1)!}\\wedge\\gamma.\\end{eqnarray*} Taking $t=0$ and using the fact that  \\begin{eqnarray*}\\frac{\\omega^{n-1}}{(n-1)!}\\wedge\\gamma = (\\Lambda_\\omega\\gamma)\\,\\frac{\\omega^n}{n!} = (\\gamma_1+\\dots + \\gamma_n)\\,\\frac{\\omega^n}{n!},\\end{eqnarray*} with the last equality holding at $x$ only, we get the first equality below at $x$: \\begin{eqnarray*}u\\wedge\\frac{d}{dt}\\bigg|_{t=0}(\\star_{\\omega+t\\gamma}\\bar{v_t}) & = & \\bigg\\langle u,\\,\\frac{dv_t}{dt}\\bigg|_{t=0}\\bigg\\rangle_\\omega\\,\\frac{\\omega^n}{n!} + \\sum\\limits_{ |I|=p \\atop |J|=q} u_{I\\bar{J}}\\,\\overline{v_{I\\bar{J}}(0)}\\,\\bigg(\\sum\\limits_{j=1}^n\\gamma_j - \\sum\\limits_{l\\in I}\\gamma_l - \\sum\\limits_{r\\in J}\\gamma_r\\bigg)\\,\\frac{\\omega^n}{n!} \\\\\n  & = & \\bigg\\langle u,\\,\\frac{dv_t}{dt}\\bigg|_{t=0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg\\rangle_\\omega\\,\\frac{\\omega^n}{n!} = u\\wedge\\star_\\omega\\overline{\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg)},\\end{eqnarray*} where the second equality follows from the standard pointwise formula (\\ref{eqn:coordinates-standard-formula-anticom}) for $[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]$ (see e.g. [Dem97, VI, $\\S.5.2$, Proposition 5.8]) that we now reproduce.\n\nWhenever $\\omega$ is a Hermitian metric and $\\gamma$ is a real $(1,\\,1)$-form that are diagonalised simultaneously at a given point $x$ by a choice of local holomorphic coordinates $(z_1,\\dots , z_n)$ centred at $x$ such that (\\ref{eqn:omega-gamma_diagonalisation}) holds, for any bidegree $(p,\\,q)$ and any $(p,\\,q)$-form $v$ written in these local coordinates as $v=\\sum_{|I|=p,\\, |J|=q}v_{I\\bar{J}}\\,dz_I\\wedge d\\bar{z}_J$, the following equality holds at $x$: \\begin{eqnarray}\\label{eqn:coordinates-standard-formula-anticom} [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v = \\sum\\limits_{ |I|=p \\atop |J|=q} v_{I\\bar{J}}\\,\\bigg(\\sum\\limits_{j=1}^n\\gamma_j - \\sum\\limits_{l\\in I}\\gamma_l - \\sum\\limits_{r\\in J}\\gamma_r\\bigg)\\,dz_I\\wedge d\\bar{z}_J.\\end{eqnarray} In the special case where $\\gamma=\\omega$, we have $\\gamma_1=\\dots = \\gamma_n=1$ and this standard formula reads: \\begin{equation}\\label{eqn:L-Lambda_comm_standard-formula}[\\Lambda_\\omega,\\,L_\\omega] = (n-p-q)\\,\\mbox{Id} \\hspace{3ex}\\mbox{on}\\hspace{1ex} (p,\\,q)-\\mbox{forms}.\\end{equation}\n\n\nSince the form $u\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ was arbitrary, formula (\\ref{eqn:1st-variation_Hodge-star}) follows.  \\hfill $\\Box$\n\n\n\\subsection{Computation of the first variation of the trace of $(p,\\,q)$-forms}\\label{subsection:1st-variation_trace_pq}\n\nIn this subsection, we generalise the result of $\\S.$\\ref{subsection:1st-variation_trace_11} to the case of forms of arbitrary bidegree $(p,\\,q)$ by using the first variation of the Hodge star operator computed in $\\S.$\\ref{subsection:1st-variation_Hodge-star}.\n\nWe will need the following preliminary result.\n\n\\begin{Lem}\\label{Lem:star-eta-wedge_commutation} Let $X$ be an $n$-dimensional complex manifold equipped with a Hermitian metric $\\omega$ and let $\\eta$ be a $(1,\\,1)$-form on $X$. The following formula holds for operators acting on differential forms of any bidegree on $X$: \\begin{equation}\\label{eqn:star-eta-wedge_commutation}\\star_\\omega(\\eta\\wedge\\cdot) = (\\bar\\eta\\wedge\\cdot)^\\star_\\omega\\,\\star_\\omega,\\end{equation} where $\\eta\\wedge\\cdot$ is the operator of multiplication by $\\eta$ and $(\\bar\\eta\\wedge\\cdot)^\\star_\\omega$ is its adjoint w.r.t. the pointwise inner product $\\langle\\,\\cdot\\,,\\,\\cdot\\,\\rangle_\\omega$ defined by $\\omega$.\n\n\\end{Lem}\n\n\\vspace{2ex}\n\nAn immediate consequence obtained when $\\eta=\\omega$ is the following well-known formula\n\n\\begin{Cor}\\label{Cor:star-L-Lambda_commutation} Under the hypotheses of Lemma \\ref{Lem:star-eta-wedge_commutation}, we have: \\begin{equation}\\label{eqn:star-L-Lambda_commutation}\\star_\\omega L_\\omega = \\Lambda_\\omega\\,\\star_\\omega,\\end{equation} where $L_\\omega:= \\omega\\wedge\\cdot$ is the Lefschetz operator of multiplication by $\\omega$ and $\\Lambda_\\omega = L_\\omega^\\star$ is its adjoint w.r.t. the pointwise inner product $\\langle\\,\\cdot\\,,\\,\\cdot\\,\\rangle_\\omega$ defined by $\\omega$.\n\n\\end{Cor}\n\n\n\\vspace{2ex}\n\n\n\\noindent {\\it Proof of Lemma \\ref{Lem:star-eta-wedge_commutation}.} Let $(p,\\,q)$ be an arbitrary bidegree and let $u\\in\\Lambda^{p+1,\\,q+1}T^\\star X$, $v\\in\\Lambda^{p,\\,q}T^\\star X$ be arbitrary forms of the indicated bidegrees.\n\nThe definition of the Hodge star operator $\\star_\\omega$ yields the first and the third equalities below:  \\begin{eqnarray}\\label{eqn:star-L-Lambda_commutation_proof_1} u\\wedge\\star_\\omega\\overline{(\\eta\\wedge v)} & = & \\langle u,\\,\\eta\\wedge v\\rangle_\\omega\\,dV_\\omega = \\langle (\\eta\\wedge\\cdot)_\\omega^\\star u,\\, v\\rangle_\\omega\\,dV_\\omega = (\\eta\\wedge\\cdot)_\\omega^\\star u\\wedge\\star_\\omega\\bar{v}.\\end{eqnarray} \n\nThe formula to prove being pointiwse, we fix an arbitrary point $x\\in X$ and we choose local holomorphic coordinates $(z_1,\\dots , z_n)$ about $x$ such that $\\omega$ is given by the identity matrix at $x$ in these coordinates. In particular, the adjoints of the multiplication operators $dz_j\\wedge\\cdot$ and $d\\bar{z}_j\\wedge\\cdot$ w.r.t. the pointwise inner product induced by $\\omega$ at $x$ are given by the contractions with the corresponding tangent vectors at $x$: \\begin{eqnarray}\\label{eqn:adjoints_dz_j_bar}(dz_j\\wedge\\cdot)^\\star_\\omega = \\frac{\\partial}{\\partial z_j}\\lrcorner\\cdot, \\hspace{5ex} (d\\bar{z}_j\\wedge\\cdot)^\\star_\\omega = \\frac{\\partial}{\\partial\\bar{z}_j}\\lrcorner\\cdot,\\end{eqnarray} for every $j$. Let $$\\eta = \\sum\\limits_{j,\\,k=1}^n\\eta_{j\\bar{k}}\\,idz_j\\wedge d\\bar{z}_k$$ be the local expression of $\\eta$.\n\nThe last term in (\\ref{eqn:star-L-Lambda_commutation_proof_1}) reads: \\begin{eqnarray}\\label{eqn:star-L-Lambda_commutation_proof_2}(\\eta\\wedge\\cdot)_\\omega^\\star u\\wedge\\star_\\omega\\bar{v} & = & - \\sum\\limits_{j,\\,k=1}^n\\bar\\eta_{j\\bar{k}}\\,i\\,\\bigg(\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\frac{\\partial}{\\partial z_j}\\lrcorner u\\bigg)\\wedge\\star_\\omega\\bar{v}.\\end{eqnarray}\n\nMeanwhile, we have:  \n\n\n\\begin{eqnarray*}\\bigg(\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\frac{\\partial}{\\partial z_j}\\lrcorner u\\bigg)\\wedge\\star_\\omega\\bar{v} & = & \\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\bigg[\\bigg(\\frac{\\partial}{\\partial z_j}\\lrcorner u\\bigg)\\wedge\\star_\\omega\\bar{v}\\bigg] - (-1)^{p+q+1}\\,\\bigg(\\frac{\\partial}{\\partial z_j}\\lrcorner u\\bigg)\\wedge\\bigg(\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\star_\\omega\\bar{v}\\bigg) \\\\\n  \\nonumber & = & (-1)^{p+q}\\,\\bigg(\\frac{\\partial}{\\partial z_j}\\lrcorner u\\bigg)\\wedge\\bigg(\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\star_\\omega\\bar{v}\\bigg) \\\\\n  \\nonumber & = & (-1)^{p+q}\\,\\frac{\\partial}{\\partial z_j}\\lrcorner\\bigg[u\\wedge\\bigg(\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\star_\\omega\\bar{v}\\bigg)\\bigg] - u\\wedge\\bigg(\\frac{\\partial}{\\partial z_j}\\lrcorner\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\star_\\omega\\bar{v}\\bigg) \\\\\n  & = & - u\\wedge\\bigg(\\frac{\\partial}{\\partial z_j}\\lrcorner\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\star_\\omega\\bar{v}\\bigg),\\end{eqnarray*} where the first term on the r.h.s. of each of the first and third lines above vanishes for bidegree reasons. Indeed, $((\\partial\/\\partial z_j)\\lrcorner u)\\wedge\\star_\\omega\\bar{v}$ is of bidegree $(n,\\,n+1)$ and $u\\wedge((\\partial\/\\partial\\bar{z}_k)\\lrcorner\\star_\\omega\\bar{v})$ is of bidegree $(n+1,\\,n)$ since $u$ is of bidegree $(p+1,\\,q+1)$ and $\\star_\\omega\\bar{v}$ is of bidegree $(n-p,\\,n-q)$. Using again (\\ref{eqn:adjoints_dz_j_bar}), this translates to\n\n\n\\begin{eqnarray}\\label{eqn:star-L-Lambda_commutation_proof_3}\\bigg(\\frac{\\partial}{\\partial\\bar{z}_k}\\lrcorner\\frac{\\partial}{\\partial z_j}\\lrcorner u\\bigg)\\wedge\\star_\\omega\\bar{v} = - u\\wedge(d\\bar{z}_k\\wedge dz_j\\wedge\\cdot)^\\star_\\omega\\,(\\star_\\omega\\bar{v}).\\end{eqnarray}\n\nPutting (\\ref{eqn:star-L-Lambda_commutation_proof_2}) and (\\ref{eqn:star-L-Lambda_commutation_proof_3}) together, we get:\n\n\\begin{eqnarray}\\label{eqn:star-L-Lambda_commutation_proof_4}\\nonumber(\\eta\\wedge\\cdot)_\\omega^\\star u\\wedge\\star_\\omega\\bar{v} & = & \\sum\\limits_{j,k=1}^n\\bar\\eta_{j\\bar{k}}\\,i\\, u\\wedge(d\\bar{z}_k\\wedge dz_j\\wedge\\cdot)^\\star_\\omega\\,(\\star_\\omega\\bar{v}) = -u\\wedge\\bigg(\\sum\\limits_{j,\\,k=1}^n\\eta_{j\\bar{k}}\\,i\\,d\\bar{z}_k\\wedge dz_j\\wedge\\cdot\\bigg)^\\star_\\omega\\,(\\star_\\omega\\bar{v}) \\\\\n & = & u\\wedge(\\eta\\wedge\\cdot)^\\star_\\omega\\,(\\star_\\omega\\bar{v}).\\end{eqnarray}\n\n\nFinally, putting (\\ref{eqn:star-L-Lambda_commutation_proof_1}) and (\\ref{eqn:star-L-Lambda_commutation_proof_4}) together, we get: \\begin{eqnarray*}u\\wedge\\star_\\omega\\overline{(\\eta\\wedge v)} = u\\wedge\\overline{(\\bar\\eta\\wedge\\cdot)^\\star_\\omega\\,(\\star_\\omega v)}\\end{eqnarray*} for all forms $u\\in\\Lambda^{p+1,\\,q+1}T^\\star X$, $v\\in\\Lambda^{p,\\,q}T^\\star X$. This proves formula (\\ref{eqn:star-eta-wedge_commutation}).  \\hfill $\\Box$\n\n\\vspace{2ex}\n\nWe are now in a position to prove the following generalisation of Lemma \\ref{Lem:1st-variation_trace_11} to the case of forms of an arbitrary bidegree $(p,\\,q)$.\n\n\n\\begin{Lem}\\label{Lem:1st-variation_trace_pq} For any bidegree $(p,\\,q)$ and any $C^\\infty$ family $(\\alpha_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $\\alpha_t\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formula holds: \\begin{eqnarray}\\label{eqn:1st-variation_trace_pq}\\frac{d}{dt}\\bigg|_{t=0}\\,(\\Lambda_{\\omega+t\\gamma}\\alpha_t) = \\Lambda_\\omega\\bigg(\\frac{d\\alpha_t}{dt}\\bigg|_{t=0}\\bigg) - (\\gamma\\wedge\\cdot)^\\star_\\omega\\,\\alpha_0,  \\hspace{10ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray}\n     \n\\end{Lem}\n\n\\noindent {\\it Proof.} From formula (\\ref{eqn:star-L-Lambda_commutation}) applied to the metric $\\omega+t\\gamma$ and from $\\star_{\\omega+t\\gamma}\\,\\star_{\\omega+t\\gamma} = (-1)^{p+q}\\,\\mbox{Id}$ on $(p,\\,q)$-forms, we get: \\begin{eqnarray*}\\Lambda_{\\omega+t\\gamma}\\alpha_t = (-1)^{p+q}\\,\\star_{\\omega+t\\gamma}\\,L_{\\omega+t\\gamma}\\,\\star_{\\omega+t\\gamma}\\alpha_t = (-1)^{p+q}\\,\\star_{\\omega+t\\gamma}\\,v_t,\\end{eqnarray*} where we put $v_t:=L_{\\omega+t\\gamma}(\\star_{\\omega+t\\gamma}\\alpha_t)$.\n\nApplying formula (\\ref{eqn:1st-variation_Hodge-star}), we get the second equality below: \\begin{eqnarray}\\label{eqn:1st-variation_trace_pq_proof_1}\\frac{d}{dt}\\bigg|_{t=0}\\,(\\Lambda_{\\omega+t\\gamma}\\alpha_t) = (-1)^{p+q}\\,\\frac{d}{dt}\\bigg|_{t=0}\\,(\\star_{\\omega+t\\gamma}\\,v_t) =  (-1)^{p+q}\\,\\star_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg).\\end{eqnarray}\n\nOn the other hand, the same formula (\\ref{eqn:1st-variation_Hodge-star}) yields the third equality below: \\begin{eqnarray}\\label{eqn:1st-variation_trace_pq_proof_2}\\nonumber\\frac{dv_t}{dt}\\bigg|_{t=0} & = & \\frac{d}{dt}\\bigg|_{t=0}\\bigg((\\omega+t\\gamma)\\wedge\\star_{\\omega+t\\gamma}\\alpha_t\\bigg) = \\gamma\\wedge\\star_\\omega\\alpha_0 + \\omega\\wedge \\frac{d}{dt}\\bigg|_{t=0}\\bigg(\\star_{\\omega+t\\gamma}\\alpha_t\\bigg) \\\\\n  & = & \\gamma\\wedge\\star_\\omega\\alpha_0 + \\omega\\wedge\\star_\\omega\\bigg(\\frac{d\\alpha_t}{dt}\\bigg|_{t=0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\alpha_0\\bigg).\\end{eqnarray}\n\nPutting (\\ref{eqn:1st-variation_trace_pq_proof_1}) and (\\ref{eqn:1st-variation_trace_pq_proof_2}) together, we get: \\begin{eqnarray}\\label{eqn:1st-variation_trace_pq_proof_3}\\nonumber\\frac{d}{dt}\\bigg|_{t=0}\\,(\\Lambda_{\\omega+t\\gamma}\\alpha_t) & = & (-1)^{p+q}\\,\\star_\\omega(\\gamma\\wedge\\star_\\omega\\alpha_0) + \\Lambda_\\omega\\bigg(\\frac{d\\alpha_t}{dt}\\bigg|_{t=0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\alpha_0\\bigg) \\\\\n  & + & (-1)^{p+q}\\,\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,L_\\omega(\\star_\\omega\\alpha_0)\\bigg),\\end{eqnarray} where the general equality $\\star_\\omega(\\omega\\wedge\\cdot) = \\Lambda_\\omega\\,\\star_\\omega$ was used to get the second term on the r.h.s. and the equality $v_0 = L_\\omega(\\star_\\omega\\alpha_0)$ lead to the third term.\n\nTo compute the last term in (\\ref{eqn:1st-variation_trace_pq_proof_3}), we notice that  \\begin{eqnarray}\\label{eqn:1st-variation_trace_pq_proof_4}\\nonumber[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,L_\\omega(\\star_\\omega\\alpha_0) & = & \\Lambda_\\omega(\\omega\\wedge\\star_\\omega\\alpha_0\\wedge\\gamma) - \\Lambda_\\omega(\\omega\\wedge\\star_\\omega\\alpha_0)\\wedge\\gamma \\\\\n  \\nonumber & = & [\\Lambda_\\omega,\\,L_\\omega]\\,(\\gamma\\wedge\\star_\\omega\\alpha_0) + \\omega\\wedge\\Lambda_\\omega(\\gamma\\wedge\\star_\\omega\\alpha_0) - [\\Lambda_\\omega,\\,L_\\omega]\\,(\\star_\\omega\\alpha_0)\\wedge\\gamma - \\omega\\wedge\\gamma\\wedge\\Lambda_\\omega(\\star_\\omega\\alpha_0) \\\\\n  \\nonumber & = & (p+q-n-2)\\,\\gamma\\wedge\\star_\\omega\\alpha_0  - (p+q-n)\\,\\gamma\\wedge\\star_\\omega\\alpha_0 + \\omega\\wedge[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,(\\star_\\omega\\alpha_0) \\\\\n  \\nonumber & = & -2\\,\\gamma\\wedge\\star_\\omega\\alpha_0 + \\omega\\wedge[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,(\\star_\\omega\\alpha_0),\\end{eqnarray} where we have used the standard formula (\\ref{eqn:L-Lambda_comm_standard-formula}). Thus, (\\ref{eqn:1st-variation_trace_pq_proof_3}) becomes: \\begin{eqnarray}\\label{eqn:1st-variation_trace_pq_proof_5}\\nonumber\\frac{d}{dt}\\bigg|_{t=0}\\,(\\Lambda_{\\omega+t\\gamma}\\alpha_t) & = & \\Lambda_\\omega\\bigg(\\frac{d\\alpha_t}{dt}\\bigg|_{t=0}\\bigg) + \\Lambda_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\alpha_0\\bigg) - (\\gamma\\wedge\\cdot)^\\star_\\omega\\,\\alpha_0 \\\\\n  & + & (-1)^{p+q}\\,\\Lambda_\\omega\\bigg(\\star_\\omega\\,[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\star_\\omega\\,\\alpha_0\\bigg),\\end{eqnarray} where we have used formula (\\ref{eqn:star-eta-wedge_commutation}) (with $\\eta=\\gamma$, a {\\it real} form) and the fact that $\\star_\\omega\\star_\\omega\\alpha_0 = (-1)^{p+q}\\,\\alpha_0$ to get $\\star_\\omega\\,(\\gamma\\wedge\\star_\\omega\\alpha_0) = (-1)^{p+q}\\,(\\gamma\\wedge\\cdot)^\\star_\\omega\\,\\alpha_0$.\n\nNext, to compute the last term in (\\ref{eqn:1st-variation_trace_pq_proof_5}), we notice the following identities:  \\begin{eqnarray}\\label{eqn:1st-variation_trace_pq_proof_6}\\nonumber \\star_\\omega\\,[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\star_\\omega\\,\\alpha_0 & = & \\star_\\omega\\,\\Lambda_\\omega\\, (\\gamma\\wedge\\cdot)\\,\\star_\\omega\\,\\alpha_0 - \\star_\\omega\\,(\\gamma\\wedge\\cdot)\\,\\Lambda_\\omega\\,\\star_\\omega\\,\\alpha_0 \\\\\n  \\nonumber & = & L_\\omega\\,\\bigg(\\star_\\omega\\,(\\gamma\\wedge\\cdot)\\bigg)\\,\\star_\\omega\\,\\alpha_0 - (\\gamma\\wedge\\cdot)^\\star_\\omega\\,\\bigg(\\star_\\omega\\,\\Lambda_\\omega\\bigg)\\,\\star_\\omega\\,\\alpha_0  \\\\\n  \\nonumber & = & (-1)^{p+q}\\, L_\\omega\\,(\\gamma\\wedge\\cdot)^\\star_\\omega\\,\\alpha_0 -  (-1)^{p+q}\\,(\\gamma\\wedge\\cdot)^\\star_\\omega\\,L_\\omega\\,\\alpha_0 \\\\\n        & = & (-1)^{p+q}\\,[L_\\omega,\\,(\\gamma\\wedge\\cdot)^\\star_\\omega]\\,\\alpha_0 = -(-1)^{p+q}\\,[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]^\\star_\\omega\\,\\alpha_0,\\end{eqnarray} where we have used several times the identities $\\star_\\omega\\,(\\gamma\\wedge\\cdot) = (\\gamma\\wedge\\cdot)^\\star_\\omega\\,\\star_\\omega$ (see formula (\\ref{eqn:star-eta-wedge_commutation})) and $\\star_\\omega\\,\\Lambda_\\omega = L_\\omega\\,\\star_\\omega$ (see formula (\\ref{eqn:star-L-Lambda_commutation})).\n\nPutting (\\ref{eqn:1st-variation_trace_pq_proof_5}) and (\\ref{eqn:1st-variation_trace_pq_proof_6}) together, we get:\n\n\\begin{eqnarray}\\label{eqn:1st-variation_trace_pq_proof_7}\\frac{d}{dt}\\bigg|_{t=0}\\,(\\Lambda_{\\omega+t\\gamma}\\alpha_t) & = & \\Lambda_\\omega\\bigg(\\frac{d\\alpha_t}{dt}\\bigg|_{t=0}\\bigg) - (\\gamma\\wedge\\cdot)^\\star_\\omega\\,\\alpha_0 + \\Lambda_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot] - [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]^\\star_\\omega\\bigg)\\,\\alpha_0.\\end{eqnarray} Now, the operator $[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]$ is of order zero, so it acts pointwise on differential forms. The standard formula (\\ref{eqn:coordinates-standard-formula-anticom}) and the fact that the eigenvalues $\\gamma_1,\\dots , \\gamma_n$ of $\\gamma$ w.r.t. $\\omega$ are {\\it real} (since $\\gamma$ is) imply at once that the operator $[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]$ is self-adjoint w.r.t. the pointwise inner product defined by $\\omega$. Therefore, the last term in (\\ref{eqn:1st-variation_trace_pq_proof_7}) vanishes, so (\\ref{eqn:1st-variation_trace_pq_proof_7}) amounts to (\\ref{eqn:1st-variation_trace_pq}). This completes the proof of Lemma \\ref{Lem:1st-variation_trace_pq}. \\hfill $\\Box$\n\n\n\\vspace{2ex}\n\nNote that, when $(p,\\,q)=(1,\\,1)$, an immediate computation in coordinates yields: \\begin{eqnarray}\\label{eqn:mult-adjoint-inner-prod_11}(\\eta\\wedge\\cdot)^\\star_\\omega = \\langle\\cdot,\\,\\eta\\rangle_\\omega  \\hspace{3ex} \\mbox{on}\\hspace{1ex} \\Lambda^{1,\\,1}T^\\star X,\\end{eqnarray} so Lemma \\ref{Lem:1st-variation_trace_pq} reproves Lemma \\ref{Lem:1st-variation_trace_11}.\n\n\n\\subsection{Computation of the first variation of the adjoints and the Laplacians}\\label{subsection:1st-variation_adjoints-Laplacians}\n\nConsider the setting of Lemma \\ref{Lem:1st-variation_Hodge-star}. We start by deriving the standard formula $\\partial^\\star_{\\omega+t\\gamma}v_t = -\\star_{\\omega+t\\gamma}\\bar\\partial\\star_{\\omega+t\\gamma}v_t = -\\star_{\\omega+t\\gamma}\\alpha_t$, where we put $\\alpha_t:=\\bar\\partial\\star_{\\omega+t\\gamma}v_t$. Applying formula (\\ref{eqn:1st-variation_Hodge-star}) with $\\alpha_t$ in place of $v_t$, we get the second equality below: \\begin{eqnarray*}\\frac{d}{dt}\\bigg|_{t=0}(\\partial^\\star_{\\omega+t\\gamma}v_t) & = & -\\frac{d}{dt}\\bigg|_{t=0}(\\star_{\\omega+t\\gamma}\\alpha_t) = -\\star_\\omega\\bigg(\\frac{d\\alpha_t}{dt}\\bigg|_{t=0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\alpha_0\\bigg) \\\\\n  & = & - \\star_\\omega\\bar\\partial\\bigg(\\frac{d}{dt}\\bigg|_{t=0}(\\star_{\\omega+t\\gamma}v_t)\\bigg) - \\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\bar\\partial\\star_\\omega v_0\\bigg).\\end{eqnarray*} Applying again formula (\\ref{eqn:1st-variation_Hodge-star}) to the first term on the last line above, using the standard formula $\\partial^\\star_\\omega = -\\star_\\omega\\bar\\partial\\star_\\omega$ and the fact that $\\star_\\omega\\star_\\omega = (-1)^k\\,\\mbox{Id}$ on $k$-forms (for any $k$), we get formula (\\ref{eqn:1st-variation_del-adjoint}) in \n\n\n\\begin{Lem}\\label{Lem:1st-variation_adjoints} (i)\\, For every bidegree $(p,\\,q)$ and every $C^\\infty$ family $(v_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $v_t\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formulae hold: \\begin{eqnarray}\\label{eqn:1st-variation_del-adjoint}\\frac{d}{dt}\\bigg|_{t=0}(\\partial^\\star_{\\omega+t\\gamma}v_t) = \\partial^\\star_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0}\\bigg) + \\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg) + (-1)^{\\deg v_0 +1}\n\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\partial^\\star_\\omega v_0,\\end{eqnarray}  \\begin{eqnarray}\\label{eqn:1st-variation_del-bar-adjoint}\\frac{d}{dt}\\bigg|_{t=0}(\\bar\\partial^\\star_{\\omega+t\\gamma}v_t) = \\bar\\partial^\\star_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0}\\bigg) + \\bar\\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg) + (-1)^{\\deg v_0 +1}\n    \\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\bar\\partial^\\star_\\omega v_0.\\end{eqnarray}\n\n  \\vspace{1ex}\n\n (ii)\\, For every degree $k$ and every $C^\\infty$ family $(v_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $v_t\\in C^\\infty_k(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formula holds:   \\begin{eqnarray}\\label{eqn:1st-variation_d-adjoint}\\frac{d}{dt}\\bigg|_{t=0}(d^\\star_{\\omega+t\\gamma}v_t) = d^\\star_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0}\\bigg) + d^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg) + (-1)^{\\deg v_0 +1}\n\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)d^\\star_\\omega v_0.\\end{eqnarray}\n\n\n\\end{Lem}\n\n\\noindent {\\it Proof.} Formula (\\ref{eqn:1st-variation_del-adjoint}) was proved just before the statement, (\\ref{eqn:1st-variation_del-bar-adjoint}) follows by conjugating (\\ref{eqn:1st-variation_del-adjoint}), while (\\ref{eqn:1st-variation_d-adjoint}) follows by adding (\\ref{eqn:1st-variation_del-adjoint}) and (\\ref{eqn:1st-variation_del-bar-adjoint}) up for each of the pure-type components of $v_t$.  \\hfill $\\Box$\n\n\n\\vspace{3ex}\n\nLet us now consider the $\\partial$-Laplacians induced by $\\omega$ and $\\omega+t\\gamma$ (with $t\\in\\R$ close to $0$): \\begin{eqnarray*}\\Delta'_\\omega = \\partial\\partial^\\star_\\omega + \\partial^\\star_\\omega\\partial, \\hspace{6ex} \\Delta'_{\\omega+t\\gamma} = \\partial\\partial^\\star_{\\omega+t\\gamma} + \\partial^\\star_{\\omega+t\\gamma}\\partial\\end{eqnarray*} and their $\\bar\\partial$- and $d$-analogues $\\Delta''_\\omega$, $\\Delta''_{\\omega+t\\gamma}$, $\\Delta_\\omega$, $\\Delta_{\\omega+t\\gamma}$. We get:\n\n\\begin{eqnarray*}\\frac{d}{dt}\\bigg|_{t=0}(\\Delta'_{\\omega+t\\gamma}v_t) = \\partial\\bigg(\\frac{d}{dt}\\bigg|_{t=0}(\\partial^\\star_{\\omega+t\\gamma}v_t)\\bigg) + \\frac{d}{dt}\\bigg|_{t=0}(\\partial^\\star_{\\omega+t\\gamma}(\\partial v_t))\\end{eqnarray*} and its analogues for $\\Delta''_\\omega$ and $\\Delta_\\omega$, $\\Delta_{\\omega+t\\gamma}$.\n\nThen, a straightforward application of formulae (\\ref{eqn:1st-variation_del-adjoint})--(\\ref{eqn:1st-variation_d-adjoint}) of Lemma \\ref{Lem:1st-variation_adjoints} yields the following\n\n  \n  \\begin{Lem}\\label{Lem:1st-variation_Laplacians} (i)\\, For every bidegree $(p,\\,q)$ and every $C^\\infty$ family $(v_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $v_t\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formulae hold:\n\n\n    \\begin{eqnarray}\\label{eqn:1st-variation_del-Laplacian}\\frac{d}{dt}\\bigg|_{t=0}(\\Delta'_{\\omega+t\\gamma}v_t) & = & \\Delta'_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0}\\bigg) + \\partial\\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg) + \\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\partial v_0\\bigg) \\\\\n      \\nonumber      & + & (-1)^{\\deg v_0 +1}\\,\\partial\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\partial^\\star_\\omega v_0 + (-1)^{\\deg v_0}\\,\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\,\\partial^\\star_\\omega\\partial v_0,\\end{eqnarray}\n\n    \\begin{eqnarray}\\label{eqn:1st-variation_del-bar-Laplacian}\\frac{d}{dt}\\bigg|_{t=0}(\\Delta''_{\\omega+t\\gamma}v_t) & = & \\Delta''_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0}\\bigg) + \\bar\\partial\\bar\\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg) + \\bar\\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\bar\\partial v_0\\bigg) \\\\\n      \\nonumber      & + & (-1)^{\\deg v_0 +1}\\,\\bar\\partial\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\bar\\partial^\\star_\\omega v_0 + (-1)^{\\deg v_0}\\,\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\,\\bar\\partial^\\star_\\omega\\bar\\partial v_0.\\end{eqnarray}\n\n     \\vspace{1ex}\n\n   (ii)\\, For every degree $k$ and every $C^\\infty$ family $(v_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $v_t\\in C^\\infty_k(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formula holds:\n\n   \\begin{eqnarray}\\label{eqn:1st-variation_d-Laplacian}\\frac{d}{dt}\\bigg|_{t=0}(\\Delta_{\\omega+t\\gamma}v_t) & = & \\Delta_\\omega\\bigg(\\frac{dv_t}{dt}\\bigg|_{t=0}\\bigg) + dd^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,v_0\\bigg) + d^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,d v_0\\bigg) \\\\\n      \\nonumber      & + & (-1)^{\\deg v_0 +1}\\,d\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)d^\\star_\\omega v_0 + (-1)^{\\deg v_0}\\,\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\,d^\\star_\\omega d v_0.\\end{eqnarray} \n\n\n\\end{Lem}   \n\n\n\\subsection{Computation of the first variation of the orthogonal projections onto the kernels of the Laplacians}\\label{subsection:1st-variation_projections-ker-Laplacians}\n\nWe continue with the setting of Lemma \\ref{Lem:1st-variation_Hodge-star}. We will need a formula for the first variation of the $L^2_\\omega$-orthogonal projection $P^{(3)}_\\omega:C^\\infty_3(X,\\,\\C)\\longrightarrow\\mbox{Im}\\,d$ induced by the three-space decomposition (\\ref{eqn:3-space-decomp_d-3}). Since $\\partial\\omega\\in\\ker d = {\\cal H}^3_\\Delta(X,\\,\\C)\\oplus\\mbox{Im}\\,d$ under our SKT assumption, we will only be dealing with the restriction of $P^{(3)}_\\omega$ to $\\ker d$, so the computation of the first variation of $P^{(3)}_\\omega$ is equivalent to the computation of the first variation of the $L^2_\\omega$-orthogonal projection $F^{(3)}_\\Delta:C^\\infty_3(X,\\,\\C)\\longrightarrow{\\cal H}^3_\\Delta(X,\\,\\C)$.\n\nAllowing for more flexibility, we consider an arbitrary bidegree $(p,\\,q)$ and the $L^2_{\\omega+t\\gamma}$-orthogonal projection \\begin{equation*}F'_t:C^\\infty_{p,\\,q}(X,\\,\\C)\\longrightarrow{\\cal H}^{p,\\,q}_{\\Delta'_{\\omega+t\\gamma}}(X,\\,\\C), \\hspace{5ex} t\\sim 0,\\end{equation*} onto the kernel ${\\cal H}^{p,\\,q}_{\\Delta'_{\\omega+t\\gamma}}(X,\\,\\C)$ of the $\\partial$-Laplacian $\\Delta'_{\\omega+t\\gamma}$. This projection is induced by the three-space $L^2_{\\omega+t\\gamma}$-orthogonal decomposition  \\begin{equation}\\label{eqn:3-space-decomp_del-pq}C^\\infty_{p,\\,q}(X,\\,\\C) = {\\cal H}^{p,\\,q}_{\\Delta'_{\\omega+t\\gamma}}(X,\\,\\C)\\oplus\\mbox{Im}\\,\\partial\\oplus\\mbox{Im}\\,\\partial^\\star_{\\omega+t\\gamma}\\end{equation} in which $\\ker\\partial = {\\cal H}^{p,\\,q}_{\\Delta'_{\\omega+t\\gamma}}(X,\\,\\C)\\oplus\\mbox{Im}\\,\\partial$ and $\\ker\\partial^\\star_{\\omega+t\\gamma} = {\\cal H}^{p,\\,q}_{\\Delta'_{\\omega+t\\gamma}}(X,\\,\\C)\\oplus\\mbox{Im}\\,\\partial^\\star_{\\omega+t\\gamma}$.\n\nWe will denote by $F''_t:C^\\infty_{p,\\,q}(X,\\,\\C)\\longrightarrow{\\cal H}^{p,\\,q}_{\\Delta''_{\\omega+t\\gamma}}(X,\\,\\C)$ and $F_t:C^\\infty_k(X,\\,\\C)\\longrightarrow{\\cal H}^k_{\\Delta_{\\omega+t\\gamma}}(X,\\,\\C)$ the analogous orthogonal projections onto the kernels of $\\Delta''_{\\omega+t\\gamma}$ and $\\Delta_{\\omega+t\\gamma}$.\n\nNote that $(\\omega+t\\gamma)_{|t|<\\varepsilon}$ is a $C^\\infty$ family of Hermitian metrics on $X$, so the family $(\\Delta'_{\\omega+t\\gamma})_{|t|<\\varepsilon}$ is a $C^\\infty$ family of elliptic operators on $C^\\infty_{p,\\,q}(X,\\,\\C)$. Moreover, we have Hodge isomorphisms: \\begin{equation*}\\ker\\Delta'_{\\omega+t\\gamma}\\simeq H^{p,\\,q}_\\partial(X,\\,\\C),  \\hspace{6ex} t\\in(-\\varepsilon,\\,\\varepsilon),\\end{equation*} so the dimension of the kernel of $\\Delta'_{\\omega+t\\gamma}$ is independent of $t$. By the classical Kodaira-Spencer Theorem 5 in [KS60], the operator $F'_t$ depends in a $C^\\infty$ way on $t\\in(-\\varepsilon,\\,\\varepsilon)$. \n\nWe now set out to compute the first variation at $t=0$ of $F'_t$ using the classical Kodaira-Spencer {\\it Cauchy-type formula} (see [KS60, p. 56] or [Kod86, chapter 7, (7.42)]):  \\begin{equation}\\label{eqn:K-S_Cauchy}F'_tv = -\\frac{1}{2\\pi i}\\,\\int_{\\zeta\\in C}(\\Delta'_{\\omega+t\\gamma}-\\zeta)^{-1}v\\,d\\zeta, \\hspace{5ex} v\\in C^\\infty_{p,\\,q}(X,\\,\\C),\\, t\\in(-\\varepsilon,\\,\\varepsilon),\\end{equation} where $C\\subset\\C$ is any Jordan curve in the complex plane that does not meet the spectrum of $\\Delta'_\\omega$. Recall that $\\mbox{Spec}\\,(\\Delta'_\\omega)$ is discrete and has $+\\infty$ as its only accumulation point thanks to $\\Delta'_\\omega$ being elliptic and to $X$ being compact. Moreover, when $\\varepsilon>0$ is small enough, $C\\subset\\C$ does not meet the spectrum of $\\Delta'_{\\omega+t\\gamma}$ for any  $t\\in(-\\varepsilon,\\,\\varepsilon)$. (See [Kod86, chapter 7, (7.41)].)\n\nIn our case, we choose $C$ to be a circle centred at the origin in $\\C$ so small that $0$ is the only eigenvalue of $\\Delta'_\\omega$ lying in the interior of $C$\n and all the other eigenvalues of $\\Delta'_\\omega$ lie outside the closed disc $\\overline{D}$ bounded by $C$. Since $\\mbox{dim}\\ker\\Delta'_{\\omega+t\\gamma}$ is independent of $t$, $0$ continues to be the only eigenvalue of $\\Delta'_{\\omega+t\\gamma}$ lying in the interior of $C$ with all the other eigenvalues lying outside $\\overline{D}$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$ if $\\varepsilon>0$ is small enough.\n\nDeriving (\\ref{eqn:K-S_Cauchy}) w.r.t. $t$ and then taking $t=0$, we get \\begin{equation}\\label{eqn:K-S_Cauchy_derived_1}\\frac{d}{dt}\\bigg|_{t=0}(F'_tv) = -\\frac{1}{2\\pi i}\\,\\int_{\\zeta\\in C}\\frac{d}{dt}\\bigg|_{t=0}\\bigg((\\Delta'_{\\omega+t\\gamma}-\\zeta)^{-1}v\\bigg)\\,d\\zeta, \\hspace{5ex} v\\in C^\\infty_{p,\\,q}(X,\\,\\C),\\, t\\in(-\\varepsilon,\\,\\varepsilon).\\end{equation}\n\nTo continue the computations, we fix a form $v\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ and we put $$u_{t,\\,\\zeta}:=(\\Delta'_{\\omega+t\\gamma}-\\zeta)^{-1}v,  \\hspace{6ex} t\\in(-\\varepsilon,\\,\\varepsilon), \\, \\zeta\\in D_1,$$ where $D_1$ is an open disc centred at the origin in $\\C$ that contains $\\overline{D}$ such that all the positive eigenvalues of $\\Delta'_{\\omega+t\\gamma}$ lie outside $D_1$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$. Thus, $$v= (\\Delta'_{\\omega+t\\gamma}-\\zeta)\\, u_{t,\\,\\zeta},  \\hspace{6ex} t\\in(-\\varepsilon,\\,\\varepsilon), \\, \\zeta\\in D_1.$$\n\nSince $v$ is independent of $t$, we get the first equality below: \\begin{eqnarray*}0 = \\frac{dv}{dt}\\bigg|_{t=0} = \\frac{d}{dt}\\bigg|_{t=0}\\bigg(\\Delta'_{\\omega+t\\gamma}u_{t,\\,\\zeta}\\bigg) - \\zeta\\, \\frac{du_{t,\\,\\zeta}}{dt}\\bigg|_{t=0} =  (\\Delta'_\\omega-\\zeta)\\,\\bigg(\\frac{du_{t,\\,\\zeta}}{dt}\\bigg|_{t=0}\\bigg) - A(u_{0,\\,\\zeta}),\\end{eqnarray*} where the last equality follows from formula (\\ref{eqn:1st-variation_del-Laplacian}) after denoting, for every $\\zeta\\in D_1$: \\begin{eqnarray}\\label{eqn:A'u_zeta_notation}A'(u_{0,\\,\\zeta}) & := & -\\partial\\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,u_{0,\\,\\zeta}\\bigg) - \\partial^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\partial u_{0,\\,\\zeta}\\bigg) \\\\\n  \\nonumber      & + & (-1)^{\\deg u_{0,\\,\\zeta}}\\,\\partial\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\partial^\\star_\\omega u_{0,\\,\\zeta} + (-1)^{\\deg u_{0,\\,\\zeta}+1}\\,\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\,\\partial^\\star_\\omega\\partial u_{0,\\,\\zeta}.\\end{eqnarray}\n\nWe infer that, for every $v\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ and every $t\\in(-\\varepsilon,\\,\\varepsilon)$, we have: \\begin{equation}\\label{eqn:K-S_Cauchy_derived_2}\\frac{d}{dt}\\bigg|_{t=0}\\bigg((\\Delta'_{\\omega+t\\gamma}-\\zeta)^{-1}v\\bigg) = \\frac{du_{t,\\,\\zeta}}{dt}\\bigg|_{t=0} = (\\Delta'_\\omega-\\zeta)^{-1}\\,\\bigg(A'(u_{0,\\,\\zeta})\\bigg).\\end{equation}\n\nFrom this, we easily get the following\n\n\n\\begin{Lem}\\label{Lem:1st-variation_projections-ker-Laplacians} (i)\\, For every bidegree $(p,\\,q)$ and every form $v\\in C^\\infty_{p,\\,q}(X,\\,\\C)$, the following formulae hold: \\begin{eqnarray}\\label{eqn:1st-variation_projection-ker-Delta'-Delta''}\\frac{d}{dt}\\bigg|_{t=0}(F'_tv) = F'_0(A'(u_{0,\\,0})) \\hspace{3ex} \\mbox{and} \\hspace{3ex} \\frac{d}{dt}\\bigg|_{t=0}(F''_tv) = F''_0(A''(u_{0,\\,0})),\\end{eqnarray} where $A'(u_{0,\\,0})$ is the form defined in (\\ref{eqn:A'u_zeta_notation}) for $\\zeta=0$ and $A''(u_{0,\\,0})$ is the analogous form defined by replacing $\\partial$ with $\\bar\\partial$.\n\n  \\vspace{1ex}\n\n   (ii)\\, For every degree $k$ and every $C^\\infty$ family $(v_t)_{t\\in(-\\varepsilon,\\,\\varepsilon)}$ of forms $v_t\\in C^\\infty_k(X,\\,\\C)$ with $\\varepsilon>0$ so small that $\\omega+t\\gamma>0$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$, the following formula holds:  \\begin{eqnarray}\\label{eqn:1st-variation_projection-ker-Delta}\\frac{d}{dt}\\bigg|_{t=0}(F_tv) = F_0(A(u_{0,\\,0})),\\end{eqnarray} where, for every $\\zeta\\in D_1$, we put: \\begin{eqnarray}\\label{eqn:Au_zeta_notation}A(u_{0,\\,\\zeta}) & := & -dd^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,u_{0,\\,\\zeta}\\bigg) - d^\\star_\\omega\\bigg([\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,d u_{0,\\,\\zeta}\\bigg) \\\\\n  \\nonumber      & + & (-1)^{\\deg u_{0,\\,\\zeta}}\\,d\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)d^\\star_\\omega u_{0,\\,\\zeta} + (-1)^{\\deg u_{0,\\,\\zeta}+1}\\,\\bigg(\\star_\\omega[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\star_\\omega\\bigg)\\,d^\\star_\\omega d u_{0,\\,\\zeta}.\\end{eqnarray}\n\n\n\\end{Lem}\n\n\\noindent {\\it Proof.} It suffices to prove the first equality in (\\ref{eqn:1st-variation_projection-ker-Delta'-Delta''}) since the second one is obtained from the first by taking conjugates and (\\ref{eqn:1st-variation_projection-ker-Delta}) follows by running the same argument with $d$ in place of $\\partial$ and using (\\ref{eqn:1st-variation_d-Laplacian}) rather than (\\ref{eqn:1st-variation_del-Laplacian}).\n\nPutting (\\ref{eqn:K-S_Cauchy_derived_1}) and (\\ref{eqn:K-S_Cauchy_derived_2}) together, we get: \\begin{equation}\\label{eqn:K-S_Cauchy_derived_3}\\frac{d}{dt}\\bigg|_{t=0}(F'_tv) = -\\frac{1}{2\\pi i}\\,\\int_{\\zeta\\in C}(\\Delta'_\\omega-\\zeta)^{-1}\\,\\bigg(A'(u_{0,\\,\\zeta})\\bigg)\\,d\\zeta, \\hspace{5ex} v\\in C^\\infty_{p,\\,q}(X,\\,\\C),\\, t\\in(-\\varepsilon,\\,\\varepsilon).\\end{equation}\n\nLet $(e'_j)_{j\\in\\N}$ be an orthonormal basis (w.r.t. the $L^2_\\omega$-inner product) of $C^\\infty_{p,\\,q}(X,\\,\\C)$ consisting of eigenvectors of the elliptic operator $\\Delta'_\\omega$ corresponding respectively to the eigenvalues $(\\lambda_j)_{j\\in\\N}$. Thus, $\\Delta'_\\omega e'_j = \\lambda_j\\,e'_j$ for every $j\\in\\N$. Considering the decomposition \\begin{equation*}A'(u_{0,\\,\\zeta}) = \\sum\\limits_{j=0}^{+\\infty}c_j(\\zeta)\\,e_j,  \\hspace{6ex} \\zeta\\in D_1, \\end{equation*} of the form $A'(u_{0,\\,\\zeta})\\in C^\\infty_{p,\\,q}(X,\\,\\C)$ w.r.t. this orthonormal basis for every fixed $\\zeta$, we notice that every coefficient $c_j(\\zeta) = \\langle\\langle A'(u_{0,\\,\\zeta}),\\,e_j\\rangle\\rangle_\\omega$ depends holomorphically on $\\zeta$ because both $u_{0,\\,\\zeta}=(\\Delta'_\\omega-\\zeta)^{-1}v$ and the expression (\\ref{eqn:A'u_zeta_notation}) for $A'(\\cdot)$ do. \n\n\nWe get: \\begin{equation*}(\\Delta'_\\omega-\\zeta)^{-1}\\,\\bigg(A'(u_{0,\\,\\zeta})\\bigg) =  \\sum\\limits_{j=0}^{+\\infty}\\frac{c_j(\\zeta)}{\\lambda_j-\\zeta}\\,e_j,  \\hspace{6ex} \\zeta\\in D_1.\\end{equation*} Hence,  \\begin{eqnarray*}-\\frac{1}{2\\pi i}\\,\\int_{\\zeta\\in C}(\\Delta'_\\omega-\\zeta)^{-1}\\,\\bigg(A'(u_{0,\\,\\zeta})\\bigg)\\,d\\zeta = \\sum\\limits_{j=0}^{+\\infty}\\bigg(\\frac{1}{2\\pi i}\\,\\int_{\\zeta\\in C}\\frac{c_j(\\zeta)}{\\zeta-\\lambda_j}\\,d\\zeta\\bigg)\\,e_j = \\sum\\limits_{j\\in J}c_j(\\lambda_j)\\,e_j,\\end{eqnarray*} where the last equality is an application of the elementary Cauchy formula to each holomorphic function $c_j$ and $J$ is the set of indices $j$ for which $\\lambda_j$ lies in the interior of the circle $C$. (Of course, the last integral above vanishes whenever $\\lambda_j$ lies in the exterior of $C$.) However, we have chosen $C$ such that $0$ is the only eigenvalue of $\\Delta'_\\omega$ lying in the interior of $C$. Thus, $c_j(\\lambda_j) = c_j(0)$ for every $j\\in J$, $(e'_j)_{j\\in J}$ is an $L^2_\\omega$-orthonormal basis of $\\ker\\Delta'_\\omega$ in bidegree $(p,\\,q)$ and the cardinality of $J$ is the dimension of the cohomology group $H^{p,\\,q}_\\partial(X,\\,\\C)$. Therefore, we conclude that \\begin{eqnarray}\\label{eqn:K-S_Cauchy_derived_4}-\\frac{1}{2\\pi i}\\,\\int_{\\zeta\\in C}(\\Delta'_\\omega-\\zeta)^{-1}\\,\\bigg(A'(u_{0,\\,\\zeta})\\bigg)\\,d\\zeta = \\sum\\limits_{j\\in J}c_j(0)\\,e_j  = F'_0(A'(u_{0,\\,0})),\\end{eqnarray} where $F'_t:C^\\infty_{p,\\,q}(X,\\,\\C)\\longrightarrow{\\cal H}^{p,\\,q}_{\\Delta'_{\\omega+t\\gamma}}(X,\\,\\C)$ is the $L^2_\\omega$-orthogonal projection onto $\\ker\\Delta'_\\omega={\\cal H}^{p,\\,q}_{\\Delta'_\\omega}(X,\\,\\C)$.\n\nPutting (\\ref{eqn:K-S_Cauchy_derived_3}) and (\\ref{eqn:K-S_Cauchy_derived_4}) together, we get the contention.  \\hfill $\\Box$\n\n\n\\section{Computation of the critical points of the functional $F$}\\label{section:critical-points-computation_functional_F} We now pick up where we left off at the end of $\\S.$\\ref{subsection:functional_skt-Delta}.\n\nWe fix an SKT metric $\\omega$ on $X$ and we vary it along the path $\\omega + t\\gamma$, where $\\gamma\\in\\ker(\\partial\\bar\\partial)\\cap C^\\infty_{1,\\,1}(X,\\,\\R)$ is an arbitrary pluriclosed real $(1,\\,1)$-form. We will be considering reals $t\\in(-\\varepsilon,\\,\\varepsilon)$ with $\\varepsilon>0$ small enough to ensure that $\\omega + t\\gamma>0$ (i.e. $\\omega + t\\gamma$ is a, necessarily SKT, metric on $X$). Thus, $\\omega + t\\gamma\\in{\\cal S}$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$.\n\n\nFor each $t\\in(-\\varepsilon,\\,\\varepsilon)$, let $\\eta_{\\omega,\\,\\gamma,\\,t}$ be the minimal $L^2_{\\omega + t\\gamma}$-norm solution of the equation \\begin{eqnarray}\\label{eqn:eta_omega-gamma-t_def}d\\eta = P^{(3)}_{\\omega + t\\gamma}(\\partial\\gamma),\\end{eqnarray} where $P^{(3)}_{\\omega + t\\gamma}:C^\\infty_3(X,\\,\\C)\\longrightarrow\\mbox{Im}\\,d$ is the orthogonal projection w.r.t. the $L^2_{\\omega + t\\gamma}$-inner product. Set $\\eta_{\\omega,\\,\\gamma}=\\eta_{\\omega,\\,\\gamma,\\,0}$.\n\nMeanwhile, for each $t\\in(-\\varepsilon,\\,\\varepsilon)$, Definition \\ref{Def:torsion-form} ascribes a unique torsion form $\\rho_{\\omega + t\\gamma}\\in C^\\infty_2(X,\\,\\C)$ to $\\omega + t\\gamma$, identified by the property $\\rho_{\\omega + t\\gamma}\\in\\mbox{Im}\\,d^\\star_{\\omega + t\\gamma}$ and the first equality below: \\begin{eqnarray}\\label{eqn:torsion-form_omega-t-gamma_1st-cond}\\nonumber d\\rho_{\\omega + t\\gamma} & = & P^{(3)}_{\\omega + t\\gamma}(\\partial\\omega + t\\,\\partial\\gamma) \\stackrel{(a)}{=} P^{(3)}_{\\omega + t\\gamma}(\\partial\\omega) + t\\,d\\eta_{\\omega,\\,\\gamma,\\,t} \\\\\n\\nonumber  & \\stackrel{(b)}{=} & P^{(3)}_\\omega(\\partial\\omega) + d(t\\,\\eta_{\\omega,\\,\\gamma}) + \\bigg(\\frac{dP^{(3)}_{\\omega + t\\gamma}}{dt}_{|t=0}\\bigg)(\\partial\\omega)\\,t + {\\cal O}(t^2) \\\\\n& \\stackrel{(c)}{=} & d(\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}) + \\bigg(\\frac{dP^{(3)}_{\\omega + t\\gamma}}{dt}_{|t=0}\\bigg)(\\partial\\omega)\\,t + {\\cal O}(t^2),  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon),\\end{eqnarray} where equality (a) follows from the linearity of $P^{(3)}_{\\omega + t\\gamma}$, while equality (b) follows from the expansion to order two w.r.t. $t$ about $0$ of the operator $P^{(3)}_{\\omega + t\\gamma}$ and of the form \\begin{eqnarray*}\\eta_{\\omega,\\,\\gamma,\\,t} = \\Delta^{-1}_{\\omega +t\\,\\gamma}d^\\star_{\\omega +t\\,\\gamma}P^{(3)}_{\\omega + t\\gamma}(\\partial\\gamma).\\end{eqnarray*} Note that both dependencies on $t$ are $C^\\infty$. Equality (c) follows from (a) of (\\ref{eqn:system_torsion-form}).\n\nRecall that $F(\\omega + t\\gamma) = ||\\rho_{\\omega + t\\gamma}||^2_{\\omega + t\\gamma}$ for every $t\\in(-\\varepsilon,\\,\\varepsilon)$. (See formula (\\ref{eqn:F_energy-functional_SKT}).) The following result constitutes the first step in the computation of $\\frac{d}{dt}_{|t=0} F(\\omega + t\\gamma) = \\frac{d}{dt}_{|t=0}||\\rho_{\\omega + t\\gamma}||^2_{\\omega + t\\gamma}$. \n\n\n\\begin{Lem}\\label{Lem:1st-derivative_rho-omega+tgamma} The following equality holds: \\begin{equation}\\label{eqn:1st-derivative_rho-omega+tgamma}\\frac{d}{dt}_{|t=0}||\\rho_{\\omega + t\\gamma}||^2_{\\omega + t\\gamma}  = \\frac{d}{dt}_{|t=0}||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||^2_{\\omega + t\\gamma} + 2\\,||\\rho_\\omega||_\\omega\\,||\\Delta_\\omega^{-1}d^\\star_\\omega A_{\\omega,\\,\\gamma,\\,0}||_\\omega,\\end{equation} where \\begin{equation}\\label{eqn:A_omega-gamma-0_notation}A_{\\omega,\\,\\gamma,\\,0}:= \\bigg(\\frac{dP^{(3)}_{\\omega + t\\gamma}}{dt}_{|t=0}\\bigg)(\\partial\\omega).\\end{equation}\n\n\\end{Lem}  \n\n\n\\noindent {\\it Proof.} First of all we note that the formula holds if $\\rho_\\omega=0$. Indeed, if this is the case the right hand side is obviously zero, whereas the vanishing of the left hand side follows directly from Lemma \\ref{Lem:vanishing-F-Kaehler}. Henceforth we shall assume that $\\rho_{\\omega}\\neq 0$.\n\n\n\n If we put $B_{\\omega,\\,\\gamma,\\,t}:= d\\rho_{\\omega + t\\gamma} - d(\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}) = \\bigg(\\frac{dP^{(3)}_{\\omega + t\\gamma}}{dt}_{|t=0}\\bigg)(\\partial\\omega)\\,t + {\\cal O}(t^2)$ (the sum of the last two terms in (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond})), (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond}) reads: \\begin{eqnarray}\\label{eqn:torsion-form_omega-t-gamma_1st-cond_bis}d(\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}) = d\\rho_{\\omega + t\\gamma} - B_{\\omega,\\,\\gamma,\\,t},  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray}\n\n\nFor every $t\\in(-\\varepsilon,\\,\\varepsilon)$, let $\\tilde\\rho_{\\omega,\\,\\gamma,\\,t}\\in\\mbox{Im}\\,d^\\star_{\\omega + t\\gamma}$ be the minimal $L^2_{\\omega + t\\gamma}$-norm solution of the equation \\begin{eqnarray}\\label{eqn:torsion-form_omega-t-gamma_1st-cond_bis_equation}d\\rho = d\\rho_{\\omega + t\\gamma} - B_{\\omega,\\,\\gamma,\\,t}.\\end{eqnarray} This means that, for $t\\in(-\\varepsilon,\\,\\varepsilon)$, we have: $d\\tilde\\rho_{\\omega,\\,\\gamma,\\,t} = d\\rho_{\\omega + t\\gamma} - B_{\\omega,\\,\\gamma,\\,t}$ and  \\begin{eqnarray*}\\tilde\\rho_{\\omega,\\,\\gamma,\\,t} = \\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}(d\\rho_{\\omega + t\\gamma} - B_{\\omega,\\,\\gamma,\\,t}) = \\rho_{\\omega + t\\gamma} - \\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}B_{\\omega,\\,\\gamma,\\,t}.\\end{eqnarray*} (To get the last equality, we used the fact that $\\rho_{\\omega + t\\gamma}\\in\\mbox{Im}\\,d^\\star_{\\omega + t\\gamma}\\subset\\ker d^\\star_{\\omega + t\\gamma}$.) Hence,  \\begin{eqnarray}\\label{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_1}||\\tilde\\rho_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}\\geq||\\rho_{\\omega + t\\gamma}||_{\\omega + t\\gamma} - ||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}B_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma},  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray}\n\nOn the other hand, the $L^2_{\\omega + t\\gamma}$-norm minimality of $\\tilde\\rho_{\\omega,\\,\\gamma,\\,t}\\in\\mbox{Im}\\,d^\\star_{\\omega + t\\gamma}$ among the solutions of equation (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond_bis_equation}), one of which is $\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}$ (cf. (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond_bis})), implies that $||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||_{\\omega + t\\gamma}\\geq||\\tilde\\rho_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}$ for each $t\\in(-\\varepsilon,\\,\\varepsilon)$. Together with (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_1}), this yileds: \\begin{eqnarray}\\label{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_2}||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||_{\\omega + t\\gamma} - ||\\rho_{\\omega + t\\gamma}||_{\\omega + t\\gamma} + ||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}B_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}\\geq 0,  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray} Note that the l.h.s. term of (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_2}) vanishes at $t=0$, so, as a function of $t\\in(-\\varepsilon,\\,\\varepsilon)$, it achieves its minimum at $t=0$. In particular, its derivative w.r.t. $t$ vanishes at $t=0$. Here, we make use of the assumption $\\rho_\\omega\\neq 0$ (otherwise, $\\omega$ is K\\\"ahler -- see Lemma \\ref{Lem:Kaehler_torsion-vanishing} -- and there is nothing to prove), so the norms are smooth and the differentiation is justified. This yields: \\begin{eqnarray}\\label{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_3}\\frac{d}{dt}_{|t=0}||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||_{\\omega + t\\gamma} = \\frac{d}{dt}_{|t=0}||\\rho_{\\omega + t\\gamma}||_{\\omega + t\\gamma} - \\frac{d}{dt}_{|t=0}||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}B_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}.\\end{eqnarray}\n\nMeanwhile, $\\frac{d}{dt}_{|t=0}||\\rho_{\\omega + t\\gamma}||_{\\omega + t\\gamma}^2 = 2\\,||\\rho_\\omega||_\\omega\\,\\frac{d}{dt}_{|t=0}\\,||\\rho_{\\omega + t\\gamma}||_{\\omega + t\\gamma}$. Thus, using (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_3}), we get:\n\n\n\\begin{eqnarray}\\label{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_4}\\nonumber\\frac{d}{dt}_{|t=0}||\\rho_{\\omega + t\\gamma}||_{\\omega + t\\gamma}^2 & = & 2\\,||\\rho_\\omega||_\\omega\\, \\frac{d}{dt}_{|t=0}||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||_{\\omega + t\\gamma} + 2\\,||\\rho_\\omega||_\\omega\\,\\frac{d}{dt}_{|t=0}\\bigg(||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}B_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}^2\\bigg)^{\\frac{1}{2}} \\\\\n  & = & \\frac{d}{dt}_{|t=0}||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||_{\\omega + t\\gamma}^2 + 2\\,||\\rho_\\omega||_\\omega\\,M(0),\\end{eqnarray} where we put $$M(t):= \\frac{\\frac{d}{dt}\\,||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}B_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}^2}{2\\,||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}B_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}},  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).$$\n\nNow, letting $A_{\\omega,\\,\\gamma,\\,t} = \\frac{1}{t}\\, B_{\\omega,\\,\\gamma,\\,t}$, we see that $A_{\\omega,\\,\\gamma,\\,0}$ is given by the expression (\\ref{eqn:A_omega-gamma-0_notation}) and we have $$A_{\\omega,\\,\\gamma,\\,t} = \\bigg(\\frac{dP^{(3)}_{\\omega + t\\gamma}}{dt}_{|t=0}\\bigg)(\\partial\\omega) + {\\cal O}(t),  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).$$ Thus, writing $B_{\\omega,\\,\\gamma,\\,t} = t\\,A_{\\omega,\\,\\gamma,\\,t}$ in the expression for $M(t)$, we get for $t\\in(0,\\,\\varepsilon)$: \\begin{eqnarray*}M(t)= \\frac{\\frac{d}{dt}\\,(t^2\\,||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}A_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}^2)}{2t\\,||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}A_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}} = ||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}A_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma} + \\frac{t}{2}\\,\\frac{\\frac{d}{dt}\\,||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}A_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}^2}{||\\Delta^{-1}_{\\omega + t\\gamma}d^\\star_{\\omega + t\\gamma}A_{\\omega,\\,\\gamma,\\,t}||_{\\omega + t\\gamma}}.\\end{eqnarray*} Hence, $M(0)= ||\\Delta^{-1}_{\\omega}d^\\star_{\\omega }A_{\\omega,\\,\\gamma,\\,0}||_{\\omega}$. Together with (\\ref{eqn:torsion-form_omega-t-gamma_1st-cond_bis_proof_4}), this proves (\\ref{eqn:1st-derivative_rho-omega+tgamma}). \\hfill $\\Box$\n\n\n\\vspace{3ex}\n\nSince $F(\\omega + t\\gamma) = ||\\rho_{\\omega + t\\gamma}||^2_{\\omega + t\\gamma}$ for every $t\\in(-\\varepsilon,\\,\\varepsilon)$ (cf. formula (\\ref{eqn:F_energy-functional_SKT})), the above Lemma \\ref{Lem:1st-derivative_rho-omega+tgamma} reduces the computation of $\\frac{d}{dt}_{|t=0} F(\\omega + t\\gamma) = \\frac{d}{dt}_{|t=0}||\\rho_{\\omega + t\\gamma}||^2_{\\omega + t\\gamma}$ to the computation of $\\frac{d}{dt}_{|t=0}||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||^2_{\\omega + t\\gamma}$. This last derivative at $t=0$ is computed in the following\n\n\\begin{Lem}\\label{Lem:1st-variation_Ft-tilde} Let $X$ be a compact complex manifold with $\\mbox{dim}_\\C X=n$. Suppose there exists an SKT metric $\\omega$ on $X$.\n\n  For any real $(1,\\,1)$-form $\\gamma\\in\\ker(\\partial\\bar\\partial)$ and for a real $t$ varying in a neighbourhood $(-\\varepsilon,\\,\\varepsilon)$ of $0$ so small that $\\omega + t\\gamma>0$, we have \\begin{eqnarray}\\label{eqn:1st-variation_Ft-tilde}\\nonumber\\frac{d}{dt}_{|t=0}||\\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}||^2_{\\omega + t\\gamma} & = & \\langle\\langle\\eta_{\\omega,\\,\\gamma}^{2,\\,0},\\,\\rho_\\omega^{2,\\,0}\\rangle\\rangle_\\omega + \\langle\\langle\\rho_\\omega^{2,\\,0},\\,\\eta_{\\omega,\\,\\gamma}^{2,\\,0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\rho_\\omega^{2,\\,0}\\rangle\\rangle_\\omega \\\\\n\\nonumber & + & \\langle\\langle\\eta_{\\omega,\\,\\gamma}^{0,\\,2},\\,\\rho_\\omega^{0,\\,2}\\rangle\\rangle_\\omega + \\langle\\langle\\rho_\\omega^{0,\\,2},\\,\\eta_{\\omega,\\,\\gamma}^{0,\\,2} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\rho_\\omega^{0,\\,2}\\rangle\\rangle_\\omega \\\\    \n     & + & \\langle\\langle\\eta_{\\omega,\\,\\gamma}^{1,\\,1},\\,\\rho_\\omega^{1,\\,1}\\rangle\\rangle_\\omega + \\langle\\langle\\rho_\\omega^{1,\\,1},\\,\\eta_{\\omega,\\,\\gamma}^{1,\\,1} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\rho_\\omega^{1,\\,1}\\rangle\\rangle_\\omega,\\end{eqnarray} where $\\rho_\\omega = \\rho_\\omega^{2,\\,0} + \\rho_\\omega^{1,\\,1} + \\rho_\\omega^{0,\\,2}$ and $\\eta_{\\omega,\\,\\gamma} = \\eta_{\\omega,\\,\\gamma}^{2,\\,0} + \\eta_{\\omega,\\,\\gamma}^{1,\\,1} + \\eta_{\\omega,\\,\\gamma}^{0,\\,2}$ are the decompositions of $\\rho_\\omega$ and $\\eta_{\\omega,\\,\\gamma}$ into pure-type forms. \n\n\\end{Lem}  \n\n\n\\noindent {\\it Proof.} For every $(-\\varepsilon,\\,\\varepsilon)$, let $\\alpha_t:= \\rho_\\omega + t\\,\\eta_{\\omega,\\,\\gamma}$. In particular, $\\alpha_0 = \\rho_\\omega$.\n\nSince pure-type forms of different types are mutually orthogonal, we have for all $t$ near $0$: \\begin{eqnarray}\\label{eqn:1st-variation_Ft-tilde_proof}\\nonumber||\\alpha_t||^2_{\\omega + t\\gamma} & = & \\int\\limits_X|\\alpha_t^{2,\\,0}|^2_{\\omega + t\\gamma}\\,dV_{\\omega + t\\gamma} + \\int\\limits_X|\\alpha_t^{0,\\,2}|^2_{\\omega + t\\gamma}\\,dV_{\\omega + t\\gamma} + \\int\\limits_X|\\alpha_t^{1,\\,1} |^2_{\\omega + t\\gamma}\\,dV_{\\omega + t\\gamma} \\\\\n  & = & \\int\\limits_X\\alpha_t^{2,\\,0}\\wedge\\star_{\\omega + t\\gamma}\\overline{\\alpha_t^{2,\\,0}} + \\int\\limits_X\\alpha_t^{0,\\,2}\\wedge\\star_{\\omega + t\\gamma}\\overline{\\alpha_t^{0,\\,2}} + \\int\\limits_X\\alpha_t^{1,\\,1}\\wedge\\star_{\\omega + t\\gamma}\\overline{\\alpha_t^{1,\\,1}}.\\end{eqnarray}\n\nNow, for all $(p,\\,q)\\in\\{(2,\\,0),\\,(1,\\,1),\\,(0,\\,2)\\}$, we have: $\\frac{d\\alpha_t^{p,\\,q}}{dt}_{|t=0} = \\eta_{\\omega,\\,\\gamma}^{p,\\,q}$ and, thanks to Lemma \\ref{Lem:1st-variation_Hodge-star},   \\begin{eqnarray*}\\frac{d(\\star_{\\omega + t\\gamma}\\overline{\\alpha_t^{p,\\,q}})}{dt}_{\\bigg|t=0} = \\star_\\omega\\bigg(\\overline{\\eta_{\\omega,\\,\\gamma}^{p,\\,q}} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\overline{\\rho_\\omega^{p,\\,q}}\\bigg).\\end{eqnarray*}\n\nAfter differentiating (\\ref{eqn:1st-variation_Ft-tilde_proof}) with respect to $t$ and then evaluating the result at $t=0$, formula (\\ref{eqn:1st-variation_Ft-tilde}) follows at once from the above equalities thanks to the equality $u\\wedge\\star_\\omega\\bar{v} = \\langle u,\\,v\\rangle_\\omega\\,dV_\\omega$ holding for all forms $u$ and $v$ of the same degree and defining the Hodge star operator.  \\hfill $\\Box$\n\n\\vspace{3ex}\n\nPutting Lemmas \\ref{Lem:1st-derivative_rho-omega+tgamma} and \\ref{Lem:1st-variation_Ft-tilde} together, we get the following\n\n\\begin{Cor}\\label{Cor:1st-variation_Ft-tilde_final} Under the assumptions of Lemma \\ref{Lem:1st-variation_Ft-tilde}, we have \\begin{eqnarray}\\label{eqn:1st-variation_Ft-tilde_final}\\nonumber\\frac{d}{dt}_{|t=0} F(\\omega + t\\gamma)  & = & \\langle\\langle\\eta_{\\omega,\\,\\gamma}^{2,\\,0},\\,\\rho_\\omega^{2,\\,0}\\rangle\\rangle_\\omega + \\langle\\langle\\rho_\\omega^{2,\\,0},\\,\\eta_{\\omega,\\,\\gamma}^{2,\\,0} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\rho_\\omega^{2,\\,0}\\rangle\\rangle_\\omega \\\\\n\\nonumber & + & \\langle\\langle\\eta_{\\omega,\\,\\gamma}^{0,\\,2},\\,\\rho_\\omega^{0,\\,2}\\rangle\\rangle_\\omega + \\langle\\langle\\rho_\\omega^{0,\\,2},\\,\\eta_{\\omega,\\,\\gamma}^{0,\\,2} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\rho_\\omega^{0,\\,2}\\rangle\\rangle_\\omega \\\\    \n \\nonumber    & + & \\langle\\langle\\eta_{\\omega,\\,\\gamma}^{1,\\,1},\\,\\rho_\\omega^{1,\\,1}\\rangle\\rangle_\\omega + \\langle\\langle\\rho_\\omega^{1,\\,1},\\,\\eta_{\\omega,\\,\\gamma}^{1,\\,1} + [\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot]\\,\\rho_\\omega^{1,\\,1}\\rangle\\rangle_\\omega\\\\\n & + & 2\\,||\\rho_\\omega||_\\omega\\,||\\Delta_\\omega^{-1}d^\\star_\\omega A_{\\omega,\\,\\gamma,\\,0}||_\\omega, \\end{eqnarray} where $A_{\\omega,\\,\\gamma,\\,0}$ is defined by formula (\\ref{eqn:A_omega-gamma-0_notation}).\n\nIn particular, if we choose $\\gamma = \\omega$, the above formula specialises to \\begin{eqnarray}\\label{eqn:1st-variation_Ft-tilde_final_special}\\frac{d}{dt}_{|t=0} F(\\omega + t\\gamma)  & = & n\\,||\\rho_\\omega||^2_\\omega.\\end{eqnarray}\n\n\\end{Cor}\n\n\\noindent {\\it Proof.} Only formula (\\ref{eqn:1st-variation_Ft-tilde_final_special}) still needs a proof. When $\\gamma = \\omega$, we have $\\eta_{\\omega,\\,\\gamma} = \\rho_\\omega$ (see formulae (\\ref{eqn:eta_omega-gamma-t_def}) and (\\ref{eqn:system_torsion-form})) and $[\\Lambda_\\omega,\\,\\gamma\\wedge\\cdot] = [\\Lambda_\\omega,\\,\\omega\\wedge\\cdot] = (n-p-q)\\,\\mbox{Id}$ on $(p,\\,q)$-forms for all bidegrees $(p,\\,q)$ (see e.g. [Voi02, Lemma 6.19] for the last, standard, equality). Thus, the quantities on the first three lines of the right-hand side of (\\ref{eqn:1st-variation_Ft-tilde_final}) are equal to $n\\,||\\rho_\\omega^{2,\\,0}||^2_\\omega$, $n\\,||\\rho_\\omega^{0,\\,2}||^2_\\omega$ and respectively $n\\,||\\rho_\\omega^{1,\\,1}||^2_\\omega$ when $\\gamma = \\omega$.\n\nOn the other hand, using (\\ref{eqn:A_omega-gamma-0_notation}), we get: \\begin{eqnarray*}A_{\\omega,\\,\\omega,\\,0} = \\bigg(\\frac{dP^{(3)}_{(1+t)\\,\\omega}}{dt}_{|t=0}\\bigg)(\\partial\\omega) = 0 \\end{eqnarray*} since $P^{(3)}_{(1+t)\\,\\omega}(\\partial\\omega) =  P^{(3)}_\\omega(\\partial\\omega)$ for every $t$ close enough to $0$. (See (\\ref{eqn:P^3_del-omega_lambda}) for this last fact.)\n\nPutting these pieces of information together, we get (\\ref{eqn:1st-variation_Ft-tilde_final_special}).  \\hfill $\\Box$\n\n\n\\vspace{3ex}\n\nNote that formula (\\ref{eqn:1st-variation_Ft-tilde_final_special}) coincides with formula (\\ref{eqn:rescaling_F-critical_1+t}) of Lemma \\ref{Lem:rescaling_F-critical}. Another conclusion of this study is the following result that was already proved in a different way in (\\ref{eqn:critical-points_F-Kaehler}) of Lemma \\ref{Lem:rescaling_F-critical}.\n\n\\begin{Cor}\\label{Cor:crit-points_SKT-primitive-torsion} Let $X$ be a compact complex SKT manifold.\n\nThen, for every SKT metric $\\omega$ on $X$, $\\omega$ is a {\\bf critical point} for the functional $F:{\\cal S}\\longrightarrow[0,\\,+\\infty)$ if and only if $\\omega$ is a {\\bf K\\\"ahler} metric.\n\n\n\\end{Cor}  \n\n\\noindent {\\it Proof.} If an SKT metric $\\omega$ is a critical point for $F$, then $(d_\\omega F)(\\gamma) = \\frac{d}{dt}_{|t=0}F(\\omega + t\\,\\gamma) = 0$ for every real $(1,\\,1)$-form $\\gamma$ lying in $\\ker(\\partial\\bar\\partial)$. In particular, choosing $\\gamma:=\\omega$ (which is allowed since $\\partial\\bar\\partial\\omega=0$), the vanishing of $\\frac{d}{dt}_{|t=0}F(\\omega + t\\,\\omega)$ implies the vanishing of $\\rho_\\omega$ thanks to (\\ref{eqn:1st-variation_Ft-tilde_final_special}). On the other hand, the equality $\\rho_\\omega=0$ implies that $\\omega$ is K\\\"ahler by Lemma \\ref{Lem:Kaehler_torsion-vanishing}.\n\nConversely, if $\\omega$ is a K\\\"ahler metric, $\\omega$ is a minimiser, hence a critical point, for the functional $F$ by Lemma \\ref{Lem:vanishing-F-Kaehler}. \n\nOne can also argue that $\\rho_\\omega=0$ if $\\omega$ is K\\\"ahler (again by Lemma \\ref{Lem:Kaehler_torsion-vanishing}), so (\\ref{eqn:1st-variation_Ft-tilde_final}) of Corollary \\ref{Cor:1st-variation_Ft-tilde_final} implies that $\\frac{d}{dt}_{|t=0}F(\\omega + t\\,\\gamma) = 0$ for every real $(1,\\,1)$-form $\\gamma$ lying in $\\ker(\\partial\\bar\\partial)$. This means that $\\omega$ is a critical point for $F$.  \\hfill $\\Box$\n\n\n\n\n\\section{Computation of the critical points of the functional $G$}\\label{section:critical-points-computation_functional_G} We now pick up where we left off at the end of $\\S.$\\ref{subsection:functional_balanced}.\n\nWe fix a balanced metric $\\omega$ on $X$ (so $\\omega_{n-1}\\in{\\cal B}$) and we vary it along the path $\\omega + t\\Omega$, where $\\Omega\\in C^\\infty_{n-1,\\,n-1}(X,\\,\\R)$ is an arbitrary real $(n-1,\\,n-1)$-form such that $\\bar\\partial\\Omega = 0$. We will be considering reals $t\\in(-\\varepsilon,\\,\\varepsilon)$ with $\\varepsilon>0$ small enough to ensure that $\\omega_{n-1} + t\\Omega>0$ (i.e. $\\omega_{n-1} + t\\Omega$ is a, necessarily balanced, metric on $X$). Thus, $\\omega_{n-1} + t\\Omega\\in{\\cal B}$ for all $t\\in(-\\varepsilon,\\,\\varepsilon)$. By [Mic83], there exists a unique $C^\\infty$ positive definite $(1,\\,1)$-form $\\gamma_{\\omega,\\,\\Omega,\\,t}$ such that \\begin{equation}\\label{eqn:root_balanced}(\\gamma_{\\omega,\\,\\Omega,\\,t})_{n-1} = \\omega_{n-1} + t\\Omega,  \\hspace{3ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{equation} Note that $\\gamma_{\\omega,\\,\\Omega,\\,0} = \\omega$. \n\nFor each $t\\in(-\\varepsilon,\\,\\varepsilon)$, let $\\eta_{\\omega,\\,\\Omega,\\,t}$ be the minimal $L^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$-norm solution of the equation \\begin{eqnarray}\\label{eqn:eta_omega-Omega-t_def}\\bar\\partial\\eta_{\\omega,\\,\\Omega,\\,t} = P^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}(\\Omega),\\end{eqnarray} where $P^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}:C^\\infty_{n-1,\\,n-1}(X,\\,\\C)\\longrightarrow\\mbox{Im}\\,\\bar\\partial$ is the orthogonal projection w.r.t. the $L^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$-inner product. Set $\\eta_{\\omega,\\,\\Omega}=\\eta_{\\omega,\\,\\Omega,\\,0}$.\n\nMeanwhile, for each $t\\in(-\\varepsilon,\\,\\varepsilon)$, Definition \\ref{Def:balanced_torsion-form} ascribes a unique torsion $(n-1,\\,n-2)$-form $\\Gamma_{\\omega_{n-1} + t\\Omega}$ to $\\omega_{n-1} + t\\Omega$, identified by the property $\\Gamma_{\\omega_{n-1} + t\\Omega}\\in\\mbox{Im}\\,\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$ and the first equality below: \\begin{eqnarray}\\label{eqn:torsion-form_omega-Omega-t_1st-cond}\\nonumber \\bar\\partial\\Gamma_{\\omega_{n-1} + t\\Omega} & = & P^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}(\\omega_{n-1} + t\\Omega) \\stackrel{(a)}{=} P^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}(\\omega_{n-1}) + t\\,\\bar\\partial\\eta_{\\omega,\\,\\Omega,\\,t} \\\\\n\\nonumber  & \\stackrel{(b)}{=} & P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1}) + t\\,\\bar\\partial\\eta_{\\omega,\\,\\Omega} + \\bigg(\\frac{dP^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}}{dt}_{|t=0}\\bigg)(\\omega_{n-1})\\,t + {\\cal O}(t^2) \\\\\n& \\stackrel{(c)}{=} & \\bar\\partial(\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}) + \\bigg(\\frac{dP^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}}{dt}_{|t=0}\\bigg)(\\omega_{n-1})\\,t + {\\cal O}(t^2),  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon),\\end{eqnarray} where equality (a) follows from the linearity of $P^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}$, while equality (b) follows from the expansion to order two w.r.t. $t$ about $0$ of the operator $P^{(3)}_{\\omega + t\\gamma}$ and of the form \\begin{eqnarray*}\\eta_{\\omega,\\,\\Omega,\\,t} = \\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}P^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}(\\Omega).\\end{eqnarray*} Note that both dependencies on $t$ are $C^\\infty$. Equality (c) follows from (a) of (\\ref{eqn:system_balanced_torsion-form}).\n\nRecall that $G(\\omega_{n-1} + t\\Omega) = ||\\Gamma_{\\omega_{n-1} + t\\Omega}||^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$ for every $t\\in(-\\varepsilon,\\,\\varepsilon)$. (See formula (\\ref{eqn:G_energy-functional_balanced}).) The following result constitutes the first step in the computation of $\\frac{d}{dt}_{|t=0} G(\\omega_{n-1} + t\\Omega) = \\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1} + t\\Omega}||^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$.\n\n\\begin{Lem}\\label{Lem:1st-derivative_Gamma_omega_n-1+tOmega} The following equality holds: \\begin{equation}\\label{eqn:1st-derivative_Gamma_omega_n-1+tOmega}\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1} + t\\Omega}||^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}  = \\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}} + 2\\,||\\Gamma_{\\omega_{n-1}}||_\\omega\\,||\\Delta_\\omega^{''-1}\\bar\\partial^\\star_\\omega A_{\\omega,\\,\\Omega,\\,0}||_\\omega,\\end{equation} where \\begin{equation}\\label{eqn:A_omega-Omega-0_notation}A_{\\omega,\\,\\Omega,\\,0}:= \\bigg(\\frac{dP^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}}{dt}_{|t=0}\\bigg)(\\omega_{n-1}).\\end{equation}\n\n\\end{Lem}  \n\n\n\\noindent {\\it Proof.} Again if $\\Gamma_{\\omega_{n-1}}=0$ the formula holds. Thus we may assume from now on that $\\Gamma_{\\omega_{n-1}}\\neq 0$.If we put $B_{\\omega,\\,\\Omega,\\,t}:= \\bar\\partial\\Gamma_{\\omega_{n-1} + t\\Omega} - \\bar\\partial(\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}) = \\bigg(\\frac{dP^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}}{dt}_{|t=0}\\bigg)(\\omega_{n-1})\\,t + {\\cal O}(t^2)$ (the sum of the last two terms in (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond})), (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond}) reads: \\begin{eqnarray}\\label{eqn:torsion-form_omega-Omega-t_1st-cond_bis}\\bar\\partial(\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}) = \\bar\\partial\\Gamma_{\\omega_{n-1} + t\\Omega} - B_{\\omega,\\,\\Omega,\\,t},  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray}\n\nFor every $t\\in(-\\varepsilon,\\,\\varepsilon)$, let $\\tilde\\Gamma_{\\omega,\\,\\Omega,\\,t}\\in\\mbox{Im}\\,\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$ be the minimal $L^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$-norm solution of the equation \\begin{eqnarray}\\label{eqn:torsion-form_omega-Omega-t_1st-cond_bis_equation}\\bar\\partial\\Gamma = \\bar\\partial\\Gamma_{\\omega_{n-1} + t\\Omega} - B_{\\omega,\\,\\Omega,\\,t}.\\end{eqnarray} This means that, for $t\\in(-\\varepsilon,\\,\\varepsilon)$, we have: $\\bar\\partial\\tilde\\Gamma_{\\omega,\\,\\Omega,\\,t} = \\bar\\partial\\Gamma_{\\omega_{n-1} + t\\Omega} - B_{\\omega,\\,\\Omega,\\,t}$ and  \\begin{eqnarray*}\\tilde\\Gamma_{\\omega,\\,\\Omega,\\,t} = \\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}(\\bar\\partial\\Gamma_{\\omega_{n-1} + t\\Omega} - B_{\\omega,\\,\\Omega,\\,t}) = \\Gamma_{\\omega_{n-1} + t\\Omega} - \\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}B_{\\omega,\\,\\Omega,\\,t}.\\end{eqnarray*} (To get the last equality, we used the fact that $\\Gamma_{\\omega_{n-1} + t\\Omega}\\in\\mbox{Im}\\,\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\subset\\ker\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$.) Hence,  \\begin{eqnarray}\\label{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_1}||\\tilde\\Gamma_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\geq||\\Gamma_{\\omega_{n-1} + t\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}} - ||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}B_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}},  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray}\n\nOn the other hand, the $L^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$-norm minimality of $\\tilde\\Gamma_{\\omega,\\,\\Omega,\\,t}\\in\\mbox{Im}\\,\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$ among the solutions of equation (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond_bis_equation}), one of which is $\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}$ (cf. (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond_bis})), implies that $||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\geq||\\tilde\\Gamma_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$ for each $t\\in(-\\varepsilon,\\,\\varepsilon)$. Together with (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_1}), this yileds: \\begin{eqnarray}\\label{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_2}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}} - ||\\Gamma_{\\omega_{n-1} + t\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}} + ||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}B_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\geq 0,  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray} Note that the l.h.s. term of (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_2}) vanishes at $t=0$, so, as a function of $t\\in(-\\varepsilon,\\,\\varepsilon)$, it achieves its minimum at $t=0$. In particular, its derivative w.r.t. $t$ vanishes at $t=0$. Once again, the assumption $\\Gamma_{\\omega_{n-1}}\\neq0$ (otherwise, $\\omega$ is K\\\"ahler and there is nothing to prove -- see Lemma \\ref{Lem:Kaehler_balanced_torsion-vanishing}) we have made justifies the differentiability. This yields: \\begin{eqnarray}\\label{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_3}\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}} = \\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1} + t\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}} - \\frac{d}{dt}_{|t=0}||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}B_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}.\\end{eqnarray}\n\nMeanwhile, $\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1} + t\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}^2 = 2\\,||\\Gamma_{\\omega_{n-1}}||_\\omega\\,\\frac{d}{dt}_{|t=0}\\,||\\Gamma_{\\omega_{n-1} + t\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$. Thus, using (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_3}), we get:\n\n\\begin{eqnarray}\\label{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_4}\\nonumber\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1} + t\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}^2 & = & 2\\,||\\Gamma_{\\omega_{n-1}}||_\\omega\\,\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}} \\\\\n\\nonumber  &  & + 2\\,||\\Gamma_{\\omega_{n-1}}||_\\omega\\,\\frac{d}{dt}_{|t=0}\\bigg(||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}B_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}^2\\bigg)^{\\frac{1}{2}} \\\\\n& = & \\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}^2 + 2\\,||\\Gamma_{\\omega_{n-1}}||_\\omega\\,M(0),\\end{eqnarray} where we put $$M(t):= \\frac{\\frac{d}{dt}\\,||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}B_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}^2}{2\\,||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}B_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}},  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).$$\n\nNow, letting $A_{\\omega,\\,\\Omega,\\,t} = \\frac{1}{t}\\, B_{\\omega,\\,\\Omega,\\,t}$, we see that $A_{\\omega,\\,\\Omega,\\,0}$ is given by the expression (\\ref{eqn:A_omega-Omega-0_notation}) and we have $$A_{\\omega,\\,\\gamma,\\,t} = \\bigg(\\frac{dP^{(n-1,\\,n-1)}_{\\omega_{n-1} + t\\Omega}}{dt}_{|t=0}\\bigg)(\\omega_{n-1}) + {\\cal O}(t),  \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).$$ Thus, writing $B_{\\omega,\\,\\Omega,\\,t} = t\\,A_{\\omega,\\,\\Omega,\\,t}$ in the expression for $M(t)$, we get for $t\\in(0,\\,\\varepsilon)$: \\begin{eqnarray*}M(t)= \\frac{\\frac{d}{dt}\\,(t^2\\,||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}A_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}^2)}{2t\\,||\\Delta^{''-1}_{\\omega + t\\gamma}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}A_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}} & = & ||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}A_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}} \\\\\n  & + & \\frac{t}{2}\\,\\frac{\\frac{d}{dt}\\,||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}A_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}^2}{||\\Delta^{''-1}_{\\gamma_{\\omega,\\,\\Omega,\\,t}}\\bar\\partial^\\star_{\\gamma_{\\omega,\\,\\Omega,\\,t}}A_{\\omega,\\,\\Omega,\\,t}||_{\\gamma_{\\omega,\\,\\Omega,\\,t}}}.\\end{eqnarray*} Hence, $M(0)= ||\\Delta^{''-1}_{\\omega}\\bar\\partial^\\star_{\\omega}A_{\\omega,\\,\\Omega,\\,0}||_{\\omega}$. Together with (\\ref{eqn:torsion-form_omega-Omega-t_1st-cond_bis_proof_4}), this proves (\\ref{eqn:1st-derivative_Gamma_omega_n-1+tOmega}). \\hfill $\\Box$\n\n\n\\vspace{3ex}\n\nSince $G(\\omega_{n-1} + t\\Omega) = ||\\Gamma_{\\omega_{n-1} + t\\Omega}||^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$ for every $t\\in(-\\varepsilon,\\,\\varepsilon)$ (cf. formula (\\ref{eqn:G_energy-functional_balanced})), the above Lemma \\ref{Lem:1st-derivative_Gamma_omega_n-1+tOmega} reduces the computation of $\\frac{d}{dt}_{|t=0} G(\\omega_{n-1} + t\\Omega) = \\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1} + t\\Omega}||^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$ to the computation of $\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||^2_{\\gamma_{\\omega,\\,\\Omega,\\,t}}$. This last derivative at $t=0$ is computed in the following\n\n\\begin{Lem}\\label{Lem:1st-variation_Gt-tilde} Let $X$ be a compact complex manifold with $\\mbox{dim}_\\C X=n$. Suppose there exists a balanced metric $\\omega$ on $X$.\n\n  For any real $(n-1,\\,n-1)$-form $\\Omega\\in\\ker(\\bar\\partial)$ and for a real $t$ varying in a neighbourhood $(-\\varepsilon,\\,\\varepsilon)$ of $0$ so small that $\\omega_{n-1,\\,n-1} + t\\Omega>0$, if we put $\\gamma_t:=\\gamma_{\\omega,\\,\\Omega,\\,t}$ and $\\rho:=\\frac{d\\gamma_t}{dt}_{|t=0}$, we have \\begin{eqnarray*}\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||^2_{\\gamma_t} & = & \\int\\limits_X\\eta_{\\omega,\\,\\Omega}\\wedge\\star_\\omega\\overline{\\Gamma}_{\\omega_{n-1}} + \\int\\limits_X\\Gamma_{\\omega_{n-1}}\\wedge\\star_\\omega\\bigg(\\bar\\eta_{\\omega,\\,\\Omega} + [\\Lambda_\\omega,\\,\\rho\\wedge\\cdot]\\,\\overline{\\Gamma}_{\\omega_{n-1}}\\bigg).\\end{eqnarray*}\n\n\\end{Lem} \n\n\\noindent {\\it Proof.} Setting $\\alpha_t:=\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}$, the quantity that we will differentiate at $t=0$ reads: \\begin{eqnarray*}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||^2_{\\gamma_t} & = & \\int\\limits_X|\\alpha_t|^2_{\\gamma_t}\\,dV_{\\gamma_t} = \\int\\limits_X\\alpha_t\\wedge\\star_{\\gamma_t}\\bar\\alpha_t, \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray*} Hence \\begin{eqnarray}\\label{eqn:1st-variation_Gt-tilde_proof_1}\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||^2_{\\gamma_t} = \\int\\limits_X\\frac{d\\alpha_t}{dt}_{|t=0}\\wedge\\star_{\\gamma_0}\\bar\\alpha_0 + \\int\\limits_X\\alpha_0\\wedge\\frac{d}{dt}_{|t=0}(\\star_{\\gamma_t}\\bar\\alpha_t).\\end{eqnarray}\n\nNote that $d\\alpha_t\/dt = \\eta_{\\omega,\\,\\Omega}$. Meanwhile, expanding $\\gamma_t$ at $t=0$ and using the fact that $\\gamma_0=\\omega$ (see (\\ref{eqn:root_balanced})), we get $\\gamma_t = \\omega + t\\rho  + O(t^2)$ for $t\\in(-\\varepsilon,\\,\\varepsilon)$ if $\\varepsilon>0$ is small enough. This implies that $(d\/dt)_{|t=0}(\\star_{\\gamma_t}\\bar\\alpha_t) = (d\/dt)_{|t=0}(\\star_{\\omega + t\\rho}\\bar\\alpha_t)$. Using Lemma \\ref{Lem:1st-variation_Hodge-star} to compute this last derivative at $t=0$, we see that (\\ref{eqn:1st-variation_Gt-tilde_proof_1}) translates to: \\begin{eqnarray*}\\frac{d}{dt}_{|t=0}||\\Gamma_{\\omega_{n-1}} + t\\,\\eta_{\\omega,\\,\\Omega}||^2_{\\gamma_t} = \\int\\limits_X\\eta_{\\omega,\\,\\Omega}\\wedge\\star_\\omega\\overline{\\Gamma}_{\\omega_{n-1}} + \\int\\limits_X\\Gamma_{\\omega_{n-1}}\\wedge\\star_\\omega\\bigg(\\frac{d\\bar\\alpha_t}{dt}_{|t=0} + [\\Lambda_\\omega,\\,\\rho\\wedge\\cdot]\\,\\overline{\\Gamma}_{\\omega_{n-1}}\\bigg),\\end{eqnarray*} which proves the contention since $(d\\bar\\alpha_t\/dt)_{|t=0} = \\bar\\eta_{\\omega,\\,\\Omega}$.  \\hfill $\\Box$\n\n\\vspace{3ex}\n\nPutting Lemmas \\ref{Lem:1st-derivative_Gamma_omega_n-1+tOmega} and \\ref{Lem:1st-variation_Gt-tilde} together, we get the following\n\n\\begin{Cor}\\label{Cor:1st-variation_Gt-tilde_final} Under the assumptions of Lemma \\ref{Lem:1st-variation_Gt-tilde}, we have \\begin{eqnarray}\\label{eqn:1st-variation_Gt-tilde_final}\\nonumber\\frac{d}{dt}_{|t=0} G(\\omega_{n-1} + t\\Omega)  & = & \\langle\\langle\\eta_{\\omega,\\,\\Omega},\\,\\Gamma_{\\omega_{n-1}}\\rangle\\rangle_\\omega + \\langle\\langle\\Gamma_{\\omega_{n-1}},\\,\\eta_{\\omega,\\,\\Omega} + [\\Lambda_\\omega,\\,\\rho\\wedge\\cdot]\\,\\Gamma_{\\omega_{n-1}}\\rangle\\rangle_\\omega \\\\\n & + & 2\\,||\\Gamma_{\\omega_{n-1}}||_\\omega\\,||\\Delta_\\omega^{''-1}\\bar\\partial^\\star_\\omega A_{\\omega,\\,\\Omega,\\,0}||_\\omega, \\end{eqnarray} where $A_{\\omega,\\,\\Omega,\\,0}$ is defined by formula (\\ref{eqn:A_omega-Omega-0_notation}).\n\nIn particular, if we choose $\\Omega = \\omega_{n-1}$, the above formula specialises to \\begin{eqnarray}\\label{eqn:1st-variation_Gt-tilde_final_special}\\frac{d}{dt}_{|t=0} G(\\omega_{n-1} + t\\omega_{n-1})  = \\frac{n+1}{n-1}\\,||\\Gamma_{\\omega_{n-1}}||^2_\\omega.\\end{eqnarray}\n\n\\end{Cor}\n\n\\noindent {\\it Proof.} Only formula (\\ref{eqn:1st-variation_Gt-tilde_final_special}) still needs a proof. (The definition of the Hodge star operator $\\star_\\omega$ was used to get the first line of (\\ref{eqn:1st-variation_Gt-tilde_final}) from Lemma \\ref{Lem:1st-variation_Gt-tilde}.) \n\nWhen $\\Omega = \\omega_{n-1}$, $\\eta_{\\omega,\\,\\Omega} = \\Gamma_{\\omega_{n-1}}$ (see formulae (\\ref{eqn:eta_omega-Omega-t_def}) and (\\ref{eqn:system_balanced_torsion-form})), while $\\rho = \\frac{1}{n-1}\\,\\omega$. This last equality follows from $\\rho = \\frac{d\\gamma_t}{dt}_{|t=0}$ after noting that $(\\gamma_t)_{n-1} = (1+t)\\,\\omega_{n-1}$, hence $\\gamma_t = (1+t)^{\\frac{1}{n-1}}\\,\\omega$, when $\\Omega = \\omega_{n-1}$.\n\nThus, in this case we get: \\begin{eqnarray*}[\\Lambda_\\omega,\\,\\rho\\wedge\\cdot]\\,\\overline{\\Gamma}_{\\omega_{n-1}} = \\frac{1}{n-1}\\,[\\Lambda_\\omega,\\,\\omega\\wedge\\cdot]\\,\\overline{\\Gamma}_{\\omega_{n-1}} = -\\frac{n-3}{n-1}\\,\\overline{\\Gamma}_{\\omega_{n-1}},\\end{eqnarray*} since $\\overline{\\Gamma}_{\\omega_{n-1}}$ is of bidegree $(n-2,\\,n-1)$ and it is well known that $[\\Lambda_\\omega,\\,\\omega\\wedge\\cdot] = (n-p-q)\\,\\mbox{Id}$ on $(p,\\,q)$-forms (see e.g. [Voi02, Lemma 6.19]). \n\nSince $||\\Gamma_{\\omega_{n-1}}||^2_\\omega = \\int_X\\Gamma_{\\omega_{n-1}}\\wedge\\star_\\omega\\overline{\\Gamma}_{\\omega_{n-1}}$, (\\ref{eqn:1st-variation_Gt-tilde_final_special}) follows at once from these remarks, from (\\ref{eqn:1st-variation_Gt-tilde_final}) and from the following observation whose first equality is a consequence of (\\ref{eqn:A_omega-Omega-0_notation}): \\begin{eqnarray*}A_{\\omega,\\,\\omega_{n-1},\\,0} = \\bigg(\\frac{dP^{(n-1,\\,n-1)}_{(1+t)\\,\\omega_{n-1}}}{dt}_{|t=0}\\bigg)(\\omega_{n-1}) = 0 \\end{eqnarray*} since $P^{(n-1,\\,n-1)}_{(1+t)\\,\\omega_{n-1}}(\\omega_{n-1}) =  P^{(n-1,\\,n-1)}_{\\omega_{n-1}}(\\omega_{n-1})$ for every $t$ close enough to $0$. (See (\\ref{eqn:P^n-1n-1_omega_n-1_lambda}) for this last fact.)     \\hfill $\\Box$\n\n\n\n\\vspace{3ex}\n\nNote that formula (\\ref{eqn:1st-variation_Gt-tilde_final_special}) coincides with formula (\\ref{eqn:rescaling_G-critical_1+t}) of Lemma \\ref{Lem:rescaling_G-critical}. Another conclusion of this study is the following result that was already proved in a different way in (\\ref{eqn:critical-points_G-Kaehler}) of Lemma \\ref{Lem:rescaling_G-critical}.\n\n\n\n\\begin{Cor}\\label{Cor:crit-points_balanced} Let $X$ be a compact complex $n$-dimensional balanced manifold. \n\n  Then, for every balanced metric $\\omega$ on $X$, $\\omega_{n-1}$ is a {\\bf critical point} for the functional $G:{\\cal B}\\longrightarrow[0,\\,+\\infty)$ if and only if $\\omega$ is a {\\bf K\\\"ahler} metric.\n\n\n\\end{Cor}  \n\n\\noindent {\\it Proof.} If a balanced metric $\\omega_{n-1}$ is a critical point for $G$, then $(d_{\\omega_{n-1}}G)(\\Omega) = \\frac{d}{dt}_{|t=0}G(\\omega_{n-1} + t\\,\\Omega) = 0$ for every real $(n-1,\\,n-1)$-form $\\Omega$ lying in $\\ker(\\bar\\partial)$. In particular, choosing $\\Omega:=\\omega_{n-1}$ (which is allowed since $\\bar\\partial\\omega_{n-1}=0$), the vanishing of $\\frac{d}{dt}_{|t=0}G(\\omega_{n-1} + t\\,\\omega_{n-1})$ implies the vanishing of $\\Gamma_{\\omega_{n-1}}$ thanks to (\\ref{eqn:1st-variation_Gt-tilde_final_special}). On the other hand, the equality $\\Gamma_{\\omega_{n-1}}=0$ implies that $\\omega$ is K\\\"ahler by Lemma \\ref{Lem:Kaehler_balanced_torsion-vanishing}.\n\nConversely, if $\\omega$ is a K\\\"ahler metric, $\\omega_{n-1}$ is a minimiser, hence a critical point, for the functional $G$ by Lemma \\ref{Lem:vanishing-G-Kaehler}. \n\nOne can also argue that $\\Gamma_{\\omega_{n-1}}=0$ if $\\omega$ is K\\\"ahler (again by Lemma \\ref{Lem:Kaehler_balanced_torsion-vanishing}), so (\\ref{eqn:1st-variation_Gt-tilde_final}) of Corollary \\ref{Cor:1st-variation_Gt-tilde_final} implies that $\\frac{d}{dt}_{|t=0}G(\\omega_{n-1} + t\\,\\Omega) = 0$ for every real $(n-1,\\,n-1)$-form $\\Omega$ lying in $\\ker(\\bar\\partial)$. This means that $\\omega_{n-1}$ is a critical point for $G$.  \\hfill $\\Box$\n\n\n\\vspace{3ex}\n\nIn view of a future study of the existence of critical points for the functional $G$, we now give an explicit formula for the $(1,\\,1)$-form $\\rho=\\frac{d\\gamma_t}{dt}_{|t=0}$ featuring in (\\ref{eqn:1st-variation_Gt-tilde_final}).\n\n\\begin{Lem}\\label{Lem:balanced_rho_formula}  Let $\\omega$ be a Hermitian metric and let $\\Omega$ be a real $(n-1,\\,n-1)$-form on a compact complex manifold $X$ with $\\mbox{dim}_\\C X=n$.\n  For a real $t$ varying in a neighbourhood $(-\\varepsilon,\\,\\varepsilon)$ of $0$ so small that $\\omega_{n-1} + t\\Omega>0$, we let $\\gamma_t:=\\gamma_{\\omega,\\,\\Omega,\\,t}$ be the unique positive definite $(1,\\,1)$-form on $X$ such that $(\\gamma_t)_{n-1} = \\omega_{n-1} + t\\Omega$ and we set $\\rho:=\\frac{d\\gamma_t}{dt}_{|t=0}$.\n\nThen, the following formula holds: \\begin{eqnarray}\\label{eqn:balanced_rho_formula}\\rho = \\frac{1}{n-1}\\,\\Lambda_\\omega(\\star_\\omega\\Omega)\\,\\omega - \\star_\\omega\\Omega.\\end{eqnarray}\n\n\\end{Lem} \n\n\\noindent {\\it Proof.} Since the result is of a pointwise nature, we fix an arbitrary point $x\\in X$ and local coordinates $z_1,\\dots , z_n$ centred at $x$ such that \\begin{eqnarray*}\\omega(x) = \\sum\\limits_{j=1}^n idz_j\\wedge d\\bar{z}_j  \\hspace{3ex} \\mbox{and} \\hspace{3ex} \\Omega(x) = \\sum\\limits_{j=1}^n\\Omega_j\\,\\widehat{idz_j\\wedge d\\bar{z}_j},\\end{eqnarray*} where, for every $j$, $\\Omega_j\\in\\R$ and $\\widehat{idz_j\\wedge d\\bar{z}_j}$ is the $(n-1,\\,n-1)$-form given as the product of all the $(1,\\,1)$-forms $idz_k\\wedge d\\bar{z}_k$ with $k\\neq j$. Then \\begin{eqnarray*}(\\omega_{n-1} + t\\Omega)(x) = \\sum\\limits_{j=1}^n(1 + t\\Omega_j)\\,\\widehat{idz_j\\wedge d\\bar{z}_j}, \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon),\\end{eqnarray*} and the $(n-1)$-st root of this form is given by \\begin{eqnarray*}\\gamma_t(x) = \\sum\\limits_{j=1}^n\\frac{\\Pi_{k=1}^n(1 + t\\Omega_k)^{\\frac{1}{n-1}}}{1 + t\\Omega_j}\\,idz_j\\wedge d\\bar{z}_j, \\hspace{5ex} t\\in(-\\varepsilon,\\,\\varepsilon).\\end{eqnarray*}\n\nDifferentiating with respect to $t$ and then evaluating at $t=0$, we get: \n\n\\begin{eqnarray*}\\frac{d\\gamma_t}{dt}_{|t=0}(x) & = & - \\sum\\limits_{j=1}^n\\frac{\\Omega_j\\,\\Pi_{k=1}^n(1 + t\\Omega_k)^{\\frac{1}{n-1}}}{(1 + t\\Omega_j)^2}_{\\bigg|t=0}\\,idz_j\\wedge d\\bar{z}_j \\\\\n & + & \\frac{1}{n-1}\\,\\sum\\limits_{j=1}^n\\frac{\\sum\\limits_{k=1}^n(1 + t\\Omega_1)^{\\frac{1}{n-1}}\\cdots\\widehat{(1 + t\\Omega_k)^{\\frac{1}{n-1}}}\\cdots(1 + t\\Omega_n)^{\\frac{1}{n-1}}\\,(1 + t\\Omega_k)^{\\frac{2-n}{n-1}}\\,\\Omega_k}{1 + t\\Omega_j}_{\\bigg|t=0}\\,idz_j\\wedge d\\bar{z}_j \\\\\n & = & -\\sum\\limits_{j=1}^n\\Omega_j\\,idz_j\\wedge d\\bar{z}_j + \\frac{1}{n-1}\\,\\sum\\limits_{j=1}^n\\bigg(\\sum\\limits_{k=1}^n\\Omega_k\\bigg)\\,idz_j\\wedge d\\bar{z}_j \\\\\n & = & \\frac{1}{n-1}\\,\\bigg(\\sum\\limits_{k=1}^n\\Omega_k\\bigg)\\,\\omega(x) - \\sum\\limits_{j=1}^n\\Omega_j\\,idz_j\\wedge d\\bar{z}_j.\\end{eqnarray*} This proves (\\ref{eqn:balanced_rho_formula}) since $(\\star_\\omega\\Omega)(x) = \\sum_{j=1}^n\\Omega_j\\,idz_j\\wedge d\\bar{z}_j$ and $\\Lambda_\\omega(\\star_\\omega\\Omega)(x) = \\sum_{k=1}^n\\Omega_k$.  \\hfill $\\Box$\n\n\n\\section{Simultaneous study of SKT and balanced metrics}\\label{section:simultaneous_SKT-bal}\n\nHaving separately studied SKT and balanced metrics in the previous sections, we indicate in this short section a possible simultaneous variational approach to these two classes of metrics in vew of a future attack on the problem proposed by Fino and Vezzoni in [FV15] asking whether or not the existence of both kinds of metrics on a compact complex manifold implies the existence of a K\\\"ahler metric.\n\n\\subsection{An SKT-balanced observation}\\label{subsection:SKT-bal_obs} We now run again the argument of [Pop15, Proposition 1.1], made even shorter in [DP20, Proposition 2.6.], but with two different metrics. The next observation will be the starting point of the subsequent considerations.\n\n\\begin{Prop}\\label{Prop:SKT-bal_obs} Let $X$ be a compact complex manifold with $\\mbox{dim}_\\C X =n$. For any {\\bf SKT} metric $\\omega$ (if any) and any {\\bf balanced} metric $\\gamma$ (if any) on $X$, we have: \\begin{equation}\\label{eqn:SKT-bal_obs}\\langle\\langle\\bar\\partial\\omega,\\,\\bar\\partial\\gamma\\rangle\\rangle_\\gamma = 0,\\end{equation} where $\\langle\\langle\\cdot\\,\\,,\\,\\,\\cdot\\rangle\\rangle_\\gamma$ is the $L^2$ inner product induced by the metric $\\gamma$. \n\n\\end{Prop}\n\n\\noindent {\\it Proof.} The {\\it SKT} assumption on $\\omega$ translates to any of the following equivalent properties: \\begin{eqnarray}\\label{eqn:pluriclosed-equiv}\\partial\\bar\\partial\\omega=0 \\Longleftrightarrow \\partial\\omega\\in\\ker\\bar\\partial \\Longleftrightarrow \\star_\\omega(\\partial\\omega)\\in\\ker\\partial^{\\star}_\\omega,\\end{eqnarray}\n\n\\noindent where the last equivalence follows from the standard formula $\\partial^{\\star}_\\omega=-\\star_\\omega\\bar\\partial\\star_\\omega$ involving the Hodge-star isomorphism $\\star_{\\omega}:\\Lambda^{p,\\,q}T^{\\star}X\\rightarrow \\Lambda^{n-q,\\,n-p}T^{\\star}X$ defined by $\\omega$ for arbitrary $p,q=0,\\dots , n$.\n\n\n Meanwhile, the {\\it balanced} assumption on $\\gamma$ translates to any of the following equivalent properties: \\begin{eqnarray*}\\label{eqn:bal-equiv}d\\gamma^{n-1}=0 \\Longleftrightarrow \\partial\\gamma^{n-1}=0 \\Longleftrightarrow \\gamma^{n-2}\\wedge\\partial\\gamma = 0   \\Longleftrightarrow \\partial\\gamma \\,\\,\\mbox{is $\\gamma$-primitive}.\\end{eqnarray*}\n\nNow, recall the following standard formula (cf. e.g. [Voi02, Proposition 6.29, p. 150]) for the Hodge star operator $\\star = \\star_\\omega$ of any Hermitian metric $\\omega$ applied to {\\it primitive} forms $v$ of arbitrary bidegree $(p, \\, q)$: \\begin{eqnarray}\\label{eqn:prim-form-star-formula-gen}\\star\\, v = (-1)^{k(k+1)\/2}\\, i^{p-q}\\, \\frac{\\omega^{n-p-q}\\wedge v}{(n-p-q)!}, \\hspace{2ex} \\mbox{where}\\,\\, k:=p+q.\\end{eqnarray}\n\n Thus, since the $(2,\\,1)$-form $\\partial\\gamma$ is $\\gamma$-primitive when $\\gamma$ is balanced, the standard formula (\\ref{eqn:prim-form-star-formula-gen}) yields: \\begin{eqnarray}\\label{eqn:consequence_balanced}\\star_\\gamma(\\partial\\gamma) = i\\,\\frac{\\gamma^{n-3}}{(n-3)!}\\wedge\\partial\\gamma = \\frac{i}{(n-2)!}\\,\\partial\\gamma^{n-2}\\in\\mbox{Im}\\,\\partial.\\end{eqnarray}\n\n\\vspace{1ex}\n\nFrom the subspaces $\\ker\\partial^{\\star}_\\omega$ and $\\mbox{Im}\\,\\partial$ of $C^\\infty_{n-1,\\,n-2}(X,\\,\\C)$ being $L^2_\\omega$-orthogonal and from (\\ref{eqn:pluriclosed-equiv}) and (\\ref{eqn:consequence_balanced}) we infer the first relation below: \\begin{eqnarray*}\\star_\\omega(\\partial\\omega)\\perp_{L^2_\\omega}\\star_\\gamma(\\partial\\gamma) & \\iff & \\langle\\langle\\star_\\gamma(\\partial\\gamma),\\,\\star_\\omega(\\partial\\omega)\\rangle\\rangle_\\omega = 0 \\iff \\int\\limits_X\\bigg\\langle\\star_\\gamma(\\partial\\gamma),\\,\\star_\\omega(\\partial\\omega)\\bigg\\rangle_\\omega\\,dV_\\omega = 0 \\\\\n  & \\iff & \\int\\limits_X\\star_\\gamma(\\partial\\gamma)\\wedge\\star_\\omega\\star_\\omega(\\bar\\partial\\omega) = 0  \\iff \\int\\limits_X\\bar\\partial\\omega\\wedge\\star_\\gamma\\overline{(\\bar\\partial\\gamma)} = 0 \\\\\n  & \\iff & \\int\\limits_X\\bigg\\langle\\bar\\partial\\omega,\\,\\bar\\partial\\gamma\\bigg\\rangle_\\gamma\\,dV_\\gamma = 0 \\iff \\langle\\langle\\bar\\partial\\omega,\\,\\bar\\partial\\gamma\\rangle\\rangle_\\gamma = 0.\\end{eqnarray*}\n\nThis proves the contention.  \\hfill $\\Box$\n\n\n\\hspace{2ex}\n\nNote that, if $\\omega=\\gamma$, we deduce from Proposition \\ref{Prop:SKT-bal_obs} the well-known fact that, if a Hermitian metric $\\omega$ is at once SKT and balanced, it is K\\\"ahler. We even get the following slight generalisation of this observation.\n\n\n\\begin{Cor}\\label{Cor:SKT-bal_obs} Let $X$ be a compact complex manifold with $\\mbox{dim}_\\C X =n$. Suppose there exists an {\\bf SKT} metric $\\omega$ and a {\\bf balanced} metric $\\gamma$ on $X$ such that $$\\bar\\partial\\omega = \\bar\\partial\\gamma.$$ Then, both $\\omega$ and $\\gamma$ are {\\bf K\\\"ahler}.\n\n\\end{Cor}  \n\n\\noindent {\\it Proof.} If $\\bar\\partial\\omega = \\bar\\partial\\gamma$, (\\ref{eqn:SKT-bal_obs}) reads $||\\bar\\partial\\omega||^2_\\gamma = ||\\bar\\partial\\gamma||^2_\\gamma=0$. Hence, $\\bar\\partial\\omega = \\bar\\partial\\gamma =0$, translating the fact that $\\omega$ and $\\gamma$ are K\\\"ahler.  \\hfill $\\Box$\n\n\n\\vspace{2ex}\n\nIt also follows from Proposition \\ref{Prop:SKT-bal_obs} that, whenever $\\omega$ is an {\\it SKT metric} and $\\gamma$ is a {\\it balanced metric}, we have: \\begin{equation}\\label{eqn:SKT-bal_difference-orth}||\\bar\\partial\\omega - \\bar\\partial\\gamma||^2_\\gamma = ||\\bar\\partial\\omega||^2_\\gamma + ||\\bar\\partial\\gamma||^2_\\gamma .\\end{equation}\n\n\n\n\n\n\\subsection{The SKT-balanced functional}\\label{subsection:functional_skt-balanced}\n\nWe now introduce a functional depending on both SKT and balanced metrics that are assumed to exist on a given compact complex manifold.\n\n\t\n\n\n\\begin{Def}\\label{Def:functional_skt-balanced} Let $X$ be a compact complex manifold with $\\mbox{dim}_\\C X=n$. Suppose both {\\bf SKT} and {\\bf balanced} metrics\n exist on $X$. For a SKT metric $\\omega$ and a $(\\gamma)_{n-1}$ of any balanced metric $\\gamma$, we define the following {\\bf energy functional}:\n  \n \\begin{equation}\\label{eqn:H_energy-functional_skt-balanced}H : {\\cal S}\\times{\\cal B} \\to [0,\\,+\\infty), \\hspace{3ex} H(\\omega,\\,\\gamma_{n-1}) = ||\\Lambda_\\omega(\\partial\\omega)||^2_\\gamma,\\end{equation}\n\n\\noindent where $||\\,\\,\\,||_\\gamma$ is the $L^2$-norm induced by $\\gamma$. \n\n\\end{Def}   \n\n\nWe have: \\begin{equation}\\label{eqn:H_energy-functional_skt-balanced_bis}H(\\omega,\\,\\gamma_{n-1}) = \\int\\limits_X|\\Lambda_\\omega(\\partial\\omega)|^2_\\gamma\\,dV_\\gamma = \\int\\limits_X\\Lambda_\\omega(\\partial\\omega)\\wedge\\star_\\gamma\\Lambda_\\omega(\\bar\\partial\\omega) = i\\int\\limits_X\\Lambda_\\omega(\\partial\\omega)\\wedge\\Lambda_\\omega(\\bar\\partial\\omega)\\wedge\\gamma_{n-1}\\end{equation} for every SKT metric $\\omega$ and every balanced metric $\\gamma$.\n\n\\vspace{2ex}\n\nThe first trivial observation that justifies the introduction of the functional $H$ is the following.\n\n\n\\begin{Lem}\\label{Lem:vanishing-H-omega-Kaehler} Let $\\omega_0$ be an SKT metric and let $\\gamma_0$ be a balanced metric on $X$. For any $\\omega\\in{\\cal S}$ the following equivalence holds: \\begin{equation}\\label{eqn:vanishing-G-omega-Kaehler}\\omega \\hspace{1ex} \\mbox{is K\\\"ahler} \\iff H(\\omega,\\,(\\gamma_0)_{n-1})=0.\\end{equation}\n\n\n\\end{Lem} \n\n\\noindent {\\it Proof.} The condition $H(\\omega,\\,(\\gamma_0)_{n-1})=0$ is equivalent to $\\Lambda_\\omega(\\partial\\omega)=0$, which in turn is equivalent to $\\partial\\omega$ being primitive w.r.t. $\\omega$. This last property translates to $\\omega^{n-2}\\wedge\\partial\\omega = 0$ (since $\\partial\\omega$ is a $3$-form), which amounts to $\\partial\\omega^{n-1} = 0$. This means that $\\omega$ satisfies the balanced condition. Being already SKT, the metric $\\omega$ is balanced if and only if it is K\\\"ahler. (See Corollary \\ref{Cor:SKT-bal_obs}.) \\hfill $\\Box$\n\n\n\n\\vspace{3ex}\n\nThe idea is to use the balanced metric $\\gamma_{n-1}$ as a reference metric and to vary the SKT metric $\\omega$. Note that, for any Hermitian metric $\\omega$ and every real $\\lambda>0$, we have $\\Lambda_{\\lambda\\omega} = (1\/\\lambda)\\,\\Lambda_\\omega$ as operators acting in any degree, hence \\begin{eqnarray*}H(\\lambda\\omega,\\,(\\gamma_0)_{n-1}) = H(\\omega,\\,(\\gamma_0)_{n-1}), \\hspace{5ex} \\lambda>0,\\end{eqnarray*} showing that the functional $H$ is constant in its first variable along the rays of the cone ${\\cal S}$ whenever its second variable $\\gamma_{n-1}$ has been fixed.\n\n\nWe now to compute the first variation of $H$ in the $\\omega$ variable. We fix an SKT metric $\\omega$ on $X$ and we vary it in $\\cal S$ along the path $\\omega + t\\eta$, where $\\eta\\in C^\\infty_{1,\\,1}(X,\\,\\R)$ is a fixed real $(1,\\,1)$-form. We will be considering reals $t\\in(-\\varepsilon,\\,\\varepsilon)$ with $\\varepsilon>0$ small enough to ensure that $\\omega + t\\eta>0$ (i.e. $\\omega + t\\eta$ is a, necessarily SKT, metric on $X$).\n\nUsing (\\ref{eqn:H_energy-functional_skt-balanced_bis}), we get: \\begin{eqnarray}\\label{eqn:H_energy-functional_skt-balanced_computation_1}\\nonumber\\frac{d}{dt}\\bigg|_{t=0} H(\\omega + t\\eta,\\,\\gamma_{n-1}) & = & i\\,\\frac{d}{dt}\\bigg|_{t=0}\\int\\limits_X\\Lambda_{\\omega + t\\eta}(\\partial\\omega + t\\,\\partial\\eta)\\wedge\\Lambda_{\\omega + t\\eta}(\\bar\\partial\\omega + t\\,\\bar\\partial\\eta)\\wedge\\gamma_{n-1}  \\\\\n  \\nonumber & = & i\\int\\limits_X\\frac{d}{dt}\\bigg|_{t=0}\\bigg(\\Lambda_{\\omega + t\\eta}(\\partial\\omega + t\\,\\partial\\eta)\\bigg)\\wedge\\Lambda_\\omega(\\bar\\partial\\omega)\\wedge\\gamma_{n-1} \\\\\n  \\nonumber & + & i\\int\\limits_X\\Lambda_\\omega(\\partial\\omega)\\wedge\\frac{d}{dt}\\bigg|_{t=0}\\bigg(\\Lambda_{\\omega + t\\eta}(\\bar\\partial\\omega + t\\,\\bar\\partial\\eta)\\bigg)\\wedge\\gamma_{n-1}.\\end{eqnarray}\n\nWe now apply Lemma \\ref{Lem:1st-variation_trace_pq} to the computation of the derivatives at $t=0$ of the two factors of the shape $\\Lambda_{\\omega + t\\eta}(\\cdot)$. Formula (\\ref{eqn:1st-variation_trace_pq}) yields: \\begin{eqnarray*}\\frac{d}{dt}\\bigg|_{t=0}\\bigg(\\Lambda_{\\omega + t\\eta}(\\partial\\omega + t\\,\\partial\\eta)\\bigg) & = & \\Lambda_\\omega(\\partial\\eta) - (\\eta\\wedge\\cdot)^\\star_\\omega(\\partial\\omega) \\\\\n \\frac{d}{dt}\\bigg|_{t=0}\\bigg(\\Lambda_{\\omega + t\\eta}(\\bar\\partial\\omega + t\\,\\bar\\partial{\\eta})\\bigg) & = & \\Lambda_\\omega(\\bar\\partial\\eta) - (\\eta\\wedge\\cdot)^\\star_\\omega(\\bar\\partial\\omega).\\end{eqnarray*}\n\n\\begin{Lem}\\label{Lem:1st-variation_H_omega} Let $X$ be a compact complex manifold with $\\mbox{dim}_\\C X=n$. Suppose there exists an SKT metric $\\omega$ and a balanced metric $\\gamma$ on $X$.\n\n  For any real $(1,\\,1)$-form  $\\eta\\in C^\\infty_{1,\\,1}(X,\\,\\R)$, and for a real $t$ varying in a small neighbourhood of $0$ such that $\\omega + t\\eta>0$, we have \\begin{eqnarray}\\label{eqn:1st-variation_G_omega}\\nonumber\\frac{d}{dt}\\bigg|_{t=0} H(\\omega + t\\eta,\\,\\gamma_{n-1}) & = & 2\\,\\mbox{Re}\\,\\bigg(i\\int\\limits_X\\Lambda_\\omega(\\partial\\eta)\\wedge\\Lambda_\\omega(\\bar\\partial\\omega)\\wedge\\gamma_{n-1}\\bigg) \\\\\n    & - & 2\\,\\mbox{Re}\\,\\bigg(i\\int\\limits_X(\\eta\\wedge\\cdot)^\\star_\\omega(\\partial\\omega)\\wedge\\Lambda_\\omega(\\bar\\partial\\omega)\\wedge\\gamma_{n-1}\\bigg).\\end{eqnarray}\n\n\\end{Lem}    \n\n\n\\vspace{3ex}\n\nNote that, if we take $\\eta=\\omega$, we get $(\\eta\\wedge\\cdot)^\\star_\\omega = \\Lambda_\\omega$, so the right-hand side term of (\\ref{eqn:1st-variation_G_omega}) vanishes. This shows that $(d\/dt)_{t=0}H((1+t)\\omega,\\,\\gamma_{n-1}) = 0$, in line with the above observation that $H$ is constant in the first variable along the rays $\\{\\lambda\\omega\\,\\mid\\,\\lambda>0\\}$ of the cone ${\\cal S}$ for any fixed second variable $\\gamma_{n-1}$.  \n\n\n\n\\vspace{3ex}\n\n\n\n\\noindent {\\bf References.} \\\\\n\n\\vspace{1ex}\n\n\\noindent[AB95]\\, L. Alessandrini, G. Bassanelli --- {\\it Modifications of compact balanced manifolds} --- C. R. Acad. Sci. Paris {\\bf 320} (1995), 1517-1522.\n\n\\vspace{1ex}\n\n\n\\noindent [BP18]\\, H. Bellitir, D. Popovici --- {\\it Positivity Cones under Deformations of Complex Structures} --- Riv. Mat. Univ. Parma, Vol. {\\bf 9} (2018), 133-176.\n\n\n\n\\vspace{1ex}\n\n\n\\noindent [Dem97]\\, J.-P. Demailly --- {\\it Complex Analytic and Algebraic Geometry} --- http:\/\/www-fourier.ujf-grenoble.fr\/~demailly\/books.html\n\n\\vspace{1ex}\n\n\\noindent [DP20]\\, S. Dinew, D. Popovici --- {\\it A Generalised Volume Invariant for Aeppli Cohomology Classes of Hermitian-Symplectic Metrics} ---  Adv. Math. {\\bf393} (2021), Paper No. 108056, 46 pp.\n\n\\vspace{1ex}\n\\noindent[FPS04] A. Fino, M. Parton, S. Salamon --- {\\it Families of strong KT structures\nin six dimensions}--- Comment. Math. Helv. {\\bf79} (2004), 317-340.\n\n\n\\vspace{1ex}\n\n\\noindent [FV15]\\, A. Fino, L. Vezzoni --- {\\it Special Hermitian metrics on compact solvmanifolds} --- J. Geom.\nPhys. {\\bf91} (2015), 40-53.\n\n\\vspace{1ex}\n\n\\noindent [Gau77a]\\, P. Gauduchon --- {\\it Le th\\'eor\\`eme de l'excentricit\\'e nulle} --- C. R. Acad. Sci. Paris, S\\'er. A, {\\bf 285} (1977), 387-390.\n\n\n\\vspace{1ex}\n\n\n\\noindent [Gau77b]\\, P. Gauduchon --- {\\it Fibr\\'es hermitiens \\`a endomorphisme de Ricci non n\\'egatif} --- Bull. Soc. Math. France {\\bf 105} (1977) 113-140.\n\n\\vspace{1ex}\n\n\n\\noindent [IP12]\\, S. Ivanov, G. Papadopoulos --- {\\it Vanishing Theorems on $(l\/k)$-strong K\\\"ahler Manifolds with Torsion} --- arXiv e-print DG 1202.6470v1.\n\n\n\n\\vspace{1ex}\n\n\\noindent [Kod86]\\, K. Kodaira --- {\\it Complex Manifolds and Deformations of Complex Structures} --- Grundlehren der Math. Wiss. {\\bf 283}, Springer (1986).\n\n\n\\vspace{1ex}\n\n\\noindent [KS60]\\, K. Kodaira, D.C. Spencer --- {\\it On Deformations of Complex Analytic Structures, III. Stability Theorems for Complex Structures} --- Ann. of Math. {\\bf 71}, no.1 (1960), 43-76.\n\n\\vspace{1ex}\n\n\\noindent [MP21]\\, {\\it Balanced Hyperbolic and Divisorially Hyperbolic Compact Complex Manifolds} --- arXiv e-print CV 2107.08972v2, to appear in Mathematical Research Letters.\n\n\n\\vspace{1ex}\n\n\\noindent [Mic83]\\, M. L. Michelsohn --- {\\it On the Existence of Special Metrics in Complex Geometry} --- Acta Math. {\\bf 143} (1983) 261-295.\n\n\\vspace{1ex}\n\n\n\\noindent [Pop13]\\, D. Popovici --- {\\it Deformation Limits of Projective Manifolds: Hodge Numbers and Strongly Gauduchon Metrics} --- Invent. Math. {\\bf 194} (2013), 515-534.\n\n\\vspace{1ex}\n\n\\noindent [Pop15]\\, D. Popovici --- {\\it Aeppli Cohomology Classes Associated with Gauduchon Metrics on Compact Complex Manifolds} --- Bull. Soc. Math. France {\\bf 143}, no. 3 (2015), 1-37.\n\n\n\\vspace{1ex}\n\n\n\\noindent [Sch07]\\, M. Schweitzer --- {\\it Autour de la cohomologie de Bott-Chern} --- arXiv e-print math.AG\/0709.3528v1.\n\n\\vspace{1ex}\n\n\\noindent[ST10]\\, J. Streets, G. Tian --- {\\it Parabolic flow of pluriclosed metrics}--- Int. Math. Res. Not. IMRN {\\bf 16} (2010), 3101-3133.\n\n\\vspace{1ex}\n\n\\noindent [Voi02]\\, C. Voisin --- {\\it Hodge Theory and Complex Algebraic Geometry. I.} --- Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002.\n\n\\vspace{1ex}\n\n\n\\vspace{6ex}\n\n\n\\noindent Department of Mathematics and Computer Science     \\hfill Institut de Math\\'ematiques de Toulouse,\n\n\n\\noindent Jagiellonian University     \\hfill  Universit\\'e Paul Sabatier,\n\n\\noindent 30-409 Krak\\'ow, Ul. Lojasiewicza 6, Poland    \\hfill  118 route de Narbonne, 31062 Toulouse, France\n\n\n\\noindent  Email: Slawomir.Dinew@im.uj.edu.pl                \\hfill     Email: popovici@math.univ-toulouse.fr\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf Plasma oscillation.}\n\nIn 1931 Sauter \\cite{sauter1931} and four years later Heisenberg and\nEuler \\cite{he1935} provided a first description of the vacuum properties in\n{\\it constant} electromagnetic fields. They identified a characteristic scale of strong\nfield $E_c=\nm^2_ec^3\/e\\hbar$, at which the field energy is sufficient\nto create electron positron pairs from the vacuum. In 1951, Schwinger\n\\cite{schw1951} gave an elegant quantum-field theoretic reformulation\nof their result in the spinor and scalar QED framework (see\nalso \\cite{bf1970}). The special attention was given for the presence of magnetic fields \\cite{Nikishov1969}. In the configuration of {\\it constant} electromagnetic fields, \nthe pair-production rate per unit volume is given by\n\\begin{equation}\n\\frac{ \\Gamma }{V}=\\frac{  \\alpha   \\varepsilon^2}{ \\pi^2 }\n\\sum_{n=1}  \\frac{1}{n^2}\n\\frac{ n\\pi\\beta \/ \\varepsilon }\n{\\tanh {n\\pi \\beta\/ \\varepsilon}}\\exp\\left(-\\frac{n\\pi E_c}{ \\varepsilon}\\right),\n\\label{probabilityeh}\n\\end{equation}\nwhere the two Lorentz invariants\n$ \\varepsilon $ and $ \\beta $ are \\begin{eqnarray}\\label{fieldinvariant}\n\\varepsilon \\equiv \\sqrt{({\\mathcal S}^2+{\\mathcal P}^2)^{1\/2}+ {\\mathcal S}}, \\quad\n\\beta  \\equiv \n\\sqrt{({\\mathcal S}^2+{\\mathcal P}^2)^{1\/2}- {\\mathcal S}}.\n\\end{eqnarray}\nIn terms of the two\nLorentz invariants, the scalar ${\\mathcal S}\\equiv ({\\bf E}^2-{\\bf B}^2)\/2=(\\varepsilon^2-\\beta^2)\/2$, and the pseudoscalar ${\\mathcal P}={\\bf E}\\,\\cdot {\\bf B}=\\varepsilon\\beta$. In order to focus on studying the phenomenon of plasma oscillations, as a model for quantitative calculations, we postulate an initial configuration of {\\it constant} electromagnetic fields: (i) The electric and magnetic fields are perpendicular to each other (${\\bf E} \\perp {\\bf B}$). (ii) Their amplitudes are different ($|{\\bf E}|>|{\\bf B}|\\not=0$) in the laboratory frame, i.e., the rest frame of electron-positron pair production.  For this electromagnetic configuration ${\\mathcal P}=0$ and leading term ($n=1$), Eq.~(\\ref{probabilityeh}) yields \n\\begin{equation}\nS=\\frac{m_e^4}{4\\pi^3}\\left(\\frac{2{\\mathcal S}}{E^2_c}\\right)\\exp\\left[-\\frac{\\pi E_c}{(2{\\mathcal S})^{1\/2}}\\right], \n\\label{srate}\n\\end{equation}\nwhere the critical field $E_c\\equiv m_e^2\/e$ and $m_e$ ($-e$) is the electron mass (charge). Note that Eq.~(\\ref{srate}) is valid only for ${\\mathcal S}> 0$, i.e., $|{\\bf E}|>|{\\bf B}|$ and ${\\bf E} \\perp {\\bf B}$. In this case $\\beta=0$ and $\\varepsilon^2=2{\\mathcal S}$, Eq.~(\\ref{srate}) is equivalent to the case for a purely electric field $E= 2{\\mathcal S}$. Equation (\\ref{srate}) approaches zero as $|{\\bf B}|$ approaches $|{\\bf E}|+0^-$. \nAs shown below, we have chosen an electric field strength $\\bf E$ that is significantly larger than the magnetic one $\\bf B$; otherwise, Eq.~(\\ref{srate}) would approximately vanish for ${\\mathcal S}\\approx 0$ and ${\\mathcal P}=0$, analogously to the field configuration of a monochromatic laser beam (plane wave ${\\mathcal S}={\\mathcal P}=0$). We will also discuss the situation in which electromagnetic fields are parallel. It is an important issue for future investigations how initial configurations are dynamically generated from the outset. We adopt $\\hbar=c=1$ and Compton units of length $\\lambda_C=\\hbar\/m_ec$, time $\\tau_C=\\hbar\/m_ec^2$, energy scale $m_ec^2$ and critical field strength $E_c$.\n\nIn the kinetic description for plasma fluids of positrons ($+$) and electrons ($-$), with single-particle spectra $p_{\\pm}^0=({\\bf p}_\\pm^2+m_e^2)^{1\/2}$,\nwe define the number densities $n_\\pm (t,{\\bf x})$\nand ``averaged''\nvelocities  ${\\bf v}_\\pm (t,{\\bf x})$ of the fluids:\n\\begin{align}\nn_\\pm \\equiv  \\int  \\frac{d^3{\\bf p}_\\pm}{(2\\pi)^3}f_\\pm,\\quad\n{\\bf v}_\\pm  \\equiv  \\frac{1}{n_\\pm}\\int \\frac{d^3{\\bf p}_\\pm}{(2\\pi)^3} \\left(\\frac{{\\bf p}_\\pm} {p^0_\\pm}\\right) f_\\pm,\n\\label{meanv}\n\\end{align}\nwhere $f_\\pm=f_\\pm(t,{\\bf p}_\\pm,{\\bf x})$ is the distribution function in phase space. The four-velocities of the electron and positron fluids are $U_\\pm^{\\mu}=\\gamma_\\pm (1,{\\bf v}_\\pm)$, the Lorentz factor $\\gamma_{\\pm}=( 1-|{\\bf v}_{\\pm}|^{2}) ^{-1\/2}$, and the comoving number densities $\\bar n_\\pm=n_\\pm\\gamma_\\pm^{-1}$.\nThe collisionless plasma fluid of electrons and positrons coupling to electromagnetic fields is governed\nby the equations of particle-number and energy-momentum conservation and the Maxwell equations:\n\\setlength\\arraycolsep{0.5pt}\n\\begin{align}\n&\\frac{\\partial\\left(  \\bar{n}_\\pm U_\\pm^{\\mu}\\right)  }{\\partial\nx^{\\mu}}\n=S;\n\\quad \\frac{\\partial T_\\pm^{\\mu\\nu}}{\\partial x^{\\nu}}\n=-F^{\\mu}_{~~\\sigma}(J_\\pm^{\\sigma }+J_{\\pm\\rm pola}^{\\sigma\n}),\\label{contp}\\\\\n&\\frac{\\partial F^{\\mu\\nu}}{\\partial x^{\\nu}}   = -4\\pi (J^{\\mu}_{\\rm cond}+J^{\\mu}_{\\rm pola}+J^{\\mu}_{\\rm ext}), \\label{me\n\\end{align}\nwhere we have an external electric current $J_{\\rm ext}^{\\mu} = (\\rho_{\\rm ext},{\\bf J}_{\\rm ext})$, electron and positron fluid currents $J_\\pm^\\mu =\\pm e\\bar\nn_\\pm U^\\mu_\\pm$, and energy-momentum tensors\n\\begin{align}\nT^{\\mu\\nu}_\\pm &= \\bar p_\\pm g^{\\mu\\nu}+(\\bar p_\\pm +\\bar \\epsilon_\\pm)U^\\mu_\\pm U^\\nu_\\pm,\\quad T^{\\mu\\nu}_{\\rm m}=\\sum_\\pm T^{\\mu\\nu}_\\pm.\n\\label{eptensor}\n\\end{align}\nHere the pressure $\\bar p_\\pm$ and energy density $\\bar \\epsilon_\\pm$ are related by the equation of state $\\bar p_\\pm=\\bar p_\\pm(\\bar \\epsilon_\\pm)$ in the fluid comoving frame.\nIn the laboratory frame, the electron and positron energy density\n$p^0_\\pm \\equiv T^{00}_\\pm$ and momentum density $p^i_{\\pm} \\equiv  T^{i0}_\\pm$ are given by $\np^0_\\pm =(\\bar\\epsilon_{\\pm}+\\bar p_\\pm {\\bf v}^2_\\pm)\\gamma^2_{\\pm}\\, {\\rm and}\\,\\,\n{\\bf p}_{\\pm} =(\\bar\\epsilon_{\\pm}+\\bar p_\\pm )\\gamma^2_{\\pm}{\\bf v}_\\pm .\n$\n\\comment{useful equations: $p^\\mu=(p^0_\\pm,{\\bf p}_{\\pm})$ and $T^{ij}_\\pm=\\bar p_\\pm\\delta_{ij} +(\\bar \\epsilon_\\pm+\\bar p_\\pm)\\gamma_\\pm^2v_{i\\pm} v_{j\\pm}$.}\nThe conducting four-current density is\n\\begin{align}\nJ_{\\rm cond}^{\\mu}  &  \\equiv e(\\bar{n}_+U_+^{\\mu} - \\bar{n}_- U_-^{\\mu}),\\quad\n\\partial_\\mu J_{\\rm cond}^{\\mu} =0,\n\\label{current}\n\\end{align}\nand the polarized four-current density $J_{\\rm pola}^{\\mu} = \\sum_\\pm\nJ_{\\pm\\rm pola}^{\\mu}$ with $J_{\\pm\\rm pola}^{\\mu}   =\n\\left(\\rho^\\pm_{\\rm pola}, {\\bf J}^\\pm_{\\rm pola} \\right)$\ndefined by \\cite{Matsui1}\n\\begin{align}\nF^\\nu_{\\,\\,\\,\\,\\mu} J_{\\pm\\rm pola}^{\\mu}=\\Sigma^\\nu_\\pm, \\quad \\Sigma^\\nu_\\pm =\n\\int\\frac{d^3{\\bf p}_\\pm}{(2\\pi)^3p_\\pm^0} p_\\pm^\\nu {\\mathcal A},\n\\label{pcurrentd}\n\\end{align}\nwhere ${\\mathcal A}$ is related to Eq.~(\\ref{srate}) by $S=\\int d^3{\\bf p}_\\pm\/[(2\\pi)^3p_\\pm^0]{\\mathcal A}$. $F^{\\mu\\nu}$ and $T^{\\mu\\nu}_{\\rm em}$ are the field strength and the energy-momentum tensor of electromagnetic fields. \n\\comment{\nuseful equations: \n\\begin{align}\nT^{\\mu\\nu}_{\\rm em} &= F^\\mu_\\rho F^{\\nu\\rho}-\\frac{1}{4}\\eta^{\\mu\\nu}F_{\\rho\\sigma}F^{\\rho\\sigma}.\n\\label{ebtensor}\n\\end{align}\nwhere $\\eta^{\\mu\\nu}=(-1,1,1,1)$, $T^{00}_{\\rm em}=({\\bf E}^2+{\\bf B}^2)\/(8\\pi)$, $T^{i0}_{\\rm em}=({\\bf E}\\times{\\bf B})\/(4\\pi)$ and $T^{ij}_{\\rm em}=-[E_iE_j+B_iB_j-\\delta_{ij}({\\bf E}^2+{\\bf B}^2)\/2)]\/(4\\pi)$.\n}\n\nWe now assume external electromagnetic fields ${\\bf E}_{\\rm ext}=E_{\\rm ext}\\hat {\\bf z}$ and ${\\bf B}_{\\rm ext}=B_{\\rm ext}\\hat {\\bf x}$, where $E_{\\rm ext}$ and $B_{\\rm ext}$ are constant fields in space and time. \nAs will be shown below, in this system, the electron-positron fluid velocities [Eq.~(\\ref{meanv})] have $\\hat {\\bf z}$ and $\\hat {\\bf y}$ components ${\\bf v}_\\pm =(v^y_\\pm \\hat {\\bf y} + v^z_\\pm \\hat {\\bf z})$ in the $y-z$ plane, and the total electromagnetic fields are ${\\bf E} =E_y \\hat {\\bf y}+E_z \\hat {\\bf z}$ and ${\\bf B}=B_x\\hat {\\bf x}$,\nwhich are the superposition of two contributions:\n\\begin{align}\nE_z&=E_{\\rm ext}+ \\tilde E_z(t,y,z),\\quad E_y =\\tilde E_y(t,y,z);\\nonumber\\\\\nB_x&=B_{\\rm ext} + \\tilde B_x(t,y,z),\\quad B_y =0, \\label{tote}\n\\end{align}\nwhere the space- and time-dependent $\\tilde E_{z,y}(t,y,z)$ and $\\tilde B_z(t,y,z)$ are the electromagnetic fields created by the motion of electron and positron pairs.\n\nWe adopt the approximations $\\bar p_\\pm\\approx 0$, $\\bar\\epsilon_\\pm\\approx m_e \\bar n_\\pm$, and $ \\epsilon_\\pm=\\bar \\epsilon_\\pm\\gamma_\\pm^2$ when the pair number density is not very large for $E\\simeq E_c$. Using Eqs.~(\\ref{meanv}) and (\\ref{pcurrentd}), we obtain\n\\comment{these equations are useful, we drop them for limiting pages\n\\begin{align}\nE_yJ^0_{\\pm\\rm pola} + B_xJ^z_{\\pm\\rm pola} &\\approx  m_e\\gamma_\\pm v^y_\\pm S,\\nonumber\\\\\nE_zJ^0_{\\pm\\rm pola} - B_xJ^y_{\\pm\\rm pola} &\\approx  m_e\\gamma_\\pm v^z_\\pm S,\\nonumber\\\\\nE_yJ^y_{\\pm\\rm pola} + E_zJ^z_{\\pm\\rm pola} &\\approx  m_e\\gamma_\\pm  S;\n\\label{pcurrent0}\n\\end{align}\nand\n}\n\\begin{align}\nJ^{z,y}_{\\rm pola}  &\\approx \\frac{E_{z,y}}{E^2}\\left(m_e\\gamma_\\pm S\\right),\\,\\,\nJ^0_{\\pm\\rm pola} \\approx \\frac{v^z_\\pm E_z+v^y_\\pm E_y}{E^2}m_e\\gamma_\\pm S,\n\\nonumbe\n\\end{align}\nwhere $E^2=E_z^2+E_y^2$.\nThe total electric current and charge densities of the  electron-positron fluid are composed by Eqs.~(\\ref{current}) and (\\ref{pcurrentd}) as\n\\begin{align}\nJ_z &= e_+ n_{+}v^z_{+} + e_-n_{-}v^z_{-}\n+J^z_{+\\rm pola}+J^z_{-\\rm pola},\\label{tcur}\n\\end{align}\n$J_y=J_z(z\\rightarrow y)$ and $\\rho = \\sum_\\pm (e_\\pm n_{\\pm}+J^0_{\\pm\\rm pola})$,\nwhere the positron and electron charge $e_\\pm \\equiv \\pm e$.\n\nIt turns out to be a $(1+2)$-dimensional problem in space-time coordinates $(t,y,z)$. Equations (\\ref{contp}) and (\\ref{me}) are reduced to\n(i) the particle-number and energy conservation,\n\\begin{align}\n\\frac{\\partial n_\\pm}{\\partial t}&\\!+\\! \\frac{\\partial n_\\pm\nv^z_\\pm}{\\partial z}\\!+\\!\\frac{\\partial n_\\pm\nv^y_\\pm}{\\partial y}=S,\\label{nudot}\\\\\n\\frac{\\partial\\epsilon_\\pm}{\\partial t}&\\!+\\!\n\\frac{\\partial p^z_\\pm}{\\partial z}\\!+\\!\\frac{\\partial p^y_\\pm}{\\partial y}= e_\\pm n_\\pm v^z_\\pm E_z \\!+\\! e_\\pm n_\\pm v^y_\\pm E_y\\!+\\!m_e\\gamma_\\pm S;\n\\nonumbe\n\\end{align}\n(ii) the momentum conservation,\n\\begin{align}\n\\!\\!\\frac{\\partial p^z_\\pm}{\\partial t}&\\!+\\!\\frac{\\partial p^z_\\pm\nv^z_\\pm}{\\partial z} \\!+\\!\\frac{\\partial p^z_\\pm\nv^z_\\pm}{\\partial y}=e_\\pm n_\\pm E_z \\!-\\! e_\\pm n_\\pm v^y_\\pm B_x\\!+\\! E_zJ^0_{\\pm\\rm pola}\n\\nonumbe\n\\end{align}\nwith $(z\\leftrightarrow y, B_x\\rightarrow -B_x)$;\n(iii) Maxwell equations $\\nabla\\cdot {\\bf E}= 4\\pi \\rho$,  $\\nabla\\cdot {\\bf B}=0$,\n\\begin{align}\n\\frac{\\partial \\tilde E_z}{\\partial t}+\\frac{\\partial \\tilde B_x}{\\partial y} =-4\\pi J_z,\n\\quad\n\\frac{\\partial \\tilde E_z}{\\partial y}-\\frac{\\partial \\tilde E_y}{\\partial z}=-\\frac{\\partial \\tilde B_x}{\\partial t},\\label{Ediv-d}\n\\end{align}\nwith $(z\\leftrightarrow y, B_x\\rightarrow -B_x)$.\nThe pair-production rate [Eq.~(\\ref{srate})] can be approximately\nused for varying electromagnetic fields [Eq.~(\\ref{tote})],\nprovided $\\tilde E(t,y,z)$ and $\\tilde B(t,y,z)$ created by\nelectron-positron pair oscillations vary very slowly compared with the rate of electron-positron pair productions ${\\mathcal O}(m_ec^2\/\\hbar)$.\nThis is justified if\nthe inverse adiabaticity\nparameter \\cite{Brezin}\n$\n\\eta=\\frac{m_e}{\\omega_p}\\frac{E}{E_{c}}\\gg 1,\n$\nwhere $\\omega_p$ is the frequency of plasma oscillations.\n\nWe are in the position of numerically integrating the basic equations (\\ref{nudot}) and (\\ref{Ediv-d}). The initial conditions\n$(t=0)$ are given by the constant electromagnetic fields $E_z=E_{\\rm ext}$ and $B_x=B_{\\rm ext}$. To simplify numerical integrations, we assume the $(z\\!-\\!y)$ homogeneity that the electron-positron fluid quantities and electromagnetic fields are independent of $y$ and $z$. As a result,\nEqs.~(\\ref{nudot}) and (\\ref{Ediv-d}) are reduced to ordinary differential equations, and  Eq.~(\\ref{Ediv-d}) leads to $\\tilde B_x=0$, i.e., the magnetic field $B_x$ of Eq.~(\\ref{tote}) is a constant in space and time.\n\\comment{\nIn this system, except electron and positron velocities are in opposite directions, their quantities, including electric currents, are the same. Therefore, we only need to solve the equations for positrons and  $(E,B)$-fields. The positron current represents the electric current\nand we hereafter omit its subscript $``+''$. In the following, we present our numerical studies of the plasma oscillations.\n}\nThe initial condition $E_y=0$ leads to the solution $\\tilde E_y=0$ and $J_y=0$ for $t\\not=0$, because $v^z_-=-v^z_+$ and $v^y_-=v^y_+>0$. This is verified in the following numerical calculations.\n\nTo illustrate the plasma oscillations of pairs and fields, we consider two cases: (i) $E_{\\rm ext}=E_c$ and $B_{\\rm ext}=0.1\\,E_c$; (ii) $E_{\\rm ext}=E_c$ and $B_{\\rm ext}=0.3\\,E_c$. Due to the presence of the magnetic field $B_x$, pairs are not only oscillating up and down in the $\\hat {\\bf z}$ direction, as first shown in Ref.~\\cite{4}, but they also move in the $\\hat {\\bf y}$ direction.  In Fig.~\\ref{figm}, we show the trajectory and velocity of pairs produced at $z=y=0$ and $t=0$. When $B_x\\not=0$ and $dv_y \\sim ev_zB_x dt$, $v_y$ increases for $v_z >0$ and decreases for $v_z <0$.\nIn the case of $B_x$ being small enough compared with $E_z$, $v_y$ does not change its sign (see Fig.~\\ref{figm}, $B_x=0.1E_c$) in the period of one circle oscillation in the ${\\bf \\hat z}$ direction; therefore pairs move forward in the ${\\bf \\hat y}$ direction. When $B_x=0.3\\, E_c$, $v_y$ changes its sign (see Fig.~\\ref{figm}, $B_x=0.3E_c$); therefore pairs also oscillate back and forth, while they are moving in the ${\\bf \\hat y}$ direction. In contrast to the case $B_x=0$, the negative $E_z$ amplitude is smaller than the positive $E_z$ amplitude (see Fig.~\\ref{figm}). The reason is that $v_y$ increases in the phase of positive decreasing $v_z$ when $E_z<0$; i.e., the electric energy goes to the kinetic energy of the motion in the $\\hat {\\bf y}$ direction.\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[height=1.25in]{yz_E1B01.eps}\n\\includegraphics[height=1.25in]{yz_E1B03.eps}\n\\includegraphics[height=1.25in]{vy_vz_E1B01.eps}\n\\includegraphics[height=1.25in]{vy_vz_E1B03.eps}\n\\includegraphics[height=1.25in]{Ez_E1B01.eps}\n\\includegraphics[height=1.25in]{Ez_E1B03.eps}\n\\caption{With initial conditions $E_z=E_c$, $B_x=0.1E_c$ (left) and $B_x=0.3E_c$ (right), we plot the electron and positron trajectories and velocities $v_y$ vs $v_z$ and the electric field $E_z$ vs the time $t$ from $t=0$ to $t= 10^4$.} \\label{figm}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{figT}, we plot\n(i) the electric current density of pairs $J_z$ as a function of the time, which is the source of electromagnetic radiation, and\n(ii)the total energy-momentum tensor of pairs and fields $T^{\\mu\\nu}=T^{\\mu\\nu}_{\\rm m}+T^{\\mu\\nu}_{\\rm em}$ as functions of the time, which are the sources of gravitational radiation.\n\nBefore ending this section, we would like to present some discussions on the role of magnetic fields. In the particular initial configuration of fields ${\\bf E} \\perp {\\bf B}$ and $|{\\bf E}|>|{\\bf B}|$\nconsidered in this paper, by integrating Eqs.~(\\ref{srate}), (\\ref{contp}) and (\\ref{me}), we show the oscillating electric field strength (see Fig.~\\ref{figm}), and the number and current densities of pairs (see Fig.~\\ref{fign}) are suppressed by magnetic fields, compared with their counterparts in the absence of magnetic fields. However, we cannot conclude that such magnetic suppression is generally true. For example, when electromagnetic fields are parallel (${\\bf E}\\times{\\bf B}=0$ and $|{\\bf E}|>|{\\bf \nB}|$), Eq.~(\\ref{probabilityeh}) yields (see Ref.~\\cite{Nikishov1969})\n\\begin{equation}\n\\frac{\\Gamma}{V}\\simeq\n\\frac{\\alpha |{\\bf B}||{\\bf E}|}{\\pi}\\coth\\left(\\frac{\\pi |{\\bf B}|}{ |{\\bf E}|}\\right)\n\\exp\\left(-\\frac{\\pi E_c}{|{\\bf E}|}\\right),\n\\label{wkbehfermion1}\n\\end{equation}\nindicating that the pair-production rate receives an enhancement $(\\pi |{\\bf B}|\/|{\\bf E}|) \\coth (\\pi |{\\bf B}|\/|{\\bf E}|)$ to the prefactor, compared with the rate in the absence of magnetic fields \\cite{Nikishov1969,supp_mag} (see also \\cite{dunne2006,report}). \nIt is worthwhile to study the phenomenon of plasma oscillations by numerically integrating Eqs.~(\\ref{contp}), (\\ref{me}) and (\\ref{wkbehfermion1}) consistently with the initial configuration of parallel electromagnetic fields \\cite{future}. \n\n\\begin{figure\n\\begin{center}\n\\includegraphics[height=1.25in]{n_E1.eps}\n\\includegraphics[height=1.25in]{jz_B01_B03.eps}\n\\caption{The pair number density $n_\\pm$ and current density $J_z$ vs the time $t$ for $E_z=E_c$ and different $B_{x}$ field values.}\n\\label{fign}\n\\end{center}\n\\end{figure}\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[height=1.25in]{jz_B00.eps}\n\\includegraphics[height=1.25in]{Tuv.eps}\n\\caption{$E_z=E_c$ and $B_x=0.0$. The charged current density $J_z$ vs time $t$ (left). The total energy-momentum tensor $T^{00}$ and $T^{zz}$ vs time $t$ (right).} \\label{figT}\n\\end{center}\n\\end{figure}\n\n\\vskip0.1cm \\noindent{\\bf Electromagnetic and gravitational radiation.}\nWe attempt to study electromagnetic and gravitational radiation generated, respectively,  by the electric current and energy-momentum tensor of pairs and fields. Suppose that we observe this radiation in the {\\it wave zone}; that is, at distances much larger than the dimension ${\\mathcal R}$ of the plasma oscillations, and also much larger than $\\omega {\\mathcal R}^2$ and $1\/\\omega$, where $\\omega$ is the typical frequency of radiation.\n\nFor definiteness we think of the electric current and energy-momentum tensor of the plasma oscillations occurring\nin the volume ${\\mathcal\nV}$ and for a finite interval of time ${\\mathcal T}$, so that the total energy radiated is finite. Thus, the electromagnetic energy radiated per unit solid angle per frequency interval is given by \\cite{Jackson}\n\\begin{align}\n\\!\\!\\frac{d^2{\\mathcal E}^{^{\\rm em}}}{d\\omega d\\Omega}\n=2\\left|\\int_{\\mathcal\nV} d^3x' \\int_{\\mathcal T} dt' e^{i\\omega t'-i{\\bf k}{\\bf x}'}\\left[\\frac{\\partial J_z({\\bf x}',t')}{\\partial t'}\\right]\\right|^2.\n\\label{intem}\n\\end{align}\nThe gravitational energy radiated per unit solid angle per frequency interval is then given by \\cite{Weinberg1972}\n\\begin{align}\n&\\frac{d^2{\\mathcal E}^{^{\\rm grav}}}{d\\omega d\\Omega} = 2G\\omega^2\\Big[T^{\\mu\\nu *}({\\bf k}, \\omega)T_{\\mu\\nu }({\\bf k},\\omega)-\\frac{1}{2}|T^\\nu_{\\,\\,\\,\\nu }({\\bf k}, \\omega)|^2\\Big],\\nonumber\\\\\n&T_{\\mu\\nu }({\\bf k}, \\omega) =\n\\int_{\\mathcal\nV} d^3x' \\int_{\\mathcal T} dt' \\,\\, T_{\\mu\\nu }({\\bf x}', t')e^{i\\omega t'-i{\\bf k}{\\bf x}'}\\label{intg}\n\\end{align}\nwhere $|{\\bf k}|=\\omega$ and $T^{\\mu\\nu }({\\bf x}', t')=T^{\\mu\\nu }_{\\rm m}({\\bf x}', t')+T_{\\rm em}^{\\mu\\nu }({\\bf x}', t')$. We consider $\\omega {\\mathcal R}\\ll 1$ and $e^{-i{\\bf k}{\\bf x}'}\\approx 1$ for dipole electromagnetic radiation in Eq.~(\\ref{intem}), and for quadrapole gravitational radiation in Eq.~(\\ref{intg}). In the calculations of Eq.~(\\ref{intg}), we set $B_{\\rm ext}=0$ and $E_{\\rm ext}=E_c$, and then the nonvanishing components are $T^{00}=T^{00}_{\\rm m}+T^{00}_{\\rm em}$ and $T^{zz}=T^{zz}_{\\rm m}+T^{zz}_{\\rm em}$.\nUsing the approximation of spatial homogeneity in Eqs.~(\\ref{intem}) and (\\ref{intg}), we can factorize out the volume ${\\mathcal V}=\\int_{\\mathcal V} d^3x'$, in which the total energy density $T^{00}=T^{00}_{\\rm m}+T^{00}_{\\rm em}=E_{\\rm ext}^2\/(8\\pi)$ is conserved (see Fig.~\\ref{figT}).\n\nLet ${\\mathcal T}$ and ${\\mathcal V}$ also be the time and volume of strong fields $E_{\\rm ext}\\gtrsim E_c$ created by coherent laser beams. Selecting different ${\\mathcal T}$ values, in Fig.~\\ref{figR} we plot the electromagnetic and gravitational radiation spectra (\\ref{intem}) and (\\ref{intg}) with ${\\mathcal V}^2$ factored out. These two energy spectra are narrow, and the locations ($\\omega_{\\rm peak}$) of their peaks are related to the coherent oscillation frequency ($\\omega_p$) of pairs and fields, which depend on ${\\mathcal T}$ and $E_{\\rm ext}$ (see Ref.~\\cite{Han1}). The peculiar energy spectrum of electromagnetic radiation is clearly distinguishable from the energy spectra of\nthe bremsstrahlung radiation, electron-positron annihilation and other possible background events. Therefore, it is sensible and distinctive to detect such peculiar radiative signatures to identify the production and\noscillation of electron-positron pairs in strong laser fields. \nAs shown in Fig.~\\ref{figR}, gravitational radiation is much smaller than the electromagnetic one for the reason that the gravitational coupling $Gm_e^2=2.5\\times 10^{-45}$ is much smaller than the electromagnetic coupling $e^2=1\/137$. \n\\comment{For example, the pair density $n\\sim 10^{31}{\\rm cm}^{-3}$ for $E_{\\rm ext} \\sim E_c$ (see Fig.~\\ref{fign}), from our results (see Fig.~\\ref{figR}) the intensities of electromagnetic and gravitational radiation are given by\n\\begin{eqnarray}\nI_{\\rm grav}&\\approx& 1.56\\cdot 10^{-29}\\,\\,{\\mathcal V}^2({\\rm ergs\/sec})=1.56\\cdot 10^{-11}({\\rm ergs\/sec});\\nonumber\\\\\nI_{\\rm em}&\\approx& 1.56\\cdot 10^{17}\\,\\,{\\mathcal V}^2({\\rm ergs\/sec})=1.56\\cdot 10^{35}({\\rm ergs\/sec}).\n\\label{ints}\n\\end{eqnarray}\nwhere ${\\mathcal V} = (3.86\\cdot 10^{-8} {\\rm cm}\/\\lambda_C)^3$.} \nIn order to achieve a sizable radiation intensity from the plasma oscillation, the volume ${\\mathcal V}$ of oscillating pairs and strong electric fields should be large enough and\/or the strength of strong fields should be enhanced ($E_{\\rm ext} \\gtrsim E_c$) to increase the pair density. \nIt is worthwhile to point out that Fig.~\\ref{figR} shows the numerical results of Eqs.~(\\ref{intem}) and (\\ref{intg}) being consistent with the approximate relation between Eqs.~(\\ref{intem}) and (\\ref{intg}) in the ultrarelativistic limit of charged particles moving in external electromagnetic fields \\cite{ritus1968}.\n \n\\begin{figure\n\\begin{center}\n\\includegraphics[height=1.25in]{empower4t_log.eps}\n\\includegraphics[height=1.25in]{gwpower4t_log.eps}\n\\caption{$E_z=E_c$ and $B_x=0.0$. By factoring ${\\mathcal V}^2$ out, the electromagnetic radiation (left) of Eq.~(\\ref{intem}) and the gravitational radiation (right) of Eq.~(\\ref{intg}) are plotted as functions of the frequency $\\omega$, for different ${\\mathcal T}$ values .} \\label{figR}\n\\end{center}\n\\end{figure}\n\n\\comment{\nWe recall the intensity of gravitational quadrapole radiation from two compact stars in non-relativistic circular motion of frequency $\\omega_{\\rm s}$ \\cite{Laudau}\n\\begin{align}\nI^{^{\\rm grav}}_{\\rm s}=\\frac{32}{5}(G \\mu^2)\\omega_{\\rm s}^6 r_{\\rm s}^4.\n\\label{gv01e}\n\\end{align}\nwhere $r_{\\rm s}$ is the distance between two stars. Similar to Eq.~(\\ref{intg}), in one period ${\\mathcal T}_{\\rm s}=2\\pi\/\\omega_{\\rm s}$ the energy radiated per unit solid angle per frequency interval is\n\\begin{align}\n\\frac{d^2{\\mathcal E}_{\\rm s}^{^{\\rm grav}}}{d\\omega_{\\rm s} d\\Omega}=16(G \\mu^2)\\omega^4_{\\rm s} r^4_{\\rm s}.\n\\label{gv01es}\n\\end{align}\n\\blue{Comparing Eq.~(\\ref{intg}) with Eq.~(\\ref{gv01es}) and its numerical result, we notice that the gravitational radiation energy is proportional to $\\omega^6$ for the ultra-relativistic motion of particles, instead, $\\omega^4$ for the non-relativistic motion of particles.} \nSuppose that the two star masses are equal to a solar mass $M_\\odot$, their reduced mass $\\mu=M_\\odot\/2$ and separation $r_{\\rm s}=10\\,G M_\\odot\\approx 15 {\\rm km}$, as well as the frequency $\\omega_{\\rm s}\\approx 10\\, {\\rm Hz}$.\nMultiplying  the energy radiation (\\ref{intg}) and (\\ref{gv01es}) by the solid angle $\\Delta\\Omega=\\Delta\\Sigma\/R^2$ subtended by the detector area $\\Delta\\Sigma$ to the source at the distance $R$, and using the numerical values of Eq.~(\\ref{intg}) presented in Fig.~\\ref{figR}, we qualitatively estimate their ratio for the same radiating time ${\\mathcal T}_{\\rm s}={\\mathcal T}=10^5\\tau_C$ and detector area $\\Delta\\Sigma$ as,\n\\begin{align}\n\\Big(\\frac{10^{-37}}{10^{50}}\\Big) \\Big(\\frac{R_{\\rm star}}{R_{\\rm pair}}\\Big)^2 {\\mathcal V}^2 \\sim 10^{-42} {\\mathcal V}^2,\n\\label{com}\n\\end{align}\nwhere $R_{\\rm star}\\approx 10\\, {\\rm kpc}\\approx 3\\cdot 10^{22} {\\rm cm}$ and\n$R_{\\rm pair}\\approx 1 {\\rm cm}$.\nThis estimation indicates that in order to achieve a sizable radiation intensity from pairs, compared with the one from binary stars, the volume ${\\mathcal V}$ of strong fields with a pair density $n\\sim 10^{31}{\\rm cm}^{-3}$ for $E_{\\rm ext}\\approx E_c$ (see Fig.~\\ref{fign}), should be large enough, \n${\\mathcal V}\\sim {\\mathcal O}(10^{-11})\\,{\\rm cm}^3$, and\/or the strength of strong fields should be increased ($E_{\\rm ext} > E_c$) to increase the pair density.\nEmitted from strong gravitational\ncircumstances, for example from compact binary coalescence, astrophysical gravitational waves are in the frequency band\n$(10^{-7}-10^{4})$Hz. Experimental efforts, for example LIGO, VERGO, eLISA and SKA, have been made to detect such astrophysical gravitational waves \\cite{binary}.}\n\nGravitational waves from an inflationary cosmos \\cite{cosmos} are in the high frequency band $(10^{8}-10^{11}$Hz).\nGravitational waves originating from some sources in ground laboratories are also in this frequency band, and several proposals have been made to detect high-frequency gravitational waves up to $5$ GHz  \\cite{high1}.  Gravitational waves generated from  the high-energy particle beam \\cite{ritus1968,hep-grav}\nin the ground experiments of the Stanford Linear\nCollider and LHC\nhave much higher frequencies of ${\\mathcal O}(10^{23})$ Hz. The frequency of gravitational wave discussed here is ${\\mathcal O}(10^{19-20})$ Hz, i.e., ${\\mathcal O}(10^{0-1})$KeV,\nor sub nanometer ${\\mathcal O}(10^{-(9-10)})$ cm. \nIt is not clear\nwhether such gravitational waves could ever be detected, or have observable effects. One would have to build an atom-sized gravitational wave detector to response incoming gravitational wave with such high frequencies (for some more details, see Ref.~\\cite{weiton2010}) \n\\comment{We try to make some speculations that If would have been realized, possible candidates for such a detector would be highly coherent systems of particles and fields, for instance, a neutral electron-proton plasma of frequency $\\omega_{pe}\\approx\\sqrt{4\\pi e^2 n_e\/m_e}\\sim 10^{19}$Hz when the electron density $n_e\\sim 10^{30}\/{\\rm cm^3}$, and\/or a system of electron-positron plasma oscillations discussed in this paper.}\n\n\nTo end this paper, we remark again that\nthe intensity of electromagnetic\nradiation emitted by the plasma oscillations is tens of orders of magnitude larger than their gravitational radiation; therefore any detectable signal is enormously more likely to result from the electromagnetic interaction. The prospect of detecting gravitational radiation of such ultrahigh frequencies looks dim. Nevertheless, our theoretical investigation of the gravitational radiation from the electron-positron plasma oscillation would be useful for the study of gravitational radiation emitted from particles and antiparticles in the very early Universe.\n\n\\comment{\nTo end this paper, we remark that \nsince such gravitational waves are emitted from a highly coherent system of particles and fields created by the advanced laser technique, it might be possible to set up two systems that emit and receive coherent gravitational waves, leading to detectable signals of resonance and\/or interference effects. \n}\n\n\n\n\\noindent{\\bf Acknowledgements:} Wen-Biao Han is supported by NSFC Grant No.11273045.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}