diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbvew" "b/data_all_eng_slimpj/shuffled/split2/finalzzbvew" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbvew" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe Epoch of Reionization is the interval of time during which the cosmic gas evolves from an almost completely neutral state (neglecting the recombination leftovers) to an ionized state.\nThis ionization process is believed to happen due to the onset of star formation at redshifts $z\\simeq 12$, and it is believed to last until $z\\simeq 6$.\nSeveral astrophysical observables (quasars~\\cite{Fan:2005es,Becker:2014oga}, Lyman $\\alpha$ emitters~\\cite{Stark:2010qj,Treu:2013ida,Pentericci:2014nia,Schenker:2014tda,Tilvi:2014oia}, $\\gamma$ ray bursts~\\cite{Wang:2015ira,Gallerani:2009aw}) seem to agree with this hypothesis.\nHowever, the precise details of the overall reionization process still remain obscure.\nThe main reason is that the currently available most precise information on the reionization period comes from Cosmic Microwave Background (CMB) measurements through a redshift-integrated quantity.\nDuring reionization, the number density of free electrons which can scatter the CMB, $n_e$, increases. As a consequence, the reionization optical depth $\\tau$ increases according to a line of-sight integral of $n_e$, generating a suppression of the CMB peaks at any scale within the horizon at the reionization period.\nThis suppression, however, can be easily compensated with an enhancement of the primordial power spectrum amplitude, $A_{\\rm s}$.\nA much better and cleaner measurement of $\\tau$ can be obtained via measurements of the CMB polarization, which is linearly affected by reionization (see e.g. Refs.~\\cite{Kaplinghat:2002vt,Haiman:2003ea,Holder:2003eb,Hu:2003gh} for seminal works and \\cite{Reichardt:2015cos} for a recent review). The latest measurements of the Planck collaboration provide a value of $\\tau = 0.055 \\pm 0.009$~\\cite{Aghanim:2016yuo, Adam:2016hgk} based exclusively on the CMB polarization spectrum.\nThis value of $\\tau$ is in a much better agreement than previous WMAP~\\cite{Hinshaw:2012aka} and Planck~\\cite{Ade:2015xua} estimates with observations of Lyman-$\\alpha$ (Ly-$\\alpha$) emitters at $z\\simeq 7$~\\cite{Stark:2010qj,Treu:2013ida,Pentericci:2014nia,Schenker:2014tda,Tilvi:2014oia}, which require that reionization is complete by $z\\simeq 6$.\nEven if now cosmological and astrophysical tests of the reionization process seem to agree, the measurement of $\\tau$ provides only integrated information on the free electron fraction $x_e$, and not on its precise redshift evolution.\nConsequently, the same measured value of $\\tau$ may correspond to very different reionization histories.\n\nTraditionally, the most commonly exploited model for the time evolution of the free electron fraction, $x_e(z)$, uses a step-like transition, implemented via a hyperbolic tangent~\\cite{Lewis:2008wr}.\nModel independent attempts have been carried out in several works in the past~\\cite{Hu:2003gh,Mortonson:2007hq,Mortonson:2007tb,Mortonson:2008rx,Mortonson:2009qv,Mortonson:2009xk,Mitra:2010sr,Lewis:2006ym,Pandolfi:2010dz,Pandolfi:2010mv} and also more recently~\\cite{Heinrich:2016ojb,Hazra:2017gtx,Mitra:2017oxx}, based either on a redshift-node decomposition of $x_e(z)$ or on a Principal Component Analysis (PCA) of the CMB polarization angular power spectrum.\nMore concretely, using the latter approach, the authors of \\cite{Heinrich:2016ojb} claimed that Planck 2015 data favors a high-redshift ($z>15$) component to the reionization optical depth.\nThe quoted $2\\sigma$ evidence would come from the excess in power in the low multipole range of the Planck 2015 CMB polarization spectrum. \nAccordingly to their results, the functional form of the usual step-like model prevents a priori for such an early component in the reionization history of our universe.\nHowever, the authors of \\cite{Hazra:2017gtx}, using a different method, which implements reionization through a non-parametric reconstruction that uses a Piecewise Cubic Hermite Interpolating Polynomial (\\texttt{PCHIP}), find only marginal evidence for extended reionization histories.\nSince an early component in the reionization history $x_e(z)$ (or, in other words, a high redshift contribution to the reionization optical depth $\\tau$) may either imply the need for a high-redshift population of ionizing sources (hypothesis that will be tested by the future James Webb Space Telescope~\\cite{Gardner:2006ky}),\nor give hints about a possible energy injection from dark matter annihilations or decays~\\cite{Pierpaoli:2003rz,Mapelli:2006ej,Natarajan:2008pk,Natarajan:2009bm,Belikov:2009qx,Huetsi:2009ex,Cirelli:2009bb,Kanzaki:2009hf,Natarajan:2010dc,Giesen:2012rp,Diamanti:2013bia,Lopez-Honorez:2013lcm,Lopez-Honorez:2016sur,Poulin:2016nat,Poulin:2015pna},\nor accreting massive primordial black holes~\\cite{Ricotti:2007au,Horowitz:2016lib,Ali-Haimoud:2016mbv,Blum:2016cjs,Poulin:2017bwe},\nit is mandatory to robustly establish what current data prefer, regardless of the model used to describe the redshift evolution of the free electron fraction. \n\nHere we first analyze several possible parameterizations for reionization (PCA with several fiducial cosmologies and the \\texttt{PCHIP}\\ method)\nand explore the corresponding constraints on the reionization history of the universe.\nWe then shall exploit tools related to model selection among competing models, using both the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), which will allow us to quantitatively decide which model is currently preferred and whether it exists or not an indication for an early reionization component in our universe.\n\nThe structure of the paper is as follows.\nWe start by discussing the different reionization approaches that we shall test against current data in Sec.~\\ref{sec:histories}.\nIn Sec.~\\ref{sec:data} we describe the cosmological observations exploited in our numerical analyses, whose results are shown in Sec.~\\ref{sec:results}.\nOur conclusions are summarized in Sec.~\\ref{sec:conclusions}.\n\n\n\\section{Reionization histories}\n\\label{sec:histories}\n\nIn the following, we will derive the constraints on the reionization history of our universe from cosmological observations exploring several possible scenarios, focusing on a possible early reionization component in our universe.\nFor that, we shall exploit the reionization optical depth:\n\\begin{equation}\n\\tau(z) = \\int_z^{\\infty} dz' \\frac{c ~dt'}{dz'} (n_{\\rm e}(z')- n_{\\rm e, 0}(z'))\\sigma_{\\rm T}\\,~,\n\\label{eq:cumtau}\n\\end{equation}\nwhere $n_{\\rm e}(z)=n_{\\rm H}(0)(1+z)^3x_{\\rm e}(z)$ and $n_{\\rm e,0}(z)=n_{\\rm H}(0)(1+z)^3x_{\\rm e, 0}(z)$, being $n_{\\rm H}(0)$ the number density of hydrogen at present, $x_{\\rm e}(z)$ the free electron fraction and $x_{e,0}(z)$ the free electron fraction leftover from the recombination epoch (see e.g.\\ \\cite{Kolb:1990vq,2009fflr,2010gfe}). Therefore, Eq.~\\eqref{eq:cumtau} just accounts for the cumulative Compton optical depth after recombination, subtracting the pre-reionization contribution.\n\n\n\\subsection{Canonical scenarios}\n\\label{subsec:canonical}\nWe start describing the free electron fraction by means of the most simple and commonly exploited parameterizations in the literature, i.e.\\ the so-called \\emph{redshift-symmetric} and \\emph{redshift-asymmetric} parameterizations (see e.g.~\\cite{Adam:2016hgk}). \n\n\\begin{itemize}\n\n\\item \\emph{Redshift-symmetric} parameterization.\n\nThe most economical and widely employed approach to describe the reionization process in our universe assumes that the free electron fraction follows a step-like function, taking the recombination leftover value at high redshifts and becoming close to one at low redshifts, and being described by the hyperbolic tangent function~\\cite{Lewis:2008wr}\n\\begin{equation}\nx_e^{\\rm tanh}(z) = \\frac{1+f_{\\rm He}}{2} \\left(1+ \\tanh \\left[ \\frac{y(z_{\\rm{re}})-y(z)}{\\Delta y} \\right] \\right),\n\\label{eqn:tanh}\n\\end{equation}\nwhere $y(z)=(1+z)^{3\/2}$, $\\Delta y=3\/2(1+z_{\\rm{re}})^{1\/2}\\Delta z$, and $\\Delta z$ is the width of the transition, fixed in the following to $\\Delta z=0.5$.\nThis parametrization is named ``redshift symmetric'' because the redshift interval between the beginning of reionization and its half completion equals the corresponding one between half completion and the reionization offset, and it is the default one implemented in Boltzmann solver codes such as \\texttt{CAMB}~\\footnote{\\href{http:\/\/camb.info}{http:\/\/camb.info}}~\\cite{Lewis:1999bs}.\nThis parameterization, as well as the following ones, also accounts for the first ionization of helium $f_{\\rm He}=n_{\\rm{He}}\/n_{\\rm{H}}$, assumed to happen at the same time than that of hydrogen.\nThe full helium reionization is modeled via another hyperbolic tangent function with $z_{\\rm{re,He}}=3.5$ and $\\Delta z=0.5$.\nTherefore, the only free parameter in this simple approach is the reionization redshift $z_{\\rm{re}}$.\nWhen this redshift-symmetric parameterization is used as the fiducial model in our PCA analyses (see next subsection), we fix $z_{\\rm{re}}=8.8$, following the results quoted in Ref.~\\cite{Adam:2016hgk}.\n\n\\item \\emph{Redshift-asymmetric} reionization.\n\nBesides the previous case, alternative reionization parameterizations with a non redshift-symmetric transition have been proposed in the literature.\nOne of the most flexible choices, which shows good agreement with current measurements from quasars, Ly$\\alpha$ emitters and star-forming galaxies, is represented by a power law, described via three parameters~\\cite{Adam:2016hgk,Douspis:2015nca}:\n\\begin{equation}\n x_e^{asym}(z) =\n \\begin{cases}\n\t1+f_{\\rm He} & \\mbox{for } z z_{\\rm early}.\n \\end{cases}\n \\label{eqn:asym}\n\\end{equation}\nFollowing Planck 2016 reionization analyses~\\cite{Adam:2016hgk}, when using this redshift-asymmetric model as a fiducial model in our PCA analyses, we shall fix the redshift at which the first sources in our universe switch on, $z_{\\rm early} = 20$, the redshift at which reionization is fully complete, $z_{\\rm end} = 6$, and the exponent $\\alpha = 6.10$. \n\\end{itemize}\n\n\\subsection{Principal Component Analysis (PCA)}\nThe second method we follow here to model the reionization process is the Principal Component Analysis (PCA) approach of Refs.~\\cite{Hu:2003gh,Mortonson:2007hq,Mortonson:2007tb,Mortonson:2008rx,Mortonson:2009qv,Mortonson:2009xk,Mitra:2010sr}, exploited more recently in Refs.~\\cite{Heinrich:2016ojb,Mitra:2017oxx}.\nFollowing these previous works, we discretize the redshift range from $z_{\\rm{min}}=6$ to $z_{\\rm{max}}=30$ in $N_z$ bins of width of $\\delta z = 0.25$.\nWe set the ionization fraction to $x_e=0$ for $z \\geq z_{\\rm{max}}$, when the reionization processes have not started yet, while for $z \\leq 6$ we assume fully ionized hydrogen and singly ionized helium, i.e.\\ $x_e=1+f_{\\rm He}$.\nThe full helium reionization is modeled as aforementioned.\nThis approach makes use of the Fisher information matrix~\\cite{Tegmark:1996bz}, that we compute as:\n\\begin{equation}\nF_{ij} = \\sum_{\\ell=2}^{\\ell_{\\rm max}}\\frac{1}{\\sigma_{ C_{\\ell}}^2}\n \\frac{\\partial C_{\\ell}}{\\partial x_e(z_i)}\n \\frac{\\partial C_{\\ell}}{\\partial x_e(z_j)} =\\sum_{\\ell=2}^{\\ell_{\\rm max}}\\left(\\ell+\\frac{1}{2}\\right)\n \\frac{\\partial \\ln C_{\\ell}}{\\partial x_e(z_i)}\n \\frac{\\partial \\ln C_{\\ell}}{\\partial x_e(z_j)} \\,,\n\\label{eq:fisher}\n\\end{equation}\nwhere the $C_{\\ell}$ are the components of the large angle $EE$ polarization spectrum.\nThe sum above is truncated at $\\ell_{\\rm max}=100$, because the reionization imprint is mostly located in the lowest modes of the CMB polarization spectrum.\nIn Eq.~\\eqref{eq:fisher} we have used the well-known result for the cosmic variance: $\\sigma_{ C_{\\ell}}^2 = C_{\\ell}^2\\, 2\/(2\\ell+1)$.\nHaving the Fisher matrix, we can diagonalize it and find that the eigenfunctions are the principal components $S_{\\mu}(z)$ and the eigenvalues are proportional to the inverse of the estimated variance of each eigenmode, $\\sigma^2_{\\mu}$.\nUsing the normalization of Ref.~\\cite{Mortonson:2007hq}, we can write the Fisher matrix as\n\\begin{equation}\nF_{ij}=\\frac{1}{(N_z+1)^2}\\sum_{\\mu=1}^{N_z}\n \\frac{1}{\\sigma^2_{\\mu}}S_{\\mu}(z_i) S_{\\mu}(z_j)~.\n\\label{eq:fisher2}\n\\end{equation}\nWe sort the different eigenfunctions in order to have the smallest uncertainties at the lowest modes, being therefore the $\\mu=1$ case the best constrained mode.\nDue to completeness and orthogonality of the principal components, the following properties are fulfilled:\n\\begin{align}\n\\int_{z_{\\rm min}}^{z_{\\rm max}} dz \\, S_{\\mu}(z)S_{\\nu}(z)&=(z_{\\rm max}-z_{\\rm min})\\delta_{\\mu\\nu} \\, , \\\\\n\\sum_{\\mu=1}^{N_z} S_{\\mu}(z_i)S_{\\mu}(z_j)&= (N_z+1) \\delta_{ij}\\,.\n\\end{align}\nSince the width of the bins is chosen to be sufficiently small, in practice we can replace the integrals over redshift by discrete sums.\nOne of the ideas behind the PCA approach is that one can write redshift-dependent quantities such as the ionization fraction as a linear combination of the principal components.\nSince the lowest modes have the smallest uncertainties, we truncate the sum, using only the first 5 principal components, following Ref.~\\cite{Mortonson:2007hq}.\nWe apply the PCA analysis to the ionization history in two different ways, which are explained below.\n\n\\begin{itemize}\n\n\\item \\textbf{Case A}\n\nIn the first PCA approach, named \\textbf{PCA-A} in what follows, the reionization history reads as\n\\begin{equation}\nx_e^A (z) = x^{\\rm{fid}}_{\\rm e} (z)+ \\sum_\\mu m_{\\mu}^{A} S_\\mu (z)~.\n\\label{eq:pca_a}\n\\end{equation}\nGiven a fiducial model $x^{\\rm{fid}}_{\\rm e} (z)$, and knowing the amplitudes derived from the Fisher matrix (see Eq.~\\eqref{eq:fisher}), one can recover an arbitrary ionization history using a PCA analysis.\nThis is the standard approach adopted in Refs.~\\cite{Mortonson:2007hq,Heinrich:2016ojb} in order to constrain the ionization history with CMB data.\nFollowing \\cite{Mortonson:2007hq}, we can derive upper and lower bounds for each amplitude $m_{\\mu}$:\n\\begin{equation}\nm_{\\mu}^{\\pm} = \\int_{z_{\\rm min}}^{z_{\\rm max} } dz \\frac{S_{\\mu}(z)[x_e^{\\rm max} -2 x_e^{\\rm fid}(z)]\n\\pm x_e^{\\rm max} | S_{\\mu}(z)|}{2(z_{\\rm max}-z_{\\rm min})}~.\n\\label{eq:mbounds}\n\\end{equation}\nAdditionally, in order to guarantee physical ionization histories, the choice of our amplitudes $m_{\\mu}$ has to fulfill the condition $0 \\leq x_e(z) \\leq 1+f_{\\rm He}$ at any redshift $z$~\\footnote{Notice that this constraint for physicality is stronger than that followed in Ref.~\\cite{Heinrich:2016ojb}, as any unphysical model will be retained for the Monte Carlo analyses.}.\n\n\\item \\textbf{Case B}\n\nIn the second of our PCA analyses, named \\textbf{PCA-B}, we choose a different approach to the standard PCA analysis described above, in which the free electron fraction is proportional to the fiducial model plus the PCA decomposition.\nHere, we exploit the functional form of the fiducial model in order to test other possible reionization parameterizations.\nFollowing this idea, for the redshift-symmetric, \\textit{tanh} description, we insert the PCA decomposition inside the argument of the hyperbolic tangent:\n\\begin{equation}\nx_e^{B,tanh}(z) = \\frac{1+f_{\\rm He}}{2} \\left(1+ \\tanh \\left[ \\frac{y(z_{\\rm{re}})-y(z)}{\\Delta y} + \\sum_\\mu m_\\mu^B S_\\mu (z) \\right] \\right)~.\n\\label{eqn:tanh_b}\n\\end{equation}\nNotice that we recover the fiducial \\textit{tanh} model by setting the amplitudes $m_{\\mu}$ to $0$.\nWe perform an analogous replacement for the redshift-asymmetric parameterization:\n\\begin{equation}\n x_e^{B,asym}(z) =\n \\begin{cases}\n\t1+f_{\\rm He} & \\mbox{for } z z_{\\rm early}.\n \\end{cases}\n \\label{eqn:asym_b}\n\\end{equation}\nWe take for the specific parameters of the \\textit{tanh} and \\textit{asym} cases the fiducial values given in Sec.~\\ref{subsec:canonical}.\n\n\\end{itemize}\n\n\\subsection{\\texttt{PCHIP}}\nThe third and last method we adopt in order to describe the reionization history is based on a non-parametric form for the free electron fraction $x_e(z)$, which is described using the function values $x_e(z_i)$ in a number $n$ of fixed redshift points $z_1,\\ \\ldots,\\ z_n$.\nFollowing the procedure adopted for the PCA analyses, we fix the function to be a constant\nboth at low\nredshifts ($z\\leq6$) and at high redshifts ($z\\geq30$).\nThe first and the last redshift nodes we use to parameterize the function at intermediate redshifts are therefore $z_1=6$ and $z_n=30$, where we also want the function to be continuous:\nas a consequence, the values $x_e(z_1)=1+f_{\\rm He}$ and $x_e(z_n)=0$ are fixed\nand the number of varying parameters that describe $x_e(z)$ is always $n-2$.\nWe consider a case with a total of $n=7$ nodes (5 free parameters),\nlocated at redshifts\n\\begin{equation}\\label{eq:nodes7}\n z_i \\in \\{6,\\,7,\\,8.5,\\,10,\\,13,\\,20,\\,30\\}\\,,\n\\end{equation}\nin order to have the same number of free parameter than in the PCA cases.\n\nThe function $x_e(z)$ at $z\\neq z_i$ is computed through an interpolation among its values in the nodes.\nWe employ the\n``piecewise cubic Hermite interpolating\npolynomial'' (\\texttt{PCHIP})~\\cite{Fritsch:1980,Fritsch:1984}\nin a very similar way to Refs.~\\cite{Gariazzo:2014dla,DiValentino:2015zta,Gariazzo:2015qea,DiValentino:2016ikp},\nwhere the \\texttt{PCHIP}\\ function was adopted to describe the power spectrum\nof initial curvature perturbations, or the more recent work of \\cite{Hazra:2017gtx}, where the \\texttt{PCHIP}\\ method has also been used to model the evolution of $x_e(z)$.\nThe idea behind the \\texttt{PCHIP}\\ function is similar to that of the natural cubic spline,\nwith the difference that the monotonicity of the series of interpolating points\nmust be preserved.\nSpurious oscillations that may be introduced by the standard spline\nare avoided by imposing a condition on the first derivative of the function in the nodes,\nwhich must be zero if there is a change in the monotonicity\nof the point series.\nA more detailed discussion on the \\texttt{PCHIP}\\ function can be found in the appendix of Ref.~\\cite{Gariazzo:2014dla}.\n\nSummarizing, the free electron fraction in the \\texttt{PCHIP}\\ case is described by:\n\\begin{equation}\\label{eq:xe_pchip}\n x_e(z) =\n \\begin{cases}\n 1+f_{\\rm He}\n & \\mbox{for } z \\leq z_1, \\\\\n \\texttt{PCHIP}(z;\\ x_e(z_1),\\ \\ldots,\\ x_e(z_n))\n & \\mbox{for } z_1 < z < z_n, \\\\\n 0\n & \\mbox{for } z \\geq z_n,\n \\end{cases}\n\\end{equation}\nwhere $n$ will be 7 and the redshifts $z_i$ are reported in Eq.~\\eqref{eq:nodes7}.\n\nFor the values of the function in the varying nodes,\nwhich are the free reionization parameters in our Markov Chain Monte Carlo analyses,\nwe impose a linear prior $0 \\leq x_e(z_i) \\leq 1+f_{\\rm He}$,\nwith $i=2,\\ \\ldots,\\ n-1$.\nThis ensures that the free electron fraction is always positive and smaller than its value today.\nThe value of the reionization optical depth $\\tau$ \nthat we report in our results is derived from Eq.~\\eqref{eq:cumtau}.\n\n\\section{Cosmological data}\n\\label{sec:data}\nWe use Planck satellite 2015 measurements of the CMB temperature,\npolarization, and cross-correlation spectra~\\cite{Adam:2015rua,Ade:2015xua}\nto derive the constraints on the possible reionization histories~\\footnote{%\nWe make use of the publicly available Planck likelihoods~\\cite{Aghanim:2015xee}, see \\href{http:\/\/www.cosmos.esa.int\/web\/planck\/pla}{www.cosmos.esa.int\/web\/planck\/pla}.\n}.\nMore precisely, we exploit both\nthe high-$\\ell$ ($30 \\leq \\ell \\leq 2508$) and\nthe low-$\\ell$ ($2 \\leq \\ell \\leq 29$) $TT$\nlikelihoods\nbased on the reconstructed CMB maps\nand\nwe include the Planck\npolarization likelihoods in the low-multipole regime\n($2 \\leq \\ell \\leq 29$), plus the high-multipole ($30 \\leq \\ell \\leq 1996$) $EE$ and $TE$ likelihoods~\\footnote{The latest reionization constraints from the Planck collaboration do not consider the TE data in the analyses, due to its larger cosmic variance and its weaker dependence on the reionization optical depth, when compared to EE measurements, see \\cite{Adam:2016hgk}.}. \nAll these CMB likelihood functions depend on several nuisance parameters\n(e.g.\\ residual foreground contamination, calibration, and\nbeam-leakage~\\cite{Ade:2015xua,Aghanim:2015xee}),\nwhich have been properly considered and marginalized over. \nTo derive constraints on the reionization history and related parameters, we have modified the Boltzmann equations solver \\texttt{CAMB} code \\cite{Lewis:1999bs} and apply\nMarkov Chain Monte Carlo (MCMC) methods by means of an adapted version of the \\texttt{CosmoMC} package~\\cite{Lewis:2002ah}.\nAs for current constraints, we consider a minimal version of the $\\Lambda$CDM model, described by the following set of parameters: \n\\begin{equation}\\label{parameterPPS}\n\\{\\omega_{\\rm{b}},\\,\\omega_{\\rm{c}},\\, \\Theta_{\\rm{s}},\\,\\ln{(10^{10} A_{\\rm{s}})},\\,n_{\\rm{s}}\\}~,\n\\end{equation}\nwhere $\\omega_{\\rm{b}}\\equiv\\Omega_{\\rm{b}}h^2$ and $\\omega_{\\rm{c}}\\equiv\\Omega_{\\rm{c}}h^2$\nrepresent the physical baryon and cold dark matter energy densities, $\\Theta_{\\rm{s}}$\nis the angular scale of recombination, $A_{\\rm{s}}$ is the primordial power spectrum amplitude and $n_{\\rm s}$ the spectral index.\nNotice that we do not have $\\tau$ among the parameters included in our analyses, as $\\tau$ is a derived parameter.\nInstead, we will add the additional parameters describing the PCA and \\texttt{PCHIP}\\ reionization models, that will lead to the constraints presented in what follows. \n\n\\section{Results}\n\\label{sec:results}\nFigure~\\ref{fig:tau} shows the most relevant results from our analyses of Planck 2015 temperature and polarization data assuming different reionization histories.\nAs aforementioned, we shall focus on the cumulative redshift distribution function of the reionization optical depth, Eq.~\\eqref{eq:cumtau}.\nA large departure from $0$ at redshifts $z>10$ would indicate evidence for an early reionization contribution, and therefore for non-standard reionization sources as, for instance, energy injection from dark matter annihilations or from matter accretion on massive primordial black holes.\nNotice that the PCA-A method of Ref.~\\cite{Heinrich:2016ojb}, in which the PCA decomposition is added linearly to a fiducial $x^{\\rm{fid}}_{\\rm{e}}(z)$, leads \\emph{always} to an early contribution to the optical depth $\\tau$, i.e.\\ $\\tau$ is significantly different from 0 at $z>10$, in contrast to standard reionization scenarios.\nFurthermore, the presence of this early contribution is independent of the fiducial model,\nas we can see from\nthe four PCA-A cases depicted in Fig.~\\ref{fig:tau}, which provide the same predictions at $z>10$, differing only mildly at small redshifts, regardless whether the fiducial model is a constant function or it depends on the redshift instead. \n\n\\begin{figure}[t]\n\\centering \n\\includegraphics[width=0.85\\textwidth]{reioPCHIP_pol_sm_bands_taue_new.pdf}\n\\caption{\\label{fig:tau} Cumulative redshift evolution of the reionization optical depth $\\tau(z)$ for several possible reionization scenarios.\nThe black thin solid and dot-dashed lines illustrate the PCA-A scenario for the case of two fiducial models constant in redshift.\nThe two upper dot-dashed lines refer also to the PCA-A parameterization but with redshift-dependent fiducial models.\nThe two lower colored solid lines depict the PCA-B scenarios, while the thick solid black line and the blue contours show the mean value and the $1$, $2$ and $3\\sigma$ allowed regions within the \\texttt{PCHIP}\\ prescription.}\n\\end{figure}\n\nIn order to unravel the origin of this early reionization component present when using the PCA-A description, several tests have been carried out.\nFirstly, we have eliminated the physical limits in the PCA amplitudes, finding very similar results.\nSecondly, we have simulated mock Planck data with the hyperbolic tangent description and then fitted these data to a PCA-A modeling, using different fiducial models.\nWe always find two bumps in the recovered $x_e$, see Fig.~\\ref{fig:xe}, one located between $z=10$ and $z=15$ and a second one located between $z=20$ and $z=25$.\nUpcoming measurements from the Planck satellite could disentangle if this early reionization component is truly indicated by the data or instead it is due to the adopted modeling or to other effects (i.e.\\ systematics).\n\nFurthermore, this early reionization component is definitely absent when other possible reionization histories are used in the analyses. \nFor instance, in the case of PCA-B parameterizations (see Eqs.~\\eqref{eqn:tanh_b} and \\eqref{eqn:asym}), there is no early reionization contribution, as $\\tau(z)$ is negligibly small for $z>10$.\nThe same happens for the \\texttt{PCHIP}\\ method, in which the mean reconstructed value of $\\tau(z)$ is also very small at high redshifts, showing little evidence for an early reionization component (see also Ref.~\\cite{Hazra:2017gtx}).\nNotice that the value of $\\tau$ today is smaller in the PCA-B approaches than in the PCA-A and \\texttt{PCHIP}\\ descriptions.\nHowever, this behavior is the expected one, as the PCA-B scenarios are very close to those explored by the Planck collaboration in Ref.~\\cite{Adam:2016hgk}, where it was found that the current value of $\\tau$ is $0.058\\pm 0.012$ for the hyperbolic tangent case, in perfect agreement with our findings here, even if we make use of the 2015 Planck likelihood only (the mean value is $\\tau=0.068$ for the very same model).\nThe differences between the PCA-A and PCA-B cases can be understood from the fact that the case B imposes a more restrictive functional form on the ionization history.\n \n\\begin{figure}\n\\centering \n\\includegraphics[width=0.85\\textwidth]{reioPCHIP_pol_sm_bands_xe_new_xe.pdf}\n\\caption{\\label{fig:xe}\nFree electron fraction as a function of the redshift for several possible reionization scenarios.\nLine styles and colors are the same as in Fig.\\ref{fig:tau}.}\n\\end{figure} \n\nThe findings above are fully consistent with our limits on the free electron fraction $x_{e}(z)$ at a given redshift.\nFigure~\\ref{fig:xe} shows the free electron fraction for the \\texttt{PCHIP}\\ parameterization together with the other PCA-A and PCA-B models explored here.\nThe color coding is identical to that used in Fig.~\\ref{fig:tau}. \nNotice that for the PCA-A models the free electron fraction is almost constant in the redshift interval $z=10-30$, as a consequence of the choice of the fiducial model, and therefore there will always be an early reionization component \\emph{within this approach}.\nHowever, when considering either the \\texttt{PCHIP}\\ or the PCA-B models, the free electron fraction is significantly smaller than $0.2$ for redshifts above $z=15$ and it is almost negligible above $z=20$.\nTherefore, the fact that current CMB observations need an early contribution to reionization is highly questionable, as it strongly depends on the framework used to analyze the data.\nUsing Planck CMB temperature and polarization data within the \\texttt{PCHIP}\\ analysis,\nwe find $x_e<0.90$, $<0.49$ and $<0.13$ at $2\\sigma$ in the nodes at $z=10$, $13$ and $20$, respectively.\nFluctuations in the lower $1\\sigma$ limits shown in Fig.~\\ref{fig:xe}\nare numerical artifacts that appear when computing the error bands at intermediate positions between the fixed \\texttt{PCHIP}\\ nodes and cannot be considered as significant.\nFigure~\\ref{fig:pchipamplitudes} shows the $68\\%$ and $95\\%$~CL allowed regions for the amplitudes of the \\texttt{PCHIP}\\ nodes, i.e.\\ the $x_{\\rm e} (z)$ at the redshifts listed in Eq.~\\eqref{eq:nodes7},\nfrom the Planck CMB measurements considered here.\nA quick inspection of Fig.~\\ref{fig:pchipamplitudes} tells us that all the amplitudes are perfectly compatible with a vanishing value.\nOnly one of them, $m_5$, the node corresponding to $z=13$, shows a very mild departure from $0$. However, this mild departure is far from being a significant effect, as it barely appears at $1\\sigma$.\nWe can therefore conclude that there is no evidence for a high redshift component in $x_{\\rm e}(z)$.\nNotice also from Fig.~\\ref{fig:pchipamplitudes} that, in general, the \\texttt{PCHIP}\\ amplitudes are anti-correlated among themselves.\nWe also illustrate the derived distribution for the value of the reionization optical depth, $\\tau_{\\rm PC}$, which is significantly correlated with the nodes at the higher redshifts.\nEven a modest increase of $x_{\\rm e}$ at $z=13$ or at $z=20$ would imply a significant shift towards larger values of the current reionization optical depth. \n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.85\\textwidth]{pchipNodes}\n\\caption{\\label{fig:pchipamplitudes} $68\\%$ and $95\\%$~CL allowed regions from the Planck CMB measurements considered here on the amplitudes in the \\texttt{PCHIP}\\ approach, together with the one-dimensional posterior probability distributions.}\n\\end{figure} \n\nIn order to further assess our findings above, we adopt here two information criteria \nwhich have been widely exploited\nin astrophysical and cosmological contexts (see Refs.~\\cite{Liddle:2007fy,Trotta:2008qt} for details), namely the frequentist Akaike Information Criterion (AIC)\n\\begin{equation}\n\\textrm{AIC}\\equiv -2 \\ln \\mathcal{L }_{\\rm{max}} +2k~,\n\\end{equation}\nwhich establishes that the penalty term between competing models is twice the number of free parameters in the model, $k$; and the Bayesian Information Criterion (BIC)\n\\begin{equation}\n\\textrm{BIC}\\equiv -2 \\ln \\mathcal{L }_{\\rm{max}} +k\\ln N~,\n\\end{equation}\nin which the penalty is proportional to the number of free parameters in the model times the logarithm of the number of data points $N$.\nThe best model is the one minimizing either the AIC or the BIC criteria. \nFollowing Ref.~\\cite{Liddle:2007fy}, the significance against a given model will be judged based on the Jeffreys' scale, which will characterize a difference $\\Delta$AIC (BIC)$>5$ ($>10$) as a strong (decisive) evidence against the cosmological model with higher AIC (BIC) value.\n\nAdopting first the AIC prescription, we shall compare the different models explored here to the standard scenario, in which reionization is described via just only one parameter, $\\tau$. \nThis \\emph{tau-only} cosmological model gives $-2\\ln \\mathcal{L }_{\\rm{max}}=12956.2$~\\cite{Ade:2015xua}.\nAs a comparison, the PCA-A case with constant fiducial model $x_{e}=0.15$ ($0.05$) provides $-2\\ln \\mathcal{L }_{\\rm{max}}=12954.0$ ($12953.2$). \nNotice that both the PCA-A cases have a higher AIC value than the \\emph{tau-only} cosmology because of the larger number of parameters.\nThe values for $\\Delta$AIC are $\\Delta$AIC $=5.8$ and $5$, respectively, and therefore there is strong evidence against these possible reionization histories.\nAlso within the PCA-A description, we get $-2\\ln \\mathcal{L }_{\\rm{max}}=12956.5$ ($12958.3$) in the PCA-A \\emph{tanh} (\\emph{asym}) fiducial approach.\nThese two models also provide a larger AIC than the \\emph{tau-only} scenario, and again, there will be strong (decisive) evidence against the PCA-A \\emph{tanh} (\\emph{asym}), in favor of the simplest and most economical \\emph{tau-only} reionization paradigm.\nIn the case of the \\texttt{PCHIP}\\ approach, our results lead to $-2\\ln \\mathcal{L}_{\\rm{max}}=12954.5$, which \nalso indicates strong preference for the \\emph{tau-only} scheme. We point out that all the reported values of $-2\\ln \\mathcal{L }_{\\rm{max}}$\nare taken from the corresponding MCMC chains, and not from a specific minimization algorithm.\nFor this reason, they may not be extremely precise and they must be considered only as fair estimates of the true values of each $-2\\ln \\mathcal{L }_{\\rm{max}}$, with possible errors of order unity, as estimated from the different parallel MCMC chains.\nIn the case of the PCA-B parameterizations,\nthe difference in the minimum $-2\\ln \\mathcal{L }_{\\rm{max}}$ from the different MCMC parallel chains is too large to give even a fair estimate of the true minimum,\nand we decide not to claim any evidence against these two descriptions, for the reasons listed above.\nHowever, we expect that these two models are equally good in fitting the CMB data, at a comparable level with respect to the \\emph{tau-only} scenario, as their reionization histories are extremely close to the standard cosmological framework, see Fig.~\\ref{fig:xe}.\nNevertheless, given the fact that the number of parameters in the PCA-B scheme is larger, the \\emph{tau-only} reionization description, with current data, will always be favored over the PCA-B parameterization.\n\nWe can also compare the different reionization descriptions among themselves using the BIC approach, as all of them have the same number of free parameters (five in total) and also the same number of data points.\nThe result of comparing the PCA-A and \\texttt{PCHIP}\\ scenarios among themselves will always give very weak or inconclusive answers, as none of them in particular is preferred over the other possible formulations.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nUnraveling the reionization period, which is still a poorly known period in the evolution of our universe, is one of the most important goals of current and future cosmological probes.\nThis is a mandatory step, not only towards a complete understanding of star formation and evolution, but also to answer questions such as the nature of the dark matter component~\\cite{Barkana:2001gr,Yoshida:2003rm,Somerville:2003sh,Yue:2012na,Pacucci:2013jfa,Mesinger:2013nua,Schultz:2014eia,Dayal:2015vca,Lapi:2015zea,Bose:2016hlz,Bose:2016irl,Corasaniti:2016epp,Menci:2016eui,Lopez-Honorez:2017csg,Villanueva-Domingo:2017lae}, constraining dark matter properties or the abundance of accreting massive primordial black holes~\\cite{Pierpaoli:2003rz,Mapelli:2006ej,Natarajan:2008pk,Natarajan:2009bm,Belikov:2009qx,Huetsi:2009ex,Cirelli:2009bb,Kanzaki:2009hf,Natarajan:2010dc,Giesen:2012rp,Diamanti:2013bia,Lopez-Honorez:2013lcm,Lopez-Honorez:2016sur,Poulin:2016nat,Poulin:2015pna,Ricotti:2007au,Horowitz:2016lib,Ali-Haimoud:2016mbv,Blum:2016cjs,Poulin:2017bwe}.\nCurrently, the most accurate measurement of the reionization period comes from Cosmic Microwave Background data through a redshift-integrated quantity: the reionization optical depth $\\tau$.\nThe latest measurements of the Planck collaboration provide a value of $\\tau = 0.055 \\pm 0.009$~\\cite{Aghanim:2016yuo, Adam:2016hgk}, which shows a very good agreement with observations of Lyman-$\\alpha$ emitters at $z\\simeq 7$~\\cite{Stark:2010qj,Treu:2013ida,Pentericci:2014nia,Schenker:2014tda,Tilvi:2014oia}.\nHowever, this measured value of $\\tau$ may correspond to very different reionization histories.\n\nThe most commonly exploited model for the time evolution of the free electron fraction, $x_e(z)$, uses a step-like transition, implemented via a hyperbolic tangent~\\cite{Lewis:2008wr}.\nRecently, there have been several studies in the literature claiming that Planck 2015 data may prefer a high-redshift ($z>15$) component to the reionization optical depth, implying a clear departure from the hyperbolic tangent picture.\nHere we consider a number of possible reionization scenarios, some of them previously explored in the literature, such as the Principal Component Analysis (PCA) approach of Refs.~\\cite{Hu:2003gh,Mortonson:2007hq,Mortonson:2007tb,Mortonson:2008rx,Mortonson:2009qv,Mortonson:2009xk,Mitra:2010sr,Heinrich:2016ojb}, or the \\texttt{PCHIP}\\ framework~\\cite{Hazra:2017gtx}. \nWe find that the claimed need for an early reionization component from present data is highly debatable, as it is only motivated by a particular set of reionization descriptions.\nIn other possible reionization prescriptions, equally allowed by data, we do not find such a preference.\nTo assess this, we have applied the frequentist Akaike Information Criterion (AIC), which provides an unbiased model comparison method.\nThe AIC results show that there is strong evidence from current data against more complicated reionization scenarios, always favoring the minimal scenario with the symmetric hyperbolic tangent function and described by one single parameter, the reionization optical depth $\\tau$. In other words, current Planck CMB analyses are unable to provide more information beyond that based on a single value of the $\\tau$. Upcoming data from the Planck mission will help in further disentangling the reionization history of our universe. \n\n\n\\acknowledgments\nThis work makes use of the publicly available \\texttt{CosmoMC}~\\cite{Lewis:2002ah} and \\texttt{CAMB}~\\cite{Lewis:1999bs} codes and of the Planck data release 2015 Likelihood Code~\\cite{Aghanim:2015xee}. OM and PVD would like to thank the Fermilab Theoretical Physics Department for hospitality.\nOM and PVD are supported by PROMETEO II\/2014\/050, by the Spanish Grants SEV-2014-0398 and FPA2014--57816-P of MINECO and by the European Union's Horizon 2020 research and innovation program under the Marie Sk\\l odowska-Curie grant agreements No.\\ 690575 and 674896. \nThe work of SG was supported by the Spanish grants\nFPA2014-58183-P,\nMultidark CSD2009-00064 and\nSEV-2014-0398 (MINECO),\nand PROMETEOII\/2014\/084 (Generalitat Valenciana).\n\nThis manuscript has been authored in part by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This work made extensive use of the NASA Astrophysics Data System and {\\tt arXiv.org} preprint server.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe ALICE experiment at the LHC is dedicated to the study of strongly interacting matter under extreme conditions, i.e. high temperature, which can be reached in heavy-ion collisions. In such collisions, the formation of a Quark-Gluon Plasma (QGP) is expected. Dielectrons are produced at all stages of the collision and therefore carry information about the whole evolution of the system. Since they do not interact strongly with the medium, they are a prime probe to study the properties of the QGP. Dielectrons stem from decays of pseudoscalar and vector mesons, from semi-leptonic decays of correlated open-charm and open-beauty hadrons and from internal conversions of direct photons. In heavy-ion collisions, additional sources are expected, i.e. thermal radiation from the QGP and hadron gas. The medium introduces modifications of the vector meson properties, in particular the short-lived $\\rho$, related to chiral symmetry restoration. In addition, the initial conditions of the collisions are expected to change compared to elementary collisions due to modifications of the parton distribution functions in nuclei. The latter can be studied in proton-lead (p--Pb) collisions, whereas pp collisions provide an important vacuum baseline. It is crucial to first understand the dielectron production in vacuum to single out the signal characteristics of the QGP. Moreover, proton-proton (pp) collisions can also be used to study the heavy-flavour (HF) and direct photon production.\n\nIn the following, the steps of the data analysis are explained and the first measurements of the dielectron production in pp collisions at $\\sqrt{s} = 7$\\,TeV are presented and discussed~\\cite{ref-ee}.\n\n\n\\section{Data analysis and results}\n\n\nThe analysis is performed with pp data taken during the first data-taking period of the LHC in 2010 with the ALICE detector. The integrated luminosity of the data sample is $L_{\\rm int} = 6.0\\pm0.2$\\,nb$^{-1}$.\nAfter identifying electrons in the ALICE detector it is not a priori clear which electrons belong to the same pair. We follow a statistical approach to obtain the final spectrum. The electrons and positrons are combined to an opposite-sign spectrum (OS), which includes not only the signal but also background, that can be purely combinatorial or have some residual correlation from jets or cross pairs from double Dalitz decays. This background is estimated by the same-sign spectrum (SS). Residual acceptance differences for OS and SS pairs are estimated with mixed events and taken into account during the subtraction of the background. Finally, the background-subtracted spectrum is corrected for tracking and particle identification inefficiencies within the ALICE acceptance ($p_{\\rm T,e} > 0.2$\\,GeV\/$c$, $ \\eta_{\\rm e}<0.8 $).\n\nIn Fig. 1 the measured dielectron cross section as a function of $m_{\\rm ee}$ is compared to a so-called hadronic cocktail, which includes all known sources of dielectron production from hadron decays and uses parameterisations of measured spectra as input when available. Where no measurements are available $m_{\\rm T}$-scaling~\\cite{ref-mt-scaling} is applied. The HF contributions are simulated using the Perugia2011 tune of PYTHIA~6~\\cite{ref-pythia,ref-pythia2011}. The resulting dielectron pairs from the hadron decays are then filtered through the ALICE acceptance.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.355]{.\/2018-09-03-2018-09-03-invmassintegrated.pdf}\n\\caption{Dielectron cross section as a function of $m_{\\rm ee}$ compared to a cocktail of known hadronic sources.}\n\\end{figure}\n\nA good agreement is observed between the cocktail and the data. The charm contribution already dominates the spectrum for $m_{\\rm ee} \\geq 0.5$\\,GeV\/$c^{2}$. The very large heavy-flavour contribution makes the measurement of thermal radiation from the medium in heavy-ion collisions very challenging at LHC energies. To separate the heavy-flavour background from thermal radiation from the QGP in a future heavy-ion run in the intermediate-mass range (IMR, $\\phi < m_{\\rm ee} < J\/\\psi$), an additional variable, the pair-distance-of-closest-approach ($\\rm DCA_{ee}$), is added to the traditional analysis as a function of $m_{\\rm ee}$ and $p_{\\rm T,ee}$. $\\rm DCA_{ee}$~is defined as:\n\\begin{equation}\n{\\rm DCA_{ee}} = \\sqrt{\\frac{({\\rm DCA_{{\\it xy},1}}\/\\sigma_{xy{ \\rm ,1}})^{2}+({\\rm DCA_{{\\it xy},2}}\/\\sigma_{xy,2})^{2}}{2}}\n\\end{equation}\n\nHere ${\\rm DCA}_{xy,i}$ is the closest distance between the reconstructed electron track and the primary collision vertex in the transverse plane. Its resolution is estimated from the covariance matrix of the track reconstruction and denoted as $\\sigma_{xy,i}$. In the case of weak decays, the decay electron candidates do not point to the vertex which leads to a broader DCA distribution than for tracks from prompt decays.\nThis can be seen in Fig. 2 and Fig. 3, where the $\\rm DCA_{ee}$~spectra are shown for two invariant mass regions. Fig. 2 shows the mass region between the $\\pi^{0}$ and the $\\phi$ mass. The light flavour template is taken from the $\\pi^{0}$ shape, normalised to the expected contribution from all light flavour sources. Fig. 3 shows the mass region around the $J\/\\psi$ mass peak. In both mass regions we can see a clear peak which can be described by the expected prompt contributions, whereas the tail of the spectrum is described by the broader contributions from charm and beauty.\n\\begin{figure}\n\\centering\n \\begin{minipage}{0.47\\textwidth}\n \\includegraphics[scale=0.35]{.\/2018-May-09-resonancedca.pdf}\n \\caption{Dielectron spectrum as a function of $\\rm DCA_{ee}$~for $0.14 < m_{\\rm ee} < 1.1$\\,GeV\/$c^2$~\\cite{ref-ee}.}\n \\end{minipage}\n \\begin{minipage}{0.47\\textwidth}\n \\includegraphics[scale=0.35]{.\/2018-May-09-jpsidca.pdf}\n \\caption{Dielectron spectrum as a function of $\\rm DCA_{ee}$~for $2.7 < m_{\\rm ee} < 3.3$\\,GeV\/$c^2$~\\cite{ref-ee}.}\n \\end{minipage}\n\\end{figure}\nIn Fig. 3 the $J\/\\psi$ from $B$-mesons can be seen in addition to the open HF contributions.\nIn the so-called intermediate mass region, located between the $\\phi$ and $J\/\\psi$ in the mass spectrum, the dominant contribution is from open HF.\nThe dielectron cross section as function of $p_{\\rm T,ee}$ and $\\rm DCA_{ee}$~is compared to a hadronic cocktail using PYTHIA 6 Perugia0~\\cite{ref-pythia2011} to estimate the $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ contributions in the left and right panels of Fig. 4, respectively.\n\\begin{figure}\n\\centering\n \\begin{minipage}{0.47\\textwidth}\n \\includegraphics[scale=0.35]{.\/2018-May-09-heavyflavourptee.pdf}\n \\end{minipage}\n \\begin{minipage}{0.47\\textwidth}\n \\includegraphics[scale=0.35]{.\/2018-May-09-heavyflavourdca.pdf}\n \\end{minipage}\n\\caption{Dielectron cross section as a function of $p_{\\rm T,ee}$ (left) and $\\rm DCA_{ee}$~(right) in the IMR compared to a cocktail calculated with PYTHIA~6~\\cite{ref-ee}.}\n\\end{figure}\nThe data are well described by the hadronic cocktail within the statistical and systematic uncertainties. The contribution from $\\rm c\\bar{c}$ dominates the dielectron yield at low $p_{\\rm T,ee}$ and relatively small $\\rm DCA_{ee}$, whereas the $\\rm b\\bar{b}$ becomes relevant at high $p_{\\rm T,ee}$ and large $\\rm DCA_{ee}$.\nTo investigate the processes of heavy-quark production we changed the generator from PYTHIA to POWHEG, switching from leading order in the HF quark generation to next-to-leading order. To quantify the differences the total $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ cross sections are extracted from the data by fitting the results two-dimensionally as a function of $p_{\\rm T,ee}$ and $m_{\\rm ee}$ and one-dimensionally as a function of $\\rm DCA_{ee}$~in the IMR allowing the contributions of the two HF components to vary. The results are shown in the left and right panels of Fig. 5 for PYTHIA and POWHEG\\cite{ref-powheg}, respectively.\nBoth fits give consistent results for a given MC event generator. The uncertainties are fully correlated between the cross sections extracted with PYTHIA and POWHEG. Significant model dependences are observed which reflect the different rapidity distribution of charm quarks and different $p_{\\rm T,ee}$ spectra of the $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ contributions predicted by the two models.\n\\begin{figure}\n\\centering\n \\begin{minipage}{0.47\\textwidth}\n \\includegraphics[trim={0, 0, 0, 1.5cm},clip,scale=0.357]{.\/2018-May-09-oneSigmaPythiaDCA0to8.pdf}\n \\end{minipage}\n \\begin{minipage}{0.47\\textwidth}\n \\includegraphics[trim={0, 0, 0, 1.5cm},clip,scale=0.357]{.\/2018-May-09-oneSigmaPowhegDCA0to8.pdf}\n \\end{minipage}\n\\caption{Total $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ cross sections with their systematic and statistical uncertainties, extracted from a fit of the measured dielectron yield from heavy-flavour hadron decays in ($m_{\\rm ee}$, $p_{\\rm T,ee}$) and in $\\rm DCA_{ee}$ with PYTHIA (left) and POWHEG (right) are compared to published cross sections (lines)~\\cite{ref-ee}.}\n\\end{figure}\nThe results are compared to independent measurements of $\\sigma_{\\rm c\\bar{c}}$\\cite{ref-ccbar} and $\\sigma_{\\rm b\\bar{b}}$\\cite{ref-bbbar} from single heavy-flavour particle spectra and found to be consistent within the large uncertainties. Once these uncertainties are reduced, the dielectron measurements can give further constraints on the MC event generators aiming to reproduce the heavy-flavour production mechanisms.\n\n\\section{Conclusion}\n\nTo summarise, ALICE measured the dielectron cross sections as a function of $m_{\\rm ee}$, $p_{\\rm T,ee}$, and $\\rm DCA_{ee}$~in pp collisions at $\\sqrt{s} = 7$\\,TeV. The hadronic cocktail is in good agreement with the measured dielectron cross sections in the three discussed observables, which suggests a good understanding of the dielectron cross section in the ALICE acceptance. We show that $\\rm DCA_{ee}$~makes it possible to separate prompt from non-prompt dielectron pairs, and thus will be a key tool to determine the average temperature of the QGP formed in heavy-ion collisions in the future. In the heavy flavour sector we can report a significant dependence of the total cross sections of charm and beauty when using PYTHIA and POWHEG, which reflects the sensitivity of the dielectron measurement to the underlying heavy-flavour production mechanisms implemented in the models.\n\n\n\n\\bibliographystyle{unsrt} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec-1}\n\nThis technical report is an extension of the paper of the same title, which is to appear at MUCOCOS'13. The technical report proves correctness of the ELB-trees operations' semantics and that the operations are lock-free.\n\nThe following is a brief summary of the design of the datastructure, which is detailed in section 3 of the paper.\nAll ELB-trees have a permanent root node $r$ with a single child.\nELB-trees are $k$-ary leaf-oriented search tree, or multiway search trees, so internal nodes have up to $k$ children and $k-1$ keys. An ELB-trees contain a set $E_r$ of integer keys in the range $(0;2^{63})$. The key 0 is reserved. Keys have an additional read-only bit: when the read-only bit is set, the key cannot be written to. ELB-trees offer 3 main operations:\n\\begin{itemize}\n\\item Search($e_1$, $e_2$) returns a key $e$ from $E_r$ satisfying $e_1 \\le e \\le e_2$, if such a key exists. Otherwise it returns $0$.\n\\item Remove($e_1$, $e_2$) removes and returns a key $e$ from $E_r$ satisfying $e_1 \\le e \\le e_2$, if such a key exists. Otherwise it returns $0$.\n\\item Insert($e$) adds $e$ to $E_r$, if $e$ was not in $E_r$ before.\n\\end{itemize}\nELB-trees can also be used as dictionaries or priority queues by storing values in the least significant bits of the keys.\n\nThe operations of ELB-trees cannot generally be expressed as atomic operations, as they occur over a time interval. As a consequence, series of concurrent operations cannot generally be expressed as ocurring serially, that is the semantics are not linearizable.\nHowever, the set $E_r$ is atomic.\n$E_r$ is the union of the keys in the leaf nodes of the ELB-tree.\nThe keys in internal nodes guide tree search.\n\nSection 2 provides formal definitions for terms used throughout the proof.\nThe proof starts in Section 3 by proving that ELB-trees are leaf-oriented search trees.\nWe prove through induction, that \nELB-trees are leaf-oriented search trees initially, and that all operations maintain that property.\nThe inductive step is assisted by two significant subproofs:\n\\begin{enumerate}\n\\item Rebalancing does not change the keys in $E_r$.\n\\item The keys in leaf nodes are within a permanent range.\n\\end{enumerate}\n\nThese properties hold due to the behavior of rebalancing.\nThe first subproof shows that rebalancing is deterministic, even when concurrent.\nThe second shows that leaf nodes have a range of keys they may contain and it never changes.\n\nGiven these properties, Section 4 derives the operations' semantics.\nSection 5 follows up by proving that the operations are lock-free.\nFirst we prove that some operation has made progress whenever a node is rebalanced.\nNext we prove that some operation has made progress whenever any part of an operation is restarted.\n\nSection 6 concludes the technical report with a summary.\n\\section{Definitions}\n\\label{sec-2}\n\nThis section introduces definitions used in the following proofs of the ELB-trees' properties. The definitions start with the terms used, before moving on to the contents and properties of nodes. Finally the intitial state of ELB-trees is formally defined.\n\nLet $L$ be the set of leaf ndoes, $I$ the set of internal nodes, and $T$ the set of points in time. The sets are disjoint.\n\nNodes contain:\n\\begin{description}\n\\item[$C_i (t)$] list of children of internal node $i$ at time $t$\n\\item[$S_i (t)$] list of keys in internal node $i$ at time $t$\n\\item[$E_n (t)$] keys represeted by the node $n$ where at time $t$:\\begin{center}$E_n (t) =$ \\begin{math} \\left\\{\n \\begin{array}{lr}\n $Non-zero keys in $l & n \\in L \\\\\n \\bigcup _{c \\in C_i (t)} E_c (t) & : n \\in I\n \\end{array}\n \\right. \\end{math}\\end{center} \n\\end{description}\n\nThe following node properties can be derived from their content:\n\\begin{description}\n\\item[$D_n (t)$] the descendants of node $n$ at time $t$: \\\\ $D_n (t)$ = \\begin{math} \\left\\{\n \\begin{array}{lr}\n \\emptyset & : n \\in L\\\\\n C_n (t) \\cup \\bigcup _{d \\in C_n(t)} D_d (t) & : n \\in I\n \\end{array}\n \\right.\\end{math} \\\\ $n$ is reachable when $reachable_n (t) \\equiv n \\in (\\{r\\} \\cup D_r (t))$\n\\item[$parent_n (t)$] the parents of node $n$: \\\\ $parent_n (t) = \\{i \\in reachable_r (t) | n \\in C_i (t)\\}, t \\in T$\n\\end{description}\n\nInitially $r$ has one child $C_r (0) = \\langle ic \\rangle$, and one grandchild $C_{ic} (0) = \\\\ \\langle ln \\rangle$.\nThe grandchild is an empty leaf node $E_{ln} (0) = \\emptyset \\wedge E_r (0) = \\emptyset$.\n\n\\section{Search tree proof}\n\\label{sec-3}\n\nThis section proves that ELB-trees are $k$-ary leaf-oriented search trees.\nIn such a tree, all nodes except the root have one parent, and all internal nodes have strictly ordered keys.\nSpecifically the $i$'th key in a node provides an upper bound for the $i$'th child of the node, and a lower bound for the $i + 1$'th child.\nThe key ordering is formally expressed as: \n\\begin{center} $W_i (t) \\equiv \\forall j \\in [0;C_i (t)). E_{{C_i(t)} _t} \\subseteq (0; {S_i}_j] \\wedge E_{{C_i (t)} _t} \\subseteq ({S_i} _j; 2^{63})$ \\end{center}\nThe tree property is formally expressed as:\n\\begin{center} $\\forall n \\in reachable_n (t). \\left\\vert parent_n (t) \\right\\vert = 1 \\vee n = r$ \\end{center}\nThe properties are proven inductively, but doing so requires several intermediate steps.\nTo begin with, we will show that the behavior of rebalancing of search trees is deterministic, and does not change $E_r$.\n\n\\begin{lemma}\\label{ro-reb}\nUnbalanced nodes and their parent are read-only while rebalancing. \\end{lemma}\n\\begin{proof} While finding the nodes involved in rebalancing, they are made read-only: internal nodes are made read-only by setting their status field, and\nleaf nodes are made read-only by setting the read-only bit of all their keys, see Figure~16 in the paper.\\end{proof}\n\n\\begin{lemma}\\label{reb-nodes}\nIf $W_r$ holds and the unbalanced nodes' parent is still reachable, all threads can find the nodes involved in a rebalancing from the status field of the unbalanced nodes grandparent, . \\end{lemma}\n\\begin{proof} The status field stores the key of the unbalanced node and its parent.\nSince $W_r$ holds, the nodes can be found by searching for the key in the grandparent and parent of the unbalanced node.\\end{proof}\n\n\\begin{lemma}\\label{inv-detreb}\nRebalancing completes deterministically exactly once, if $W_r$ holds. \\end{lemma}\n\\begin{proof} Rebalancing finds the involved nodes (Lemma \\ref{reb-nodes}) and decides how to rebalance (Lemma \\ref{ro-reb}) determinstically. The parent is replaced, and the grandparent's status field is cleared using ABA safe CAS operations, see Section~3b of the paper. The grandparent has the status field \\{*,*,*,STEP2\\} when replacing the parent, ensuring that the grandparent is reachable when replacing the parent node.\\end{proof}\n\n\\begin{lemma}\\label{Er-reb}\n$E_r (t)$ does not change when rebalancing, if $W_r$ holds. \\end{lemma}\n\\begin{proof} The content of balanced nodes and their new parent is copied from the old nodes, while their content is read-only (Lemma \\ref{ro-reb}).\\end{proof}\n\nThe preceding lemmas show that rebalancing is well-behaved in search trees. The following lemmas will show that all operations maintain the tree property and $W_r$.\n\n\\begin{lemma}\\label{inv-tree}\nAll operations maintain the tree property, if $W_r$ holds. \\end{lemma}\n\\begin{proof} $descendants_n$ only changes when rebalancing. Specifically, $descendants_n$ changes when replacing an internal node $op$ with a new node $np$. \nThe children of $op$ had $op$ as their only parent, so all the children $np$ and $op$ share, will have $np$ as their only parent after rebalancing. The new children have $np$ as their only parent, because they have just been introduced, and the descendants of the new nodes have their parents replaced. Formally: \\begin{center} $(\\forall c \\in C_{op} (t_1). parent_c (t_1) = \\{op\\}) \\Rightarrow \\forall c \\in C_{np} (t_2). parent_c (t_2) = \\{np\\}$ \\end{center}\\end{proof}\n\n\\begin{lemma}\\label{lrange}\nLeaf nodes $l$ have a permanent range $R_l$ of keys they may contain, if $W_r$ holds.\\end{lemma}\n\\begin{proof} The lower bound is given by the keys of its ancestors. The ancestors change deterministically when $W_r$ holds (Lemma \\ref{inv-detreb}). Although the ancestors may change, their replacements use the same keys. \nInternal node keys are only introduced or removed when splitting and merging nodes, which results in two or three new nodes. \nWhen rebalancing results in two new nodes, the new parent has one less key. When rebalancing results in three new nodes, the new parent has one updated or additional key, which the old parent did not have. The updated or new key is copied from its the unbalanced nodes, so it only affects the new nodes. \\end{proof}\n\n\\begin{lemma}\\label{res-si}\nIf $W_r$ holds, the leaf node $l$ reached by $Search(e, e)$ satisfies: $W_r \\Rightarrow e \\in R_l$. \\end{lemma}\n\\begin{proof} Search visiting a node $n$ where $\\neg reachable_n (t)$ eventually restarts, so a terminating search only visits reachable nodes in the tree (Lemma \\ref{inv-tree}). Search of reachable nodes when $W_r$ holds is regular $k$-ary tree search.\\end{proof}\n\n\\begin{lemma}\\label{res-sl}\nIf $W_r$ holds, searching the leaf node $l$ from $t_{l1}$ to $t_{l2}$ must read the keys $O(t_{l1}, t_{l2}) \\cap R_l$. \\end{lemma}\n\\begin{proof} $l$ is read after a memory barrier, ensuring that $O(t_{l1}, t_{l2}) \\cap R_l$ are read.\\end{proof}\n\n\n\\begin{lemma}\\label{inv-Wr}\nAll writes to the tree maintain $W_r$. Formally: \\begin{center} $\\forall t_1, t_2 \\in T . (t_1 \\le t_2 \\wedge W_r (t_1)) \\Rightarrow W_r (t_2)$ \\end{center}\\end{lemma}\n\\begin{proof} Writes to the tree can be classified into: key insertion, key removal, and rebalancing. \nRebalancing maintains $W_r$ (Lemma \\ref{lrange}).\nKey removal and insertion only affects the keys in the tree.\n$remove(e_1, e_2, t_1, t_2)$ removes an key from a leaf node $l$, which maintain $W_r$. \n$insert(e, t_1, t_2)$ inserts into leaf nodes for which $\\forall t \\in T. W_r (t) \\Rightarrow e \\in R_l$ (Lemma \\ref{res-si}), which maintain $W_r$. \\end{proof}\n\n\\begin{theorem}\\label{lost}\nELB-trees are leaf-oriented search trees. \\end{theorem}\n\\begin{proof} ELB-trees are trees and $W_r$ holds initially. All operation on ELB-trees maintains the tree property (Lemma \\ref{inv-tree}) and $W_r$ (Lemma \\ref{inv-Wr}).\\end{proof}\n\nThis section proves that ELB-trees are leaf-oriented search trees. Such proofs are sufficient to derive the semantics of concurrent searches and serial insertions and removals. The next section will derive the semantics of the concurrent operations, which requires a few additional lemmas.\n\n\\section{Correctness}\n\\label{sec-4}\n\nThis section derives the semantics of the operations. But first we will introduce some terms to reason about the results of such operations. Let: \n\\begin{description}\n\\item[$search(e_1, e_2, t_1, t_2)$] be the result of a search operation matching against keys $e \\in [e_1;e_2]$ starting at $t_1$ and ending at $t_2$;\n\\item[$remove(e_1, e_2, t_1, t_2)$] be the result of a remove operation matching against keys $e \\in [e_1;e_2]$ starting at $t_1$ and ending at $t_2$;\n\\item[$insert(e, t_1, t_2)$] be an insert $e$ operation starting at $t_1$ and ending at $t_2$;\n\\item[$O(t_1, t_2)$] be the keys that were in $E_r$ at all times during $[t_1;t_2)$: \\begin{center} $O(t_1, t_2) = \\left\\{ e | \\forall t \\in [t_1;t_2) . e \\in E_r (t) \\right\\}$; and \\end{center}\n\\item[$U(t_1, t_2)$] be the keys that were in $E_r$ at some time during $[t_1;t_2)$: \\begin{center} $U(t_1, t_2) = \\left\\{ e | \\exists t \\in [t_1;t_2). e \\in E_r (t) \\right\\}$. \\end{center}\n\\end{description}\n\nWe first prove properties of search operations, then derive the operations' semantics:\n\n\\begin{lemma}\\label{sl}\nSearching a set of leaf nodes $RL$ from $t_1$ to $t_2$ reads the keys $\\bigcup _{l \\in RL} R_l \\cap O(t_1, t_2)$. \\end{lemma}\n\\begin{proof}\nThe search reads the keys $\\bigcup _ {l \\in RL} R_l \\cap O(t_{l1}, t_{l2})$ (Lemma \\ref{res-sl}). $\\forall l \\in RL . O(t_{l1}, t_{l2}) \\subseteq O(t_1, t_2)$ holds, as any key in the tree during $t_1$ to $t_2$ must have been in the tree for all fragments of that duration.\n\\end{proof}\n\n\\begin{theorem} $search(e_1, e_2, t_1, t_2)$ can only return $0$ (fail) if there are no matching entries in $E_r$ at all times during $[t_1, t_2)$: \\begin{center} $search(e_1, e_2, t_1, t_2) = 0 \\Rightarrow [e_1; e_2] \\cap O(t_1, t_2) = \\emptyset$ \\end{center} \\end{theorem}\n\\begin{proof}\n$search(e_1, e_2, t_1, t_2) = 0$ implies that a set of leaf nodes $RL$ have been searched, where $[e_1;e_2] \\subseteq \\bigcup _{l \\in RL} R_l$. If there was an key in $[e_1;e_2] \\cap O(t_1, t_2)$ it would have been read (Theorem \\ref{lost}, Lemma \\ref{sl}).\\end{proof}\n\n\\begin{theorem} Successful searches return a matching key that was in $E_r$ at some point in time during $[t_1;t_2)$: \\begin{center}$e = search(e_1, e_2, t_1, t_2) \\Rightarrow (e \\in U(t_1, t_2) \\wedge e \\in [e_1; e_2])$\\end{center} \\end{theorem}\n\\begin{proof} Successful searches return a key $e$ that was read from a leaf. Since $e$ was read it must have been in $E_r$ (Lemma \\ref{sl}).\\end{proof}\n\n\n\\begin{theorem} Remove can only return $0$ (fail) if there are no matching entries in $E_r$ at all times during $[t_1, t_2)$:\n\\begin{center} $remove(e_1, e_2, t_1, t_2) = 0 \\Rightarrow O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset$. \\end{center}\\end{theorem}\n\\begin{proof} Terminating remove operations that return $0$ have searched a set of leafs $RL$ satisfying $[e_1; e_2] \\subseteq \\bigcup _{l \\in RL} R_l$ (Lemma \\ref{sl}), so any keys in $O(t_1, t_2) \\cup [e_1;e_2]$ would have been read.\\end{proof}\n\n\\begin{theorem} Successful remove operations remove matching a key $e$ from $E_r$ that was in $E_r$ at some point in time during $[t_1;t_2)$: \\begin{center} $e = remove(e_1, e_2, t_1, t_2) \\ne 0 \\Rightarrow$ \\ $(e_1 \\le e \\le min(O(t_1, t_2) \\cap [e_1 ; e_2]) \\le e_2 \\wedge e \\in U(t_1, t_2))$ \\end{center} \\end{theorem}\n\\begin{proof} Terminating remove operations have searched a set of leafs $RL$ satisfying $[e_1; e] \\subseteq \\bigcup _{l \\in RL} R_l$ (Lemma \\ref{sl}). Any keys smaller than $e$ in $O(t_1, t_2) \\cup [e_1;e_2]$ would have been read.\\end{proof}\n\n\\begin{theorem} $insert(e, t_1 , t_2)$ adds $e$ to the $E_r$, if $e \\notin U (t_1 , t_2 )$. \\end{theorem}\n\\begin{proof} Insert operations terminate when they use a successful CAS operation to write the key into an empty key of a leaf node $l$ where $e \\in R_l$ (Lemma \\ref{res-si}). The CAS operations success implies the key is not read-only, and hence $reachable_l (t_2)$.\\end{proof}\n\nTheorem 2-6 can be summarized as: \\\\\n$e = search( e_1, e_2, t_1, t_2 ) \\Rightarrow$ \\begin{math} \\left\\{\n \\begin{array}{lr}\n O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n e_1 \\le e \\le e_2 \\wedge e \\in U(t_1, t_2) & : e \\neq 0\n \\end{array}\n \\right. \\end{math}\n\\\\$e = remove(e_1, e_2, t_1, t_2) \\Rightarrow$ \\begin{math} \\left\\{\n \\begin{array}{lr}\n O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n e_1 \\le e \\le min([e_1; e_2] \\cap O(t_1, t_2)) \\\\ ~ \\wedge e \\in U(t_1, t_2) & \\raisebox{11pt}{$: e \\neq 0$}\n \\end{array}\n \\right. \\end{math}\n\\ $insert(e, t_1, t_2)$ adds $e$ to $E_r$, if $e \\notin U(t_1, t_2)$.\n\n\n\\section{Lock-freedom}\n\\label{sec-5}\n\nLock-freedom guarantees that as long as some thread is working on an operation $o_1$, some operation $o_2$ is coming closer to terminating. In this case we say $o_1$ is causing progress, and $o_2$ is making progress. The operations $o_1$ and $o_2$ can be different.\nFor ELB-trees, this means that whenever a thread is searching, inserting, or removing, some thread must be making progress. \nThe following is proof that the operations are lock-free:\n\n\\begin{lemma}\\label{ter-op}\nOperations eventually terminate or restart part of their operation. \\end{lemma}\n\\begin{proof} The operations' algorithms have loops in the following for: node search, tree search, rebalancing, and updating keys in leafs. The algorithms are given in the paper~\\cite{bkp13}. Without concurrency, they iterate up to K, tree height, tree height, and 1 times. With concurrency, tree search, rebalancing, and key update loops may restart part of their operation.\\end{proof}\n\n\\begin{lemma}\\label{lf-rebl}\nRebalancing leaf nodes cause progress. \\end{lemma}\n\\begin{proof} If the nodes are written to between deciding to rebalance and rebalancing, some operation has made progress.\nIf there are no writes, the size of the first node is either D or S, resulting in balanced nodes of $size \\in [min(2 S, 0.5 D); D-1]$. Such nodes can be removed from and inserted into at least once before requiring additional rebalancing. As such, every time a rebalancing completes, one operation has made progress.\\end{proof}\n\n\\begin{lemma}\\label{lf-rebi}\nRebalancing internal nodes cause progress. \\end{lemma}\n\\begin{proof} Rebalancing internal nodes leads to child nodes that can be rebalanced at least one. Each leaf rebalancing cause progress (Lemma \\ref{lf-rebl}), hence each internal rebalancing cause progress. \\end{proof}\n\n\\begin{theorem}\\label{lf-s}\nSearch causes progress. \\end{theorem}\n\\begin{proof} Search eventually terminates, similar to $k$-ary tree search, or rebalances a node (Lemma \\ref{res-si}). In the first case the search operation is making progress. In the second case some operation is making progress (Lemma \\ref{lf-rebl}, Lemma \\ref{lf-rebi}).\\end{proof}\n\n\\begin{theorem}\\label{lf-ri}\nRemove and insert operations cause progress. \\end{theorem}\n\\begin{proof} The operations proceed as searches followed by writes to leaf nodes. The leaf node write takes a bounded number of steps, as each key may be read once, but the steps can be restarted due to rebalancing, or other insertions and removals terminating. In the first case, some operation is nearing termination, and in the second case some operation terminated (Lemma \\ref{lf-rebl}, Lemma \\ref{lf-rebi}).\\end{proof}\n\n\\section{Conclusion}\n\\label{sec-6}\n\nThis technical report has introduced, proved, and derived properties of ELB-trees. ELB-trees have been proven to be leaf-oriented search trees. Their operations' semantics have been derived as:\\\\\n$e = search( e_1, e_2, t_1, t_2 ) \\Rightarrow$ \\begin{math} \\left\\{\n \\begin{array}{lr}\n O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n e_1 \\le e \\le e_2 \\wedge e \\in U(t_1, t_2) & : e \\neq 0\n \\end{array}\n \\right. \\end{math}\n\\\\$e = remove(e_1, e_2, t_1, t_2) \\Rightarrow$ \\begin{math} \\left\\{\n \\begin{array}{lr}\n O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n e_1 \\le e \\le min([e_1; e_2] \\cap O(t_1, t_2)) \\\\ ~ \\wedge e \\in U(t_1, t_2) & \\raisebox{11pt}{$: e \\neq 0$}\n \\end{array}\n \\right. \\end{math}\n\\ $insert(e, t_1, t_2)$ adds $e$ to $E_r$, if $e \\notin U(t_1, t_2)$.\nFinally the operations have been proven to be lock-free.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOur notation is standard (e.g., see \\cite{Bol98}, \\cite{CDS80}, and\n\\cite{HoJo88}); in particular, all graphs are defined on the vertex set\n$\\left\\{ 1,2,...,n\\right\\} =\\left[ n\\right] $ and $G\\left( n,m\\right) $\nstands for a graph with $n$ vertices and $m$ edges. We write $\\Gamma\\left(\nu\\right) $ for the set of neighbors of the vertex $u$ and set $d\\left(\nu\\right) =\\left\\vert \\Gamma\\left( u\\right) \\right\\vert .$ Given a graph $G$\nof order $n,$ we assume that the eigenvalues of the adjacency matrix of $G$\nare ordered as $\\mu\\left( G\\right) =\\mu_{1}\\left( G\\right) \\geq...\\geq\n\\mu_{n}\\left( G\\right) $. As usual, $\\overline{G}$ denotes the complement of\na graph $G$ and $\\omega(G)$ stands for the clique number of $G.$\n\nNosal \\cite{Nos70} showed that for every graph $G$ of order $n,$\n\\begin{equation}\nn-1\\leq\\mu\\left( G\\right) +\\mu\\left( \\overline{G}\\right) <\\sqrt{2}n.\n\\label{Nosin}%\n\\end{equation}\nQuite of attention has been given to second of these inequalities. In\n\\cite{Nik02} it was shown that%\n\\begin{equation}\n\\mu\\left( G\\right) +\\mu\\left( \\overline{G}\\right) \\leq\\sqrt{\\left(\n2-\\frac{1}{\\omega(G)}-\\frac{1}{\\omega(\\overline{G})}\\right) n\\left(\nn-1\\right) }, \\label{Nikin}%\n\\end{equation}\nimproving earlier results in \\cite{Hon95}, \\cite{HoSh00}, \\cite{Li96}, and\n\\cite{Zho97}. Unfortunately inequality (\\ref{Nikin}) is not much better then\n(\\ref{Nosin}) when both $\\omega(G)$ and $\\omega(\\overline{G})$ are large\nenough. Thus, it is natural to ask whether $\\sqrt{2}$ in (\\ref{Nosin}) can be\nreplaced by a smaller absolute constant for $n$ sufficiently large. In this\nnote we answer this question in the positive but first we state a more general problem.\n\n\\begin{problem}\nFor every $1\\leq k\\leq n$ find%\n\\[\nf_{k}\\left( n\\right) =\\max_{v\\left( G\\right) =n}\\left\\vert \\mu_{k}\\left(\nG\\right) \\right\\vert +\\left\\vert \\mu_{k}\\left( \\overline{G}\\right)\n\\right\\vert .\n\\]\n\n\\end{problem}\n\nIt is difficult to determine precisely $f_{k}\\left( n\\right) $ for every $n$\nand $k,$ so at this stage it seems more practical to estimate it\nasymptotically. In this note we show that\n\\begin{equation}\n\\frac{4}{3}n-2\\leq f_{1}\\left( n\\right) <\\left( \\sqrt{2}-c\\right) n\n\\label{mainin1}%\n\\end{equation}\nfor some $c>8\\times10^{-7}$ independent of $n.$ For $f_{2}\\left( n\\right) $\nwe give the following tight bounds%\n\\begin{equation}\n\\frac{\\sqrt{2}}{2}n-3\\left(\n\\sqrt{2}-\\varepsilon\\right) n.\n\\]\nWriting $A\\left( G\\right) $ for the adjacency matrix of $G,$ we have\n\\begin{equation}\n\\sum_{i=1}^{n}\\mu_{i}^{2}\\left( G\\right) =tr\\left( A^{2}\\left( G\\right)\n\\right) =2e\\left( G\\right) , \\label{basin}%\n\\end{equation}\nimplying that%\n\\[\n\\mu_{1}^{2}\\left( G\\right) +\\mu_{n}^{2}\\left( G\\right) +\\mu_{1}^{2}\\left(\n\\overline{G}\\right) +\\mu_{n}^{2}\\left( \\overline{G}\\right) \\leq2e\\left(\nG\\right) +2e\\left( \\overline{G}\\right) \\left( 1-\\frac{\\varepsilon}{\\sqrt{2}}\\right)\n^{2}n^{2}>\\left( 1-2\\varepsilon\\right) n^{2}%\n\\]\nwe find that\n\\begin{equation}\n\\left\\vert \\mu_{n}\\left( G\\right) \\right\\vert +\\left\\vert \\mu_{n}\\left(\n\\overline{G}\\right) \\right\\vert \\leq\\sqrt{2\\left( \\mu_{n}^{2}\\left(\nG\\right) +\\mu_{n}^{2}\\left( \\overline{G}\\right) \\right) }<\\sqrt\n{4\\varepsilon}n^{2}, \\label{in1}%\n\\end{equation}\nand so, $\\mu_{n}\\left( G\\right) +\\mu_{n}\\left( \\overline{G}\\right)\n>-\\sqrt{4\\varepsilon}n.$ We thus have $\\sqrt{4\\varepsilon}n^{4}\\geq\ns^{2}\\left( G\\right) .$ On the other hand, by (\\ref{prpin1}) and in view of\n$s\\left( G\\right) =s\\left( \\overline{G}\\right) ,$ we see that\n\\[\n\\mu_{1}\\left( G\\right) +\\mu_{1}\\left( \\overline{G}\\right) \\leq\nn-1+2\\sqrt{s\\left( G\\right) }\\frac{4n}{3}-2.\n\\]\nThis gives some evidence for the following conjecture.\n\n\\begin{conjecture}%\n\\[\nf_{1}\\left( n\\right) =\\frac{4n}{3}+O\\left( 1\\right) .\n\\]\n\n\\end{conjecture}\n\nWe conclude this section with an improvement of the lower bound in\n(\\ref{Nosin}). Using the first of inequalities (\\ref{prpin1}) we obtain\n\\begin{align*}\n\\mu_{1}\\left( G\\right) +\\mu_{1}\\left( \\overline{G}\\right) & \\geq\nn-1+\\frac{s^{2}\\left( G\\right) }{2n^{2}}\\left( \\frac{1}{\\sqrt{2e\\left(\nG\\right) }}+\\frac{1}{\\sqrt{2e\\left( \\overline{G}\\right) }}\\right) \\geq\\\\\n& \\geq n-1+\\sqrt{2}\\frac{s^{2}\\left( G\\right) }{n^{3}}.\n\\end{align*}\n\n\n\\section{A class of graphs}\n\nIn this section we shall describe a class of graphs that give the right order\nof $f_{2}\\left( G\\right) $ and, we believe, also of $f_{n}\\left( G\\right)\n.$\n\nLet $n\\geq4$. Partition $\\left[ n\\right] $ in $4$ classes $A,B,C,D$ so that\n$\\left\\vert A\\right\\vert \\geq\\left\\vert B\\right\\vert \\geq\\left\\vert\nC\\right\\vert \\geq\\left\\vert D\\right\\vert \\geq\\left\\vert A\\right\\vert -1.$ Join\nevery two vertices inside $A$ and $D,$ join each vertex in $B$ to each vertex\nin $A\\cup C,$ join each vertex in $D$ to each vertex in $C.$ Write $G\\left(\nn\\right) $ for the resulting graph.\n\nNote that if $n$ is divisible by $4,$ the sets $A,B,C,D$ have equal\ncardinality and we see that $G\\left( n\\right) $ is isomorphic to its complement.\n\nOur main goal to the end of this section is to estimate the eigenvalues of\n$G\\left( n\\right) .$ Write $ch\\left( A\\right) $ for the characteristic\npolynomial of a matrix $A.$ The following general theorem holds.\n\n\\begin{theorem}\n\\label{thch}Suppose $G$ is a graph and $V\\left( G\\right) =\\cup_{i=1}%\n^{k}V_{i}$ is a partition in sets of size $n$ such that\n\n(i) for all $1\\leq i\\leq k,$ either $e\\left( V_{i}\\right) =\\binom{n}{2}$ or\n$e\\left( V_{i}\\right) =0$;\n\n(ii) for all $1\\leq i\\frac{\\sqrt{2}}{2}n-3,\n\\]\nso all we need to prove is that $f_{2}\\left( n\\right) \\leq n\/\\sqrt{2}$.\n\nBy (\\ref{basin}) we have\n\\begin{equation}\n\\mu_{1}^{2}\\left( G\\right) +\\mu_{2}^{2}\\left( G\\right) +\\mu_{n}^{2}\\left(\nG\\right) +\\mu_{1}^{2}\\left( \\overline{G}\\right) +\\mu_{2}^{2}\\left(\n\\overline{G}\\right) +\\mu_{n}^{2}\\left( \\overline{G}\\right) \\leq n\\left(\nn-1\\right) . \\label{in2}%\n\\end{equation}\nBy Weyl's inequalities (\\cite{HoJo88}, p. 181), for every graph $G$ of order\n$n,$ we have\n\\[\n\\mu_{2}\\left( G\\right) +\\mu_{n}\\left( \\overline{G}\\right) \\leq\\mu\n_{2}\\left( K_{n}\\right) =-1.\n\\]\nHence, using $\\mu_{2}\\geq0$ and $\\mu_{n}\\leq-1$ we obtain\n\\[\n\\mu_{2}^{2}\\left( G\\right) \\leq\\mu_{n}^{2}\\left( \\overline{G}\\right)\n+2\\mu_{n}\\left( \\overline{G}\\right) +1<\\mu_{n}^{2}\\left( \\overline\n{G}\\right) .\n\\]\nHence, from (\\ref{in2}) and $\\mu_{1}\\left( G\\right) +\\mu_{1}\\left(\n\\overline{G}\\right) \\geq n-1,$ we find that\n\\[\n\\frac{\\left( n-1\\right) ^{2}}{2}+2\\mu_{2}^{2}\\left( G\\right) +2\\mu_{2}%\n^{2}\\left( \\overline{G}\\right) \\leq\\mu_{1}^{2}\\left( G\\right) +\\mu_{2}%\n^{2}\\left( G\\right) +\\mu_{n}^{2}\\left( G\\right) +\\mu_{1}^{2}\\left(\n\\overline{G}\\right) +\\mu_{2}^{2}\\left( \\overline{G}\\right) +\\mu_{n}%\n^{2}\\left( \\overline{G}\\right) \\leq n\\left( n-1\\right) .\n\\]\nAfter some algebra, we deduce that\n\\[\n\\mu_{2}\\left( G\\right) +\\mu_{2}\\left( \\overline{G}\\right) \\leq\\frac\n{\\sqrt{2}}{2}n,\n\\]\ncompleting the proof of inequalities (\\ref{mainin2}).\n\n\\section{Bounds on $f_{n}\\left( n\\right) $}\n\nIn this section we shall prove inequalities (\\ref{mainin3}). From (\\ref{in4}),\nas above, we have%\n\\[\nf_{n}\\left( n\\right) >\\frac{\\sqrt{2}}{2}n-3.\n\\]\nWe believe that, in fact, the following conjecture is true.\n\n\\begin{conjecture}%\n\\[\nf_{n}\\left( G\\right) =\\frac{\\sqrt{2}n}{2}+O\\left( 1\\right) .\n\\]\n\n\\end{conjecture}\n\nHowever we can only prove that $f_{n}\\left( G\\right) <\\left( \\sqrt\n{3}\/2\\right) n$ which is implied by the following theorem.\n\n\\begin{theorem}\nFor every graph $G$ of order $n,$\n\\[\n\\mu_{n}^{2}\\left( G\\right) +\\mu_{n}^{2}\\left( \\overline{G}\\right)\n\\leq\\frac{3}{8}n^{2}.\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nIndeed, suppose $\\left( u_{1},...,u_{n}\\right) $ and $\\left( w_{1}%\n,...,w_{n}\\right) $ are eigenvectors to $\\mu_{n}\\left( G\\right) $ and\n$\\mu_{n}\\left( \\overline{G}\\right) .$ Let\n\\[\nU=\\left\\{ i:u_{i}>0\\right\\} ,\\text{ \\ \\ \\ }W=\\left\\{ i:w_{i}>0\\right\\} .\n\\]\nSetting $V=\\left[ n\\right] ,$ we clearly have $\\mu_{n}^{2}\\left( G\\right)\n\\leq E_{G}\\left( U,V\\backslash U\\right) $ and $\\mu_{n}^{2}\\left(\n\\overline{G}\\right) \\leq E_{\\overline{G}}\\left( W,V\\backslash W\\right) $.\nSince $E_{G}\\left( U,V\\backslash U\\right) \\cap E_{\\overline{G}}\\left(\nW,V\\backslash W\\right) =\\varnothing,$ we see that the graph\n\\[\nG^{\\prime}=\\left( V,E_{G}\\left( U,V\\backslash U\\right) \\cup E_{\\overline\n{G}}\\left( W,V\\backslash W\\right) \\right)\n\\]\nis at most $4$-colorable and hence $G^{\\prime}$ contains no $4$-cliques. By\nTur\\'{a}n's theorem (e.g., see \\cite{Bol98}), we obtain $e\\left( G^{\\prime\n}\\right) \\leq\\left( 3\/8\\right) n^{2},$ completing the proof.\n\\end{proof}\n\n\\section{Bounds on $f_{k}\\left( n\\right) ,$ $2\\sqrt{2m\/k}$ then\n\\[\n\\sum_{i=1}^{n}\\mu_{i}^{2}\\left( G\\right) \\geq\\left( n-k\\right) \\mu_{k}%\n^{2}\\left( G\\right) >2m\\frac{n-k}{k}>2m,\n\\]\na contradiction. Hence, $\\left\\vert \\mu_{k}\\left( G\\right) \\right\\vert\n\\leq\\sqrt{2e\\left( G\\right) \/k},$ and, by symmetry, $\\left\\vert \\mu\n_{k}\\left( \\overline{G}\\right) \\right\\vert \\leq\\sqrt{2e\\left( \\overline\n{G}\\right) \/k}.$ Now\n\\[\n\\left\\vert \\mu_{k}\\left( G\\right) \\right\\vert +\\left\\vert \\mu_{k}\\left(\n\\overline{G}\\right) \\right\\vert \\leq\\sqrt{2e\\left( G\\right) \/k}%\n+\\sqrt{2e\\left( \\overline{G}\\right) \/k}\\leq\\sqrt{\\frac{2}{k}n\\left(\nn-1\\right) }<\\sqrt{\\frac{2}{k}}n,\n\\]\nproving inequality (\\ref{in5}). The proof of inequality (\\ref{in6}) goes along\nthe same lines, so we will omit it.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction: Janus and Hades}\n\n\nJanus solutions in string\/M-theory were originally introduced in the context of type IIB supergravity as a simple deformation of the $\\,\\textrm{AdS}_{5} \\times \\textrm{S}^{5}\\,$ background involving a non-trivial dilaton profile \\cite{Bak:2003jk}. The deformation breaks the $\\,\\textrm{SO}(2,4)\\,$ isometries of $\\,\\textrm{AdS}_{5}\\,$ to the $\\,\\textrm{SO}(2,3)\\,$ isometries of $\\,\\textrm{AdS}_{4}\\,$, but preserves the $\\,\\textrm{SO}(6)\\,$ isometries of the round $\\,\\textrm{S}^{5}\\,$. Soon after, a holographic interpretation of the solutions in \\cite{Bak:2003jk} was proposed in terms of a planar $\\,(1+2)$-dimensional interface in super Yang--Mills (SYM) separating two half-spaces with different coupling constants \\cite{Clark:2004sb}. The supersymmetric Janus was constructed in \\cite{Clark:2005te} using a 5D effective SO(6) gauged supergravity approach. Its ten-dimensional incarnation was put forward in \\cite{DHoker:2006vfr}, which provided the gravity dual of the $\\,\\mathcal{N}=1\\,$ supersymmetric interface first anticipated in \\cite{Clark:2004sb} and then constructed in \\cite{DHoker:2006qeo}. The $\\,\\mathcal{N}=1\\,$ supersymmetric Janus turns out to break the symmetry of the original (non-supersymmetric) Janus down to at least $\\,\\textrm{SU(3)} \\subset \\textrm{SO}(6)\\,$. The $\\,\\mathcal{N}=4\\,$ Janus solution with $\\,\\textrm{SO}(4)\\,$ symmetry was constructed in \\cite{DHoker:2007zhm}. However it was only recently that the $\\,\\mathcal{N}=2\\,$ supersymmetric Janus with $\\,\\textrm{SU}(2) \\times \\textrm{U}(1)\\,$ symmetry was constructed in five and ten dimensions \\cite{Bobev:2020fon}, thus completing the list of Janus solutions dual to the SYM interfaces scrutinised in \\cite{DHoker:2006qeo}.\n\n\nJanus solutions have been much less investigated in the context of M-theory. The first examples were constructed in \\cite{DHoker:2009lky} (and generalised in \\cite{Bachas:2013vza}) as $\\,\\mathcal{N}=4\\,$ deformations of the $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ background preserving a subgroup $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4) \\subset \\textrm{SO}(8)\\,$ of the isometries of the round $\\,\\textrm{S}^{7}\\,$. This time the deformation breaks the $\\,\\textrm{SO}(2,3)\\,$ isometries of $\\,\\textrm{AdS}_{4}\\,$ to the $\\,\\textrm{SO}(2,2)\\,$ isometries of $\\,\\textrm{AdS}_{3}\\,$. The M-theory Janus solutions can still be holographically understood as $\\,(1+1)$-dimensional interfaces in ABJM theory \\cite{Aharony:2008ug} despite the absence of a dilaton field in the M-theory context \\cite{DHoker:2009lky,Bobev:2013yra,Kim:2018qle}. Interestingly, it was shown in \\cite{Bobev:2013yra} that the $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetric Janus can be alternatively found using a 4D effective SO(8) gauged supergravity description. Using this 4D approach, an $\\,\\mathcal{N}=1\\,$ supersymmetric and $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetric Janus was constructed in \\cite{Bobev:2013yra} using numerical methods, for which 11D uplift formuli were provided in \\cite{Pilch:2015dwa}. More numerical Janus solutions were also presented in \\cite{Bobev:2013yra} by studying the $\\,\\textrm{G}_{2}$-invariant sector of the SO(8) gauged supergravity.\\footnote{See \\cite{Karndumri:2020bkc} for a numerical study of Janus solutions in the one-parameter family of $\\omega$-deformed SO(8) gauged supergravities \\cite{Dall'Agata:2012bb}. See also \\cite{Suh:2018nmp,Karndumri:2021pva} for a similar study in the context of massive IIA compactified on $\\,\\textrm{S}^{6}\\,$ and its effective description in terms of the ISO(7) gauged supergravity \\cite{Guarino:2015jca,Guarino:2015qaa,Guarino:2015vca}.}\n\n\nAmongst the various interesting questions raised in the discussion section of \\cite{DHoker:2009lky} we will provide a positive answer to that of whether exact M-theory Janus solutions exist with no supersymmetry. We will use the four-dimensional SO(8) gauged supergravity that arises upon reduction of eleven-dimensional supergravity on $\\,\\textrm{S}^{7}\\,$ \\cite{deWit:1982ig,deWit:1986oxb} and construct non-supersymmetric, yet analytic and regular, AdS$_{3}$-sliced domain-wall solutions of the form\n\\begin{equation}\n\\label{metric_ansatz_intro}\nds_{4}^{2} = d\\mu^{2} + e^{2 A(\\mu)} \\, ds_{\\textrm{AdS}_{3}}^{2} \\ ,\n\\end{equation}\nfor which the metric function $\\,A(\\mu)\\,$ depends arbitrarily on three real constants $\\,\\alpha_{i} \\in \\mathbb{R}\\,$ with $\\,{i=1,2,3}\\,$. The geometry is supported by three complex scalar fields $\\,z_{i}(\\alpha_{i},\\beta_{i};\\mu)\\,$ which depend on three additional compact parameters, or phases $\\,\\beta_{i} \\in [0,2\\pi]\\,$, and develop non-trivial profiles along the radial coordinate $\\,\\mu\\,$ transverse to the domain-wall. The effective 4D gauge coupling $\\,g\\,$ -- which relates to the inverse radius of $\\,\\textrm{S}^{7}\\,$ -- and the set of real parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ fully determine a particular Janus configuration.\n\n\n\nThe Janus parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ specify the boundary values of the complex scalars at $\\,{\\mu \\rightarrow \\pm \\infty}\\,$. In particular, the parameters $\\,\\beta_{i}\\,$ encode the source\/VEV and bosonic\/fermionic nature of the dual operators turned on on each side of the interface living at the boundary. A generic choice of Janus parameters breaks all the supersymmetries and the $\\,\\textrm{S}^{7}\\,$ isometry group down to its Cartan subgroup $\\,\\textrm{U}(1)^4 \\subset \\textrm{SO}(8)\\,$. On the contrary, the very special choice $\\,\\alpha_{i}=0\\,$ $\\forall i\\,$ trivialises the Janus and the maximally supersymmetric AdS$_{4}$ vacuum of the SO(8) supergravity that uplifts to the $\\,\\textrm{AdS}_4 \\times \\textrm{S}^7\\,$ Freund--Rubin background of eleven-dimensional supergravity with a round $\\,\\textrm{S}^{7}\\,$ metric is recovered \\cite{Freund:1980xh}. Interestingly, (super) symmetry enhancements occur upon suitable identifications between the parameters. For instance, the supersymmetric Janus of \\cite{DHoker:2009lky,Bobev:2013yra} with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetry is recovered upon setting two of the $\\,\\alpha_{i}\\,$ parameters to zero. In this work we will pay special attention to the Janus with $\\,{\\alpha_{1}=\\alpha_{2}=\\alpha_{3}}\\,$ and $\\,{\\beta_{1}=\\beta_{2}=\\beta_{3}}\\,$ which is non-supersymmetric and features an $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetry enhancement. We will present the uplift of this 4D Janus to eleven-dimensions providing, to the best of our knowledge, the first example of an exact M-theory Janus with no supersymmetry. \n\n\n\n\nIn addition to the Janus, we will construct another class of analytic solutions -- we refer to them as flows to Hades following standard terminology in the literature -- which are non-supersymmetric and display a singularity at $\\,\\mu =0\\,$ where the $\\,e^{2 A(\\mu)}\\,$ factor in (\\ref{metric_ansatz_intro}) shrinks to zero size and the complex scalars run to the boundary of moduli space. Some similar curved-sliced \\cite{Bobev:2013yra} and flat-sliced \\cite{Cvetic:1999xx,Pope:2003jp,Pilch:2015vha,Pilch:2015dwa} singular flows have been constructed within the $\\,\\textrm{SO}(8)\\,$ gauged supergravity and argued to holographically describe an interface between a superconformal ABJM phase and a non-conformal phase with potentially interesting physics.\\footnote{The scalar potential of the maximal $\\,\\textrm{SO}(8)\\,$ gauged supergravity is bounded above by its value at the maximally supersymmetric AdS$_{4}$ vacuum thus satisfying the \\textit{good} condition of \\cite{Gubser:2000nd}.} In their simplest realisation, these flat-sliced singular flows in M-theory are the analogue of the type IIB flows to the Coulomb branch of $\\,\\mathcal{N}=4\\,$ SYM investigated in \\cite{Cvetic:1999xx,Freedman:1999gp,Freedman:1999gk,Gubser:2000nd}. \n\n\n\nThere are similarities and differences between the Janus and the Hades. As for the Janus, the Hades solutions depend on a set of six parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. Unlike for the Janus, no supersymmetric limit can be taken on the Hades parameters, and the very special choice $\\,\\alpha_{i}=0\\,$ $\\forall i\\,$ does not trivialise the Hades to recover AdS$_{4}$. Instead, a special class of Hades flows -- we will refer to them as \\textit{ridge flows} adopting the terminology of \\cite{Pilch:2015vha} -- appears in this limit. As before, we will concentrate on the simple case with $\\,{\\alpha_{1}=\\alpha_{2}=\\alpha_{3}} \\equiv \\alpha\\,$ and $\\,{\\beta_{1}=\\beta_{2}=\\beta_{3}}\\equiv \\beta\\,$ for which the flows to Hades feature an $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetry, and present their uplift to eleven-dimensional supergravity. \n\n\n\nSpecial attention will then be paid to the $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetric ridge flows with $\\,\\alpha=0\\,$ for which there is just one free parameter left, \\textit{i.e.} the phase $\\,\\beta \\in [0,2\\pi]\\,$. This phase specifies the boundary values of the complex scalars at $\\,\\mu \\rightarrow \\infty\\,$ and, therefore, the source\/VEV and bosonic\/fermionic nature of the dual operators turned on on the ultraviolet (UV) side of the conformal interface. In the infrared (IR) side $\\,\\mu \\rightarrow 0\\,$ of the interface, the four-dimensional solution becomes singular and the dual field theory is expected to enter the non-conformal phase. Interestingly, the parameter $\\,\\beta\\,$ determining the boundary conditions of the complex scalars is associated with a $\\,\\textrm{U}(1)_{\\xi}\\,$ duality symmetry of the four-dimensional supergravity Lagrangian. However, as originally noticed in \\cite{Pope:2003jp} for a class of conventional flat-sliced RG-flows (see also \\cite{Pilch:2015vha,Pilch:2015dwa}), the $\\,\\textrm{U}(1)_{\\xi}\\,$ changes the physics of the ridge flows once they are uplifted to eleven dimensions: it takes metric modes into three-form gauge field modes. \n\n\nWe will illustrate this phenomenon by analysing in some detail the simple cases of setting $\\,\\beta= \\frac{\\pi}{2}\\,$ and $\\,\\beta=0\\,$. The corresponding ridge flows are triggered from the UV solely by bosonic VEV's or fermionic sources, respectively. The resulting M-theory ridge flows will be shown to be drastically different as far as the persistence of the singularity and the presence of magnetic M5-branes sources in the background are concerned. Setting $\\,\\beta= \\frac{\\pi}{2}\\,$ produces a singular M-theory background without magnetic M5-branes sources akin the (flat-sliced) Coulomb branch flows constructed in \\cite{Cvetic:1999xx}. Modifying the phase $\\,\\beta\\,$ by acting with $\\,\\textrm{U}(1)_{\\xi}\\,$ turns out to induce a transformation on the eleven-dimensional backgrounds that parallels the dielectric rotation of Coulomb branch flows investigated in \\cite{Pope:2003jp,Pilch:2015vha,Pilch:2015dwa}. We will look in detail at the limiting case $\\,\\beta=0\\,$ and conclude that the $\\,\\textrm{U}(1)_{\\xi}\\,$ transformation totally polarises M2-branes into M5-branes when flowing from the UV to the IR, leaving no M2-branes. We will provide some evidence for this phenomenon to occur also at generic values of $\\,\\beta\\,$.\n\n\nThe paper is organised in four sections plus appendices. In Section~\\ref{sec:Janus} we present our multi-parametric $(\\alpha_{i},\\beta_{i})$-families of analytic Janus and Hades solutions and discuss the ridge flow limit of the latter. We investigate the various possibilities of (super) symmetry enhancement depending on the choice of $\\,\\alpha_{i}\\,$ parameters, as well as the various possibilities of boundary conditions for the complex scalars (sources\/VEV's of dual operators) depending on the choice of $\\,\\beta_{i}\\,$ parameters. In Section~\\ref{sec:Uplift_11D} we present the uplift of the Janus and Hades solutions with $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetry to eleven-dimensional supergravity. We then focus on the ridge flows and discuss some eleven-dimensional aspects of the solutions, like the presence of singularities or the characterisation of the M2\/M5-brane sourcing the backgrounds, as a function of the parameter $\\,\\beta\\,$. We summarise the results and conclude in Section~\\ref{sec:conclusions}. Two additional appendices accompany the main text which contain technical results regarding the BPS equations as well as some relevant uplift formuli for the STU model. This is the subsector of the four-dimensional maximal $\\,\\textrm{SO}(8)\\,$ gauged supergravity within which we have constructed all the solutions presented in this work.\n\n\n\n\n\\section{Four-dimensional Janus and Hades}\n\\label{sec:Janus}\n\n\n\\subsection{The model}\n\n\nOur starting point is the $\\mathcal{N}=2$ gauged STU supergravity in four dimensions \\cite{Cvetic:1999xp}. This theory has a gauge group $\\textrm{U}(1)^4$, the maximal Abelian subgroup of $\\textrm{SO}(8)$, and can be embedded into the maximal $\\mathcal{N}=8$ $\\textrm{SO}(8)$-gauged supergravity \\cite{deWit:1982ig} as its $\\textrm{U}(1)^4$ invariant sector \\cite{Cvetic:1999xp}. The field content consists of the $\\mathcal{N}=2$ supergravity multiplet coupled to three vector multiplets. Upon setting vector fields to zero, the bosonic Lagrangian reduces to an Einstein-scalar model given by\n\\begin{equation}\n\\label{Lagrangian_model_U1^4_Einstein-scalars}\n\\begin{array}{lll}\n\\mathcal{L} & = & \\left( \\dfrac{R}{2} - V \\right) * 1 - \\dfrac{1}{4} \\displaystyle\\sum_{i=1}^3 \\left[\nd\\varphi_{i} \\wedge* d\\varphi_{i} + e^{2 \\varphi_{i}} \\, d\\chi_{i} \\wedge* d\\chi_{i} \\right] \\\\[4mm]\n& = & \\left( \\dfrac{R}{2} - V \\right) * 1 - \\displaystyle\\sum_{i=1}^{3}\n\\dfrac{1}{\\left( 1-|\\tilde{z_{i}}|^{2} \\right) ^{2}} \\, d\\tilde{z}_{i}\n\\wedge* d\\tilde{z}_{i}^{*} \\ .\n\\end{array}\n\\end{equation}\nIn passing from the first line to the second one in (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) we have changed the parameterisation of the scalar fields $\\,z_{i}\\,$ ($i=1,2,3$) in the vector multiplets -- which serve as coordinates in the scalar coset geometry $[\\textrm{SL}(2)\/\\textrm{SO}(2)]^3$ -- from the upper-half plane to the unit-disk parameterisation via the field redefinition\n\\begin{equation}\n\\label{ztilde&z}\n\\tilde{z}_{i}=\\frac{z_{i}-i}{z_{i}+i} \n\\hspace{10mm} \\textrm{ with } \\hspace{10mm} \nz_{i} = -\\chi_{i} + i \\, e^{-\\varphi_{i}} \\ .\n\\end{equation}\nThe non-trivial scalar potential in the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) is given by\n\\begin{equation}\n\\label{V_U1^4}\nV = - \\tfrac{1}{2} \\, g^{2} \\sum_{i} \\left( 2 \\, \\cosh\\varphi_{i} + \\chi\n_{i}^{2} \\, e^{\\varphi_{i}} \\right) = g^{2} \\, \\left( 3 - \\sum_{i} \\frac{2}{1-|\\tilde{z}_{i}|^{2}} \\right) \\ ,\n\\end{equation}\nwhere $\\,g\\,$ is the gauge coupling in the gauged four-dimensional supergravity. From (\\ref{V_U1^4}) one immediately sees that only $|\\tilde{z}_{i}|$ enter the\npotential. As a result, the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) is\ninvariant under the three $\\,\\textrm{U}(1)_{\\xi_{i}}\\,$ shifts of $\\,\\arg\\tilde{z}_{i}\\,$, namely, $\\,\\delta_{\\xi_{i}}\\tilde{z}_{i} = i \\, \\xi_{i} \\, \\tilde{z}_{i}\\,$, with constant parameters $\\,\\xi_{i}\\,$. However, as we will see shortly, the phases $\\,\\arg\\tilde{z}_{i}\\,$ will play a central role when discussing boundary conditions for Janus- and Hades-like solutions in this supergravity model.\n\n\n\nIn this work we will investigate Janus-like solutions for which the space-time metric takes the form\n\\begin{equation}\n\\label{metric_ansatz}\nds_{4}^{2} = d\\mu^{2} + e^{2 A(\\mu)} \\, d\\Sigma^{2} \\ ,\n\\end{equation}\nwith $\\,\\mu\\in (-\\infty, \\infty)\\,$ or $\\,\\mu\\in [0, \\infty)\\,$ being the coordinate along which space-time is foliated with $\\Sigma$ slices, and $A(\\mu)$ being a scale function. The line element $\\,d\\Sigma^{2}\\,$ describes a globally AdS$_{3}$ space-time of radius $\\,\\ell= 1\\,$. The second-order Euler-Lagrange equations for the scalar fields that follow from the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) read\n\\begin{equation}\n\\label{EOM_scalars}\n\\begin{array}{rll}\n\\varphi_{i} '' - e^{2 \\varphi_{i}} \\,(\\chi_{i}')^2 + 3 \\, A' \\, \\varphi_{i}' + g^2 \\, \\left( 2 \\, \\sinh\\varphi_{i} + e^{\\varphi_{i}} \\, \\chi_{i}^2\\right) & = & 0 \\ , \\\\[2mm]\n\\chi_{i}'' + \\left( 3 \\, A' + 2 \\, \\varphi_{i}' \\right) \\chi_{i}' + 2 \\, g^2 \\, e^{-\\varphi_{i}} \\, \\chi_{i} & = & 0 \\ ,\n\\end{array}\n\\end{equation}\nwith $i=1,2,3$ and where primes denote derivatives with respect to the coordinate $\\,\\mu\\,$. The Einstein equations impose two additional independent equations given by\n\\begin{equation}\n\\label{EOM_Einstein}\n\\begin{array}{rll}\n1 - e^{2 A} \\Big[ A'' + \\frac{1}{4} \\displaystyle\\sum_{i} \\Big( \\, (\\varphi_{i}')^2 + e^{2 \\varphi_{i}} \\, (\\chi_{i}')^2 \\Big) \\, \\Big] &=& 0 \\ , \\\\[2mm]\n2 + e^{2 A} \\Big[ A'' + 3 \\, (A')^2 - \\tfrac{1}{2} \\, g^{2} \\displaystyle\\sum_{i} \\left( 2 \\, \\cosh\\varphi_{i} + \\chi_{i}^2 \\, e^{\\varphi_{i}} \\right) \\Big] & = & 0 \\ .\n\\end{array}\n\\end{equation}\nWe will now present analytic and multi-parametric families of Janus and Hades solutions to this system of second-order differential equations.\n\n\n\n\\subsection{Multi-parametric Janus solutions}\n\nThe second-order equations of motion in (\\ref{EOM_scalars}) and (\\ref{EOM_Einstein}) have a multi-parametric family of analytic Janus solutions. The scale factor in the space-time metric is given by\n\\begin{equation}\n\\label{A(mu)_func_U1^4}\ne^{2A(\\mu)} = {(g k)}^{-2} \\cosh^2(g\\mu) \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{k_factor}\nk^2= 1 + \\sum_{i}\\sinh^{2}\\alpha_{i} \\, \\ge \\, 1\n\\hspace{10mm} \\text{ with } \\hspace{10mm} \\alpha_{i} \\in \\mathbb{R} \\ .\n\\end{equation}\nUsing the unit-disk parameterisation in (\\ref{ztilde&z}) to describe the scalar fields in the three vector multiplets, they acquire simple $\\mu$-dependent profiles of the form\n\\begin{equation}\n\\label{Janus_solution_U1^4_ztil}\n\\tilde{z}_{i}(\\mu) = e^{i \\beta_{i}}\\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\sinh(g \\mu) } \n\\hspace{10mm} \\text{ with } \\hspace{10mm} \\beta_{i} \\in [0,2\\pi] \\ ,\n\\end{equation}\nso that $\\,|\\tilde{z}_{i}(0)|=\\tanh\\alpha_{i}\\,$. Eqs (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) describe a multi-parametric family of Janus solutions parameterised by $3+3$ arbitrary real constants $(\\alpha_{i},\\beta_{i})$. Importantly, the presence of non-trivial axions $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ (spin $0$ pseudo-scalars) turns out to be crucial for the existence of regular Janus solutions, as first noticed in \\cite{Bobev:2013yra}. Parametric plots of the complex scalars $\\,\\tilde{z}_{i}(\\mu)\\,$ in (\\ref{Janus_solution_U1^4_ztil}) are displayed in Figure~\\ref{fig:ztilde_U1^4}. The real $\\,\\textrm{Re}\\tilde{z}_{i}\\,$ and imaginary $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ components of $\\,\\tilde{z}_{i}\\,$ are shown in Figure~\\ref{fig:Rez&Imz}. Note the special limiting case of $\\,\\alpha_{i} \\gg 1\\,$ (\\textit{i.e.} $\\tanh\\alpha_{i} \\approx 1$) for which the flows become singular. In this limit, the complex scalar $\\,\\tilde{z}_{i}\\,$ gets to the boundary of the moduli space, which is located at $\\,|\\tilde{z}_{i}|=1\\,$ in the unit-disk parameterisation of the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}), and the scalar potential in (\\ref{V_U1^4}) diverges.\n\n\n\nOn the other hand, the value $\\,\\alpha_{i}=0\\,$ is certainly special. At this value an AdS$_{4}$ maximally supersymmetric solution with radius $\\,L_{\\text{AdS}_{4}}={g}^{-1}\\,$ is recovered with the scalars being fixed at the constant value $\\,\\tilde{z}_{i}=0\\,$. This AdS$_{4}$ vacuum uplifts to the $\\,\\textrm{AdS}_4 \\times \\textrm{S}^7\\,$ Freund--Rubin background of eleven-dimensional supergravity with a round $\\,\\textrm{S}^{7}\\,$ metric \\cite{Freund:1980xh}. Moreover, it describes the near-horizon geometry of a stack of M2-branes and is holographically dual to the three-dimensional ABJM theory \\cite{Aharony:2008ug}. When evaluated at this AdS$_{4}$ vacuum, the three $\\textrm{U}(1)^{4}$ invariant complex scalars have a normalised mass\n\\begin{equation}\n\\label{m^2L^2_AdS4}\nm_{i}^2 L^2 = -2 \\ ,\n\\end{equation}\nthus lying within the mass range $\\, -9\/4 < m_{i}^2 \\, L^2 < - 5\/4\\,$ for wich two possible quantisations of scalar fields in AdS$_{4}$ exist \\cite{Klebanov:1999tb}: the mode with conformal dimension $\\,\\Delta_{i}=\\Delta_{-}=1\\,$ and the mode with conformal dimension $\\,\\Delta_{i}=\\Delta_{+}=2\\,$ (where $\\,\\Delta_{\\pm}\\,$ are the two roots of $\\,m_{i}^2 \\, L^2=(\\Delta_{i}-3)\\Delta_{i}\\,$) can be interpreted as the source and the VEV of the corresponding dual operators (standard quantisation) or \\textit{viceversa} (alternative quantisation). However, as shown in \\cite{Breitenlohner:1982bm}, proper scalars $\\,\\textrm{Re}\\tilde{z}_{i}\\,$ and pseudo-scalars $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ must be quantised in exactly opposite ways in order to preserve maximal supersymmetry. And, moreover, only the choice of proper scalars having alternative quantisation yields a perfect matching between the scaling dimensions of the supergravity modes and those of the dual operators in the M2-brane theory \\cite{Bobev:2011rv} (see footnote~\\ref{footnote:operators}). \n\n\n\nThe class of Janus solutions in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) depends on the set of parameters $\\,g\\,$ and $\\,(\\alpha_{i},\\beta_{i})\\,$. As discussed in \\cite{Bobev:2013yra}, the four-dimensional gauge coupling $\\,g\\,$ sets the scale of the asymptotic AdS$_{4}$ vacuum and, via the AdS\/CFT correspondence, the number of M2-branes as well as the rank of the Chern--Simons gauge groups in ABJM theory. The parameters $\\,\\alpha_{i}\\,$ set the height of the bump, \\textit{i.e.} $\\,|\\tilde{z}_{i}(0)|=\\tanh\\alpha_{i}\\,$, and therefore the strength of the coupling between the (1+1)-dimensional defect and the three-dimensional ambient field theory. The parameters $\\,\\beta_{i}\\,$ set the boundary conditions of the bulk scalars at $\\,\\mu \\rightarrow \\pm \\infty\\,$ and, again via the AdS\/CFT correspondence (see footnote~\\ref{footnote:operators}), the specific linear combinations of bosonic and fermionic bilinear operators that are activated in the field theory. We will analyse the possible choices of boundary conditions in detail in Section~\\ref{sec:boundary conditions}.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.50\\textwidth]{Plots\/ztilde_plot.pdf} \n\\put(0,90){$\\textrm{Re}\\tilde{z}_{i}$} \\put(-105,210){$\\textrm{Im}\\tilde{z}_{i}$}\n\\put(-114,101){{\\color{red}{$\\bullet$}}}\n\\end{center}\n\\caption{Parametric plot of $\\,\\tilde{z}_{i}(\\mu)\\,$ in (\\ref{Janus_solution_U1^4_ztil}) for the Janus solutions with $\\,\\alpha_{i}=1\\,$ and $\\,\\beta_{i}=\\frac{n\\pi}{4}\\,$ with $\\,n=0,\\ldots,7\\,$. The central red point at $\\,\\tilde{z}_{i}=0\\,$ $\\,\\forall i\\,$ corresponds to the maximally supersymmetric AdS$_{4}$ vacuum and describes the asymptotic values at $\\,\\mu\\rightarrow \\pm\\infty\\,$.}\n\\label{fig:ztilde_U1^4}\n\\end{figure}\n\n\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=1_b=0.pdf} \n\\hspace{5mm}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=10_b=0.pdf}\n\\put(-307,15){\\small{$\\alpha_{i}=1 \\, , \\, \\beta_{i}=0$}}\\put(-87,15){\\small{$\\alpha_{i} \\gg 1 \\, , \\, \\beta_{i}=0$}}\n\\\\[5mm]\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=1_b=Pi4.pdf} \n\\hspace{5mm}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=10_b=Pi4.pdf} \n\\put(-307,22){\\small{$\\alpha_{i}=1 \\, , \\, \\beta_{i}=\\frac{\\pi}{4}$}}\\put(-87,22){\\small{$\\alpha_{i} \\gg 1 \\, , \\, \\beta_{i}=\\frac{\\pi}{4}$}}\n\\\\[5mm]\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=1_b=Pi2.pdf} \n\\hspace{5mm}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=10_b=Pi2.pdf} \n\\put(-307,15){\\small{$\\alpha_{i}=1 \\, , \\, \\beta_{i}=\\frac{\\pi}{2}$}}\\put(-87,15){\\small{$\\alpha_{i} \\gg 1 \\, , \\, \\beta_{i}=\\frac{\\pi}{2}$}}\n\\put(-412,-8){\\scriptsize{$-\\infty$}}\\put(-323,-8){\\scriptsize{$0$}}\\put(-238,-8){\\scriptsize{$\\infty$}}\n\\put(-330,-22){\\small{$g \\mu$}}\n\\put(-190,-8){\\scriptsize{$-\\infty$}}\\put(-101,-8){\\scriptsize{$0$}}\\put(-16,-8){\\scriptsize{$\\infty$}}\n\\put(-109,-22){\\small{$g \\mu$}}\n\\end{center}\n\\caption{Plots of $\\,\\textrm{Re}\\tilde{z}_{i}\\,$ (blue dotted line), $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ (orange dashed line) and $\\,A'(\\mu)\\,$ (green solid line) as a function of the radial coordinate $\\,g \\mu \\in (-\\infty , \\infty)\\,$ for different values of the Janus parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. The limit $\\,\\alpha_{i} \\gg 1\\,$ (\\textit{i.e.} $\\tanh\\alpha_{i} \\approx 1$) renders the Janus solution singular. In this limit, $\\,\\tilde{z}_{i}\\,$ gets to the boundary of the moduli space which is located at $\\,|\\tilde{z}_{i}|=1\\,$ in the unit-disk parameterisation of (\\ref{ztilde&z}).}\n\\label{fig:Rez&Imz}\n\\end{figure}\n\n\n\nLastly, a study of the supersymmetry preserved by this family of solutions is presented in the Appendix~\\ref{app:susy}. The BPS equations (\\ref{BPS_A}) and (\\ref{BPS_scalars}) are not satisfied by the Janus solution in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) for generic values of $(\\alpha_{i} , \\beta_{i})$ thus implying that such a solution is generically non-supersymmetric. However, as we will see in a moment, some supersymmetry can be restored upon suitable choice of $(\\alpha_{i} , \\beta_{i})$, namely, upon suitable adjustment of the Janus boundary conditions.\n\n\n\n\\subsection{Janus with (super) symmetry enhancements}\n\\label{sec:Janus_sym_enhancement}\n\nSpecific choices of the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ translate into various (super) symmetry enhancements of the general Janus solution in (\\ref{A(mu)_func_U1^4}) and (\\ref{Janus_solution_U1^4_ztil}).\n\n\\subsubsection{\\texorpdfstring{$\\text{SO}(4) \\times\\text{SO}(4)$}{SO(4)xSO(4)} symmetry enhancement}\n\\label{sec:Janus_SO4xSO4}\n\nSetting two vector multiplets to zero, \\textit{e.g.} $\\,\\tilde{z}_{2}=\\tilde{z}_{3}=0$, by setting\n\\begin{equation}\n\\label{alpha_2_3=0}\n\\alpha_{2} = \\alpha_{3} = 0 \\ ,\n\\end{equation}\nand renaming $\\tilde{z}_{1} \\equiv\\tilde{z}\\,$, the $\\text{SO}(4) \\times\\text{SO}(4)$ invariant sector of the SO(8) gauged supergravity investigated in Section~$5$ of \\cite{Bobev:2013yra} is recovered upon the identification $\\tilde{z}=z_{\\text{there}}$. The Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) reduces to\n\\begin{equation}\n\\label{Lagrangian_model_SO4xSO4}\n\\begin{array}{lll}\n\\mathcal{L} & = & \\left( \\frac{R}{2} - V \\right) * 1 - \\frac{1}{4} \\left[ (d\\varphi)^{2} + e^{2 \\varphi} \\, (d\\chi)^{2} \\right] \\\\[2mm]\n& = & \\left( \\frac{R}{2} - V \\right) * 1 - \\dfrac{1}{\\left( 1-|\\tilde{z}|^{2} \\right) ^{2}} \\, d\\tilde{z} \\wedge* d\\tilde{z}^{*} \\ ,\n\\end{array}\n\\end{equation}\nand the scalar potential in (\\ref{V_U1^4}) simplifies to\n\\begin{equation}\n\\label{V_SO4xSO4}\nV = - \\tfrac{1}{2} \\, g^{2} \\left( 4 + 2 \\cosh\\varphi+ \\chi^{2} \\, e^{\\varphi} \\right) = - g^{2} \\, \\dfrac{3 - |\\tilde{z}|^{2}}%\n{1-|\\tilde{z}|^{2}} \\ .\n\\end{equation}\nThe Janus solution then reads\n\\begin{equation}\n\\label{Janus_solution_SO4xSO4^4_alt}\nds_{4}^{2} = d\\mu^{2}+ e^{2 A(\\mu)} \\, d\\Sigma^{2} \n\\hspace{10mm} , \\hspace{10mm} \n\\tilde{z}(\\mu) = e^{i \\beta} \\,\n\\frac{\\sinh\\alpha}{\\cosh\\alpha + i \\, \\sinh(g \\mu) } \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{A(mu)_func_SO4xSO4}\ne^{2A(\\mu)} = (g k)^{-2} \\cosh^2(g\\mu)\n\\hspace{8mm} \\textrm{ and } \\hspace{8mm}\nk= \\cosh \\alpha \\ge 1\\ ,\n\\end{equation}\nwhere $(\\alpha, \\beta) = (\\alpha_{1} , \\beta_{1})$. This solution precisely matches the one presented in Section~$5$ of \\cite{Bobev:2013yra} upon the identification $\\cosh\\alpha=(1-a^{2}_{\\text{there}})^{-\\frac{1}{2}}$. As noticed therein, the Janus solution is half-PBS and preserves $\\,16\\,$ real supercharges. From a holographic perspective, the $(1+1)$-dimensional defect dual to the AdS$_{3}$ factor in the geometry features $(4,4)$ supersymmetry and therefore has an $\\,\\textrm{SO}(4)_{\\textrm{R}} \\times \\textrm{SO}(4)_\\textrm{R}\\,$ R-symmetry group. We have explicitly verified that, when selecting the minus sign in (\\ref{Janus_solution_SO4xSO4^4_alt}), the Janus solution satisfies the 1\/2-BPS equations (\\ref{BPS_A}) and (\\ref{BPS_scalars}) for the eight gravitino mass terms (superpotentials) of the maximal theory (see Footnote~\\ref{Footnote:axions}). Finally, the original M-theory supersymmetric Janus with $\\text{SO}(4) \\times\\text{SO}(4)$ symmetry was presented in \\cite{DHoker:2009lky}.\n\n\n\n\n\\subsubsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2$}{SU(3)xU(1)2} symmetry enhancement}\n\nIdentifying the three vector multiplets, namely $\\,\\tilde{z}_{1}=\\tilde{z}_{2}=\\tilde{z}_{3} \\equiv\\tilde{z}\\,$, so that\n\\begin{equation}\n\\alpha_{1} = \\alpha_{2} = \\alpha_{3} \\equiv\\alpha\n\\hspace{5mm} , \\hspace{5mm}\n\\beta_{1} = \\beta_{2} = \\beta_{3} \\equiv\\beta\\ ,\n\\end{equation}\nthe $\\text{SU}(3) \\times\\text{U}(1)^2$ invariant sector of Section~$6$ of \\cite{Bobev:2013yra} (see also \\cite{Pilch:2015dwa} for the 11D uplift) is recovered upon the identification $\\tilde{z}=z_{\\text{there}}$. The\nLagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) simplifies to\n\\begin{equation}\n\\label{Lagrangian_model_SU3xU1xU1}\n\\begin{array}{lll}\n\\mathcal{L} & = & \\left( \\frac{R}{2} - V \\right) * 1 - \\frac{3}{4} \\left[(d\\varphi)^{2} + e^{2 \\varphi} \\, (d\\chi)^{2} \\right] \\\\[2mm]\n& = & \\left( \\frac{R}{2} - V \\right) * 1 - \\dfrac{3}{\\left( 1-|\\tilde{z}|^{2} \\right) ^{2}} \\, d\\tilde{z} \\wedge* d\\tilde{z}^{*} \\ ,\n\\end{array}\n\\end{equation}\nand the scalar potential in (\\ref{V_U1^4}) reduces to\n\\begin{equation}\n\\label{V_SU3xU1^2}\nV = - \\tfrac{3}{2} \\, g^{2} \\left( 2 \\cosh\\varphi+ \\chi^{2} \\, e^{\\varphi} \\right) = - 3 \\, g^{2} \\, \\dfrac{1+|\\tilde{z}|^{2}}{1-|\\tilde{z}|^{2}} \\ .\n\\end{equation}\nThe Janus solution takes the form\n\\begin{equation}\n\\label{Janus_solution_SU3xU1xU1}\nds_{4}^{2} = d\\mu^{2}+ e^{2 A(\\mu)} \\, d\\Sigma^{2} \n\\hspace{10mm} , \\hspace{10mm} \n\\tilde{z}(\\mu) = e^{i \\beta} \\, \\frac{\\sinh\\alpha}{\\cosh\\alpha + i \\, \\sinh(g \\mu) } \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{A(mu)_func_SU3xU1xU1}\ne^{2A(\\mu)} = (g k)^{-2} \\cosh^2(g\\mu) \n\\hspace{8mm} \\textrm{ and } \\hspace{8mm}\nk^2= 1 + 3 \\sinh^{2}\\alpha \\ge 1\\ .\n\\end{equation}\nThis provides an analytic solution in the $\\text{SU}(3) \\times\\text{U}(1)^2 $ invariant sector of the SO(8) maximal supergravity investigated in Section~$6$ of \\cite{Bobev:2013yra}. The solution (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}) satisfies the second-order equations of motion in (\\ref{EOM_scalars}) and (\\ref{EOM_Einstein}). However, we have verified that the BPS equations (\\ref{BPS_A}) and (\\ref{BPS_scalars}) are not satisfied for any of the eight gravitino mass terms (superpotentials) in the maximal SO(8) gauged supergravity, so the solution is non-supersymmetric.\n\n\n\n\\subsection{Janus geometry and boundary conditions}\n\nLet us discuss the geometry of the multi-parametric family of Janus solutions presented in the previous sections. Introducing embedding coordiantes in $\\mathbb{R}^{2,3}$, the $k$-family of Janus metrics in (\\ref{metric_ansatz}) and (\\ref{A(mu)_func_U1^4}) corresponds to\n\\begin{equation}\n\\label{embedding_coordinates}\n\\begin{array}{lll}\nX_{0} &=& (g k)^{-1} \\, \\dfrac{\\cos\\tau}{\\cos\\eta} \\, \\cosh(g \\mu) \\ , \\\\[6mm]\nX_{4} &=& (g k)^{-1} \\, \\dfrac{\\sin\\tau}{\\cos\\eta} \\, \\cosh(g \\mu) \\ , \\\\[6mm]\nX_{1} &=& (g k)^{-1} \\, \\tan\\eta \\cos\\theta \\, \\cosh(g \\mu) \\ , \\\\[6mm]\nX_{2} &=& (g k)^{-1} \\, \\tan\\eta \\sin\\theta \\, \\cosh(g \\mu) \\ , \\\\[6mm]\nX_{3} &=& g^{-1} \\, i \\, \\textrm{E}(i g \\mu \\, ; \\, k^{-2}) \\ ,\n\\end{array}\n\\end{equation}\nwith \n\\begin{equation}\nk^2 = 1 + \\sum_{i}\\sinh^{2}\\alpha_{i} \\, \\ge \\, 1 \\ ,\n\\end{equation}\nand $\\textrm{E}(i g \\mu \\, ; \\, k^{-2})$ being the incomplete elliptic integral of the second kind. The solution describes the hyper-surface\n\\begin{equation}\n\\label{hypersurface_X}\n-X_{0}^2 - X_{4}^2 + X_{1}^2 + X_{2}^2 +(g k)^{-2} \\sinh^2(g \\mu) = - (g k)^{-2} \\ ,\n\\end{equation}\nwhere the term $\\,(g k)^{-2} \\sinh^2(g \\mu)\\,$ is implicitly given in terms of $X_{3}$ by the last relation in (\\ref{embedding_coordinates}). For $\\,k=1\\,$ one has that $\\,i \\, \\textrm{E}(i g \\mu \\, ; \\,1) = - \\sinh(g \\mu)\\,$ and (\\ref{hypersurface_X}) reduces to the hyperboloid describing AdS$_{4}$.\n\n\\subsubsection{Global coordinates and boundary structure}\n\n\nLet us perform a change of coordinates that will help us to understand the Janus geometry in (\\ref{metric_ansatz}) and (\\ref{A(mu)_func_U1^4})-(\\ref{k_factor}), especially its boundary structure. We start by performing a change of radial coordinate to make its range compact\n\\begin{equation}\n\\tilde{\\mu} = 2 \\, k \\, \\textrm{arctan} \\left[ \\tanh \\left( \\frac{g \\, \\mu}{2} \\right)\\right] \\ ,\n\\end{equation}\nand then choose global coordinates to describe the AdS$_{3}$ slicing in (\\ref{metric_ansatz}). The Janus metric in (\\ref{metric_ansatz}) and (\\ref{A(mu)_func_U1^4})-(\\ref{k_factor}) then becomes conformal to $\\,\\mathbb{R} \\times \\textrm{S}^{3}\\,$\n\\begin{equation}\n\\label{Janus_metric_original}\nds_{4}^{2} = \\frac{(g k)^{-2} }{\\cos^2\\left( \\frac{\\tilde{\\mu}}{k}\\right) \\cos^{2}\\eta} \\, \\left( - d\\tau^2 + \\cos^{2}\\eta \\, d\\tilde{\\mu}^{2} + d\\eta^2 + \\sin^2\\eta \\, d\\theta^2\\right) \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{global_coords_ranges}\n\\tau \\in (-\\infty \\, , \\infty)\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\tilde{\\mu} \\in [-\\frac{\\pi k}{2} \\, , \\frac{\\pi k}{2}]\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\eta \\in [0 \\, , \\frac{\\pi}{2}] \n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\theta \\in [0 \\, , 2 \\pi] \\ .\n\\end{equation}\nThese are the global coordinates used to describe the original type IIB Janus solution in \\cite{Bak:2003jk,Clark:2004sb}. The geometry (\\ref{Janus_metric_original}) has a boundary that consists of two hemi-spheres of $\\,\\textrm{S}^2\\,$ at $\\,\\tilde{\\mu} = \\pm \\tilde{\\mu}_{0}\\,$, with $\\, \\tilde{\\mu}_{0}=\\frac{\\pi k}{2} \\,$, joined at the $\\textrm{S}^1$ equator at $\\,\\eta = \\frac{\\pi}{2}\\,$. Lastly, using the new radial coordinate $\\,\\tilde{\\mu}\\,$, the profiles for the complex scalars in (\\ref{Janus_solution_U1^4_ztil}) become\n\\begin{equation}\n\\label{Janus_solution_U1^4_new}\n\\tilde{z}_{i}(\\tilde{\\mu}) = e^{i \\beta_{i}}\\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\tan\\left(\\frac{\\tilde{\\mu}}{k}\\right) } \\ ,\n\\end{equation}\nso that $\\,\\tilde{z}_{i}(\\tilde{\\mu}) \\rightarrow 0\\,$ when approaching the two hemi-spheres of $\\,\\textrm{S}^2\\,$ at $\\,\\tilde{\\mu} \\rightarrow \\pm \\tilde{\\mu}_{0} \\,$ in the Janus boundary. Note that $\\,\\textrm{arg}\\left[\\tilde{z}_{i}(\\tilde{\\mu}_{0})\\right] - \\textrm{arg}\\left[\\tilde{z}_{i}(-\\tilde{\\mu}_{0})\\right] = \\pi$, thus creating an interface discontinuity at the $\\,\\textrm{S}^{1}\\,$ equator where the defect lives.\n\n\n\\subsubsection{\\texorpdfstring{AdS$_{3}$}{AdS3} slicing and boundary conditions}\n\\label{sec:boundary conditions}\n\n\nIn order to investigate the boundary conditions of the the family of Janus solution in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) we will perform a regular change of radial coordinate\n\\begin{equation}\n\\label{new_coordinate_Janus}\n\\rho= \\sinh({g} \\mu) \n\\hspace{10mm} , \\hspace{10mm}\nd\\mu = g^{-1} \\, \\dfrac{d\\rho}{\\sqrt{\\rho^2+1}} \\ ,\n\\end{equation}\nso that the family of Janus solutions in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) becomes\\footnote{The Ricci scalar constructed from the metric (\\ref{Janus_U1^4_rho_1}) reads\n\\begin{equation}\n\\label{Janus_Ricci}\nR(\\rho) = - 6 \\, g^{2} \\left( \\, 1 + \\frac{\\rho^2 + k^2}{\\rho^2 + 1} \\, \\right) \\ ,\n\\end{equation}\nthus ensuring regularity of the Janus geometry within the whole range $\\,\\rho \\in ( -\\infty , \\infty)\\,$.}\n\\begin{equation}\n\\label{Janus_U1^4_rho_1}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left( \\frac{d\\rho^{2}}{\\rho^{2}+1} + \n\\frac{\\rho^{2}+1 }{ k^2} \\, d\\Sigma^{2} \\right)\n\\hspace{5mm} , \\hspace{5mm}\nd\\Sigma^{2} = \\frac{1}{\\cos^{2}\\eta}\\left( - d\\tau^2 + d\\eta^2 + \\sin^2\\eta \\, d\\theta^2\\right) \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{global_coords_ranges_2}\n\\tau \\in (-\\infty \\, , \\infty)\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\rho \\in (-\\infty \\, , \\infty)\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\eta \\in [0 \\, , \\frac{\\pi}{2}] \n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\theta \\in [0 \\, , 2 \\pi] \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Janus_U1^4_rho_2}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}} \\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\rho} \\ .\n\\end{equation}\nThe Janus geometry (\\ref{Janus_U1^4_rho_1}) has a three-dimensional conformal boundary at $\\,\\rho \\rightarrow \\pm \\infty\\,$ that is conformal to $\\,\\mathbb{R} \\times \\textrm{S}^{2}\\,$ with a $k$-dependent prefactor $\\,(g k)^{-2} \\, \\rho^2\\,$. This is the geometry we will use to analyse the asymptotic behaviour of the $\\,\\textrm{U}(1)^4\\,$ invariant complex scalars (\\ref{Janus_U1^4_rho_2}).\n\n\n\nWhen approaching the maximally supersymmetric AdS$_4$ vacuum dual to ABJM theory\\footnote{\\label{footnote:operators}The 35 pseudo-scalars and 35 proper scalars of the maximal supergravity multiplet are dual to single-trace deformations of ABJM theory \\cite{Aharony:2008ug}. More concretely, pseudo-scalars are dual to fermionic bilinears $\\,\\mathcal{O}_{F}=\\textrm{Tr}(\\psi^{\\dot{A}} \\psi^{\\dot{B}}) - \\frac{1}{8} \\delta^{\\dot{A}\\dot{B}} \\textrm{Tr}(\\psi^{\\dot{C}} \\psi^{\\dot{C}} )\\,$ with $\\,\\dot{A}=1,\\ldots,8\\,$ and $\\,\\textrm{dim}(\\mathcal{O}_{F})=2\\,$. Proper scalars are dual to bosonic bilinears $\\,\\mathcal{O}_{B}=\\textrm{Tr}(X^{A} X^{B}) - \\frac{1}{8} \\delta^{AB} \\textrm{Tr}(X^{C}X^{C})\\,$ with $\\,A=1,\\ldots,8\\,$ and $\\,\\textrm{dim}(\\mathcal{O}_{B})=1\\,$.}, the asymptotic behaviour of (\\ref{Janus_U1^4_rho_2}) around the endpoints $\\,\\rho \\rightarrow \\pm \\infty\\,$ of the Janus solution reads\n\\begin{equation}\n\\label{source&vevs_zt}\n\\tilde{z}_{i}(\\rho) = \\dfrac{\\tilde{z}_{i,0}}{\\rho} + \\dfrac{\\tilde{z}_{i,1}}{\\rho^2} + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) \n\\hspace{10mm} \\textrm{ with } \\hspace{10mm} \ni=1,2,3 \\ , \n\\end{equation}\nin terms of normalisable modes $\\,\\tilde{z}_{i,0}\\,$ with $\\,\\Delta_{i}=1\\,$ specified by the parameters $\\,(\\alpha_{i} , \\beta_{i})\\,$,\n\\begin{equation}\n\\label{zt_0}\n\\tilde{z}_{i,0} = \\sinh\\alpha_{i} \\, e^{i (\\beta_i - \\frac{\\pi}{2})} \\ ,\n\\end{equation}\nas well as normalisable modes $\\,\\tilde{z}_{i,1}\\,$ with $\\,\\Delta_{i}=2\\,$. These modes satisfy a set of $\\alpha_{i}$-dependent algebraic relations\n\\begin{equation}\n\\label{zt_1}\n\\tilde{z}_{i,1} - i \\cosh\\alpha_{i} \\, \\tilde{z}_{i,0} = 0 \\ .\n\\end{equation}\nThe on-shell relations (\\ref{zt_1}) will help us to characterise the deformations in the field theory dual of the Janus solution upon appropriate manipulation of boundary terms and finite counterterms. \n\n\n\nIn order to discuss the boundary conditions (\\ref{source&vevs_zt})--(\\ref{zt_1}) in more detail, we will resort to an expansion of $\\,\\text{Re}\\tilde{z}_{i}\\,$ (proper scalars) and $\\,\\text{Im}\\tilde{z}_{i}\\,$ (pseudo-scalars) around $\\rho\\rightarrow\\pm\\infty$. This yields\n\\begin{equation}\n\\label{source&vevs_rho_f}\n\\begin{array}{llll}\n\\text{Re}\\tilde{z}_{i}(\\rho) & = & \\dfrac{a^{(v)}_{i,0}}{\\rho} + \\dfrac{a^{(s)}_{i,1}}{\\rho^2} + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & , \\\\[4mm]\n\\text{Im}\\tilde{z}_{i}(\\rho) & = & \\dfrac{b^{(s)}_{i,0}}{\\rho} + \\dfrac{b^{(v)}_{i,1}}{\\rho^2} + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & ,\n\\end{array}\n\\end{equation}\nso that\n\\begin{equation}\n\\label{zToab}\n\\tilde{z}_{i,0} = a^{(v)}_{i,0} + i \\, b^{(s)}_{i,0}\n\\hspace{5mm} , \\hspace{5mm}\n\\tilde{z}_{i,1} = a^{(s)}_{i,1} + i \\, b^{(v)}_{i,1} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{ab_definition}\na^{(v)}_{i,0}=\\sinh\\alpha_{i} \\, \\sin\\beta_i\n\\hspace{5mm} , \\hspace{5mm}\nb^{(s)}_{i,0} = - \\sinh\\alpha_{i} \\, \\cos\\beta_i\n\\ .\n\\end{equation}\nThe algebraic relations in (\\ref{zt_1}) then become\n\\begin{equation}\n\\label{ab_algebraic}\na^{(s)}_{i,1} + \\cosh\\alpha_{i} \\,\\, b^{(s)}_{i,0} = 0\n\\hspace{5mm} , \\hspace{5mm}\nb^{(v\t)}_{i,1} - \\cosh\\alpha_{i} \\,\\, a^{(v)}_{i,0} = 0 \\ .\n\\end{equation}\nNote that the independent parameters specifying the boundary conditions in (\\ref{ab_definition}) are $\\,(\\alpha_{i} , \\beta_{i})\\,$. As a consequence, the coefficients in the expansions (\\ref{source&vevs_rho_f}) obey the following two sets of algebraic relations \n\\begin{equation}\n\\dfrac{\\left(a^{(s)}_{i,1}\\right)^2}{\\left(b^{(s)}_{i,0}\\right)^2} = 1+ |\\tilde{z}_{i,0} |^2\n\\hspace{8mm} , \\hspace{8mm}\n\\dfrac{\\left(b^{(v)}_{i,1}\\right)^2}{\\left(a^{(v)}_{i,0}\\right)^2} = 1+ |\\tilde{z}_{i,0} |^2 \\ .\n\\end{equation}\nLastly, following \\cite{Bobev:2011rv} (see also \\cite{Bobev:2013yra}), we have attached the labels ``source\" $\\,^{(s)}\\,$ and ``VEV\" $\\,^{(v)}\\,$ to the modes in (\\ref{source&vevs_rho_f}) to highlight that, in order to preserve maximal supersymmetry, proper scalars should feature the alternative quantisation and pseudo-scalars the standard quantisation. Note that setting $\\,\\beta_{i}=\\pm\\frac{\\pi}{2}\\,$ switches off the sources in (\\ref{source&vevs_rho_f}) leaving only the VEV's. This is in agreement with the standard AdS\/CFT prescription and renders $\\,\\tilde{z}_{i,0}\\,$ in (\\ref{zt_0}) real.\n\n\n\n\n\n\\subsubsection{Janus solutions and boundary conditions}\n\n\n\nLet us compute the on-shell variation of the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}). A standard computation yields the boundary term\n\\begin{equation}\n\\label{deltaS}\n\\delta S = \\displaystyle\\sum_{i} \\delta S_{i} = \\displaystyle\\sum_{i} \\int d^{4}x \\,\\, \\partial_{\\mu} \\theta^{\\mu}_{i} = - \\displaystyle\\sum_{i} \\int_{\\partial M}d^{3}x \\frac{\\sqrt{-h}}{\\left( 1-\\left\\vert\n\\tilde{z}_{i}\\right\\vert ^{2}\\right)^{2}} \\, N^{\\mu} \\, ( \\partial_{\\mu } \\tilde{z}_{i} \\, \\delta\\tilde{z}_{i}^{\\ast } + \\textrm{c.c.} ) \\ ,\n\\end{equation}\nwhere \n\\begin{equation}\n\\theta^{\\mu}_{i} \\equiv - \\frac{\\sqrt{-g}}{\\left( 1-\\left\\vert\n\\tilde{z}_{i}\\right\\vert^{2}\\right)^{2}} \\, g^{\\mu \\nu } \\left( \\partial_{\\nu} \\tilde{z}_{i} \\,\\,\n\\delta\\tilde{z}_{i}^{\\ast} + \\textrm{c.c.} \\right) \\ , \n\\end{equation}\nand c.c stands for complex conjugation. In (\\ref{deltaS}) we have introduced the standard foliation $\\,g_{\\mu \\nu }=h_{\\mu \\nu }+N_{\\mu} N_{\\nu }\\,$ with $\\,N_{\\mu }=\\sqrt{g_{\\rho \\rho }} \\, \\delta _{\\mu }^{\\rho }\\,$ being the vector normal to the AdS$_{3}$ leaves. \n\nPlugging into (\\ref{deltaS}) the asymptotic expansion of the scalars in (\\ref{source&vevs_zt}) around $\\,\\rho \\rightarrow \\pm \\infty\\,$, and using the asymptotic form of the metric (\\ref{Janus_U1^4_rho_1}), we encounter the well known linearly divergent term. In order to regularise the above boundary action and have a well-defined variational principle we introduce, for each complex field $\\,\\tilde{z}_{i}\\,$, the counter-term\n\\begin{equation}\n\\label{counterterm}\nS_{\\textrm{ct},i} = - \\, g \\lim_{\\rho \\rightarrow \\pm\\infty } \\, \\int_{\\partial M} \nd^{3}x \\sqrt{-h} \\,\\,\n\\tilde{z}_{i} \\, \\tilde{z}_{i}^{\\ast } \\ ,\n\\end{equation}\nso that\n\\begin{equation}\n\\label{boundary_contributions}\n\\delta S_{i} + \\delta S_{\\textrm{ct},i} = g^{-2} \\, k^{-3} \\int_{\\partial M} \\left( \\tilde{z}_{i,1} \\, \\delta \\tilde{z}_{i,0}^{\\ast } + \\tilde{z}_{i,1}^{\\ast} \\, \\delta \\tilde{z}_{i,0} \\right) \\, d\\Sigma \\ ,\n\\end{equation}\nin terms of the volume element at the boundary $\\,d\\Sigma=\\sqrt{-\\gamma} \\, d^{3}x \\,$ with $\\, \\sqrt{-\\gamma} =\\sin\\eta \\, \\cos^{-3}\\eta\\,$. Substituting the scalar mode parameterisation of (\\ref{zToab}) into the boundary contributions in (\\ref{boundary_contributions}) one obtains\n\\begin{equation}\n\\label{boundary_contributions_ab}\n\\delta S_i + \\delta S_{\\textrm{ct},i} = 2 \\, g^{-2} \\, k^{-3} \\int_{\\partial M} \\left( a^{(s)}_{i,1} \\, \\delta a^{(v)}_{i,0} + b^{(v)}_{i,1} \\, \\delta b^{(s)}_{i,0} \\right) \\, d\\Sigma \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{k_factor_alt}\nk^2= 1 + \\sum_{i}\\sinh^{2}\\alpha_{i} = 1 + \\sum_{i} |\\tilde{z}_{i,0} |^2 \\, \\ge \\, 1 \\ .\n\\end{equation}\nIn order to remove the $\\,k^{-3}\\,$ factor in (\\ref{boundary_contributions_ab}) we could rescale the radial coordinate as $\\,\\hat{\\rho} = k \\, \\rho\\,$ or, instead, perform the non-linear mode redefinitions\n\\begin{equation}\n\\label{rescaled_ab}\n\\hat{a}^{(v)}_{i,0} = k^{-1} \\, a^{(v)}_{i,0} \n\\hspace{5mm} , \\hspace{5mm}\n\\hat{a}^{(s)}_{i,1} = k^{-2} \\, a^{(s)}_{i,1}\n\\hspace{5mm} , \\hspace{5mm}\n\\hat{b}^{(s)}_{i,0} = k^{-1} \\, b^{(s)}_{i,0} \n\\hspace{5mm} , \\hspace{5mm}\n\\hat{b}^{(v)}_{i,1} = k^{-2} \\, b^{(v)}_{i,1} \\ .\n\\end{equation}\nFollowing the latter prescription, the boundary contribution in (\\ref{boundary_contributions_ab}) becomes\n\\begin{equation}\n\\label{boundary_contributions_ab_hat}\n\\delta S_{i} + \\delta S_{\\textrm{ct},i} \\, = \\, 2 \\, g^{-2} \\int_{\\partial M} \\left( \\hat{a}^{(s)}_{i,1} \\, \\delta \\hat{a}^{(v)}_{i,0} + \\hat{b}^{(v)}_{i,1} \\, \\delta \\hat{b}^{(s)}_{i,0} \\right) \\, d\\Sigma \\ ,\n\\end{equation}\nand, due to the alternative quantisation featured by the proper scalars, we must add an extra boundary term such that\n\\begin{equation}\n\\label{boundary_contributions_ab_hat_final}\n\\delta S_{i} + \\delta S_{\\textrm{ct},i} - \\delta \\left( 2 \\, g^{-2} \\int_{\\partial M} \\hat{a}^{(s)}_{i,1} \\, \\hat{a}^{(v)}_{i,0} \\right) \\, = \\, 2 \\, g^{-2} \\int_{\\partial M} \\left( \\hat{b}^{(v)}_{i,1} \\, \\delta \\hat{b}^{(s)}_{i,0} - \\hat{a}^{(v)}_{i,0} \\, \\delta \\hat{a}^{(s)}_{i,1} \\right) \\, d\\Sigma \\ .\n\\end{equation}\nTherefore, having a well-defined variational principle therefore requires $\\,\\delta \\hat{b}^{(s)}_{i,0} = \\delta \\hat{a}^{(s)}_{i,1} = 0\\,$. Recalling from (\\ref{ab_definition})-(\\ref{ab_algebraic}) that \n\\begin{equation}\nb^{(s)}_{i,0} = - \\sinh\\alpha_{i} \\, \\cos\\beta_i \n\\hspace{10mm} \\textrm{ and } \\hspace{10mm}\na^{(s)}_{i,1} = - \\cosh\\alpha_{i} \\,\\, b^{(s)}_{i,0} \\ ,\n\\end{equation}\nwe conclude that sources are generically present at the boundary theory of the Janus ($\\alpha_{i} \\neq 0$) except for the particular choice of boundary conditions $\\,\\beta_{i}= \\pm \\frac{\\pi}{2}\\,$. This implies that every choice of $\\,(\\alpha_{i}, \\beta_{i})\\,$ with $\\,\\beta_{i} \\neq \\pm \\frac{\\pi}{2}\\,$ corresponds to a different theory with a different value of the sources in the variational principle. On the contrary, when $\\,\\beta_{i}= \\pm \\frac{\\pi}{2}\\,$, the sources are zero on-shell and the boundary theory is unique.\n\n\n\n\n\n\n\n\\subsection{Multi-parametric Hades solutions}\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.50\\textwidth]{Plots\/boundary_Hades_straight.pdf} \n\\put(10,95){$\\textrm{Re}\\tilde{z}_{i}$} \\put(-105,228){$\\textrm{Im}\\tilde{z}_{i}$}\n\\put(-112.5,107.5){{\\color{red}{$\\bullet$}}}\n\\end{center}\n\\caption{Parametric plot of $\\,\\tilde{z}_{i}(\\rho)\\,$ in (\\ref{Hades_U1^4_rho_2}) for the Hades solutions with $\\,\\alpha_{i}=1\\,$ (blue-solid lines) and the ridge flows with $\\,\\alpha_{i}=0\\,$ (brown-dashed lines) upon setting $\\,\\beta_{i}=\\frac{n\\pi}{4}\\,$ with $\\,n=0,\\ldots,7\\,$. The central red point at $\\,\\tilde{z}_{i}=0\\,$ $\\,\\forall i\\,$ corresponds to the maximally supersymmetric AdS$_{4}$ vacuum and describes the asymptotic values at $\\,\\rho\\rightarrow \\infty\\,$. The boundary circle at $\\,|\\tilde{z}_{i}|=1\\,$ corresponds to the singularity at $\\,\\rho = 1\\,$.}\n\\label{fig:Hades_ztilde_U1^4}\n\\end{figure}\n\n\n\n\nStarting from the field equations in (\\ref{EOM_scalars})-(\\ref{EOM_Einstein}) and performing a change of radial coordinate\n\\begin{equation}\n\\label{new_coordinate_Hades}\n\\rho= \\cosh({g} \\mu) \n\\hspace{10mm} , \\hspace{10mm}\nd\\mu = g^{-1} \\, \\dfrac{d\\rho}{\\sqrt{\\rho^2-1}} \\ ,\n\\end{equation}\nwe find a new class of singular solutions of the form\n\\begin{equation}\n\\label{Hades_U1^4_rho_1}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left( \\frac{d\\rho^{2}}{\\rho^{2}-1} + \n\\frac{\\rho^{2}-1 }{ k^2} \\, d\\Sigma^{2} \\right)\n\\hspace{10mm} \\textrm{ with } \\hspace{10mm}\nk^2= - 1 + \\sum_{i} \\cosh^2\\alpha_{i} \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Hades_U1^4_rho_2}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}}\\, \\frac{\\cosh\\alpha_{i}}{\\sinh\\alpha_{i} + i \\rho} \\ .\n\\end{equation}\nThese solutions are defined in the domain $\\,\\rho \\in [1,\\infty)\\,$ and feature a singularity at $\\,\\rho=1\\,$ where the change of radial coordinate in (\\ref{new_coordinate_Hades}) is ill-defined, the warping factor in front of the AdS$_{3}$ piece in the geometry collapses to zero size and $\\,|\\tilde{z}_{i}(1)|=1\\,$ (see Figure~\\ref{fig:Hades_ztilde_U1^4}). More concretely, the Ricci scalar constructed from the metric (\\ref{Hades_U1^4_rho_1}) reads\n\\begin{equation}\n\\label{Hades_Ricci}\nR(\\rho) = - 6 \\, g^{2} \\left( \\, 1 + \\frac{\\rho^2 + k^2}{\\rho^2 - 1} \\, \\right) \\ ,\n\\end{equation}\nand becomes singular at $\\,\\rho= 1\\,$. An analysis of the BPS equations (\\ref{BPS_scalars}) shows that the flows in (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}) turn out to be non-supersymmetric. We will refer to these singular solutions as flows to Hades. This term was coined for singular (flat-sliced) domain-walls dual to conventional RG-flows in \\cite{Freedman:1999gp,Gubser:2000nd}.\n\n\nAs previously done for the Janus solution, let us expand $\\,\\text{Re}\\tilde{z}_{i}\\,$ (proper scalars) and $\\,\\text{Im}\\tilde{z}_{i}\\,$ (pseudo-scalars) around $\\,\\rho\\rightarrow \\infty\\,$. One finds\n\\begin{equation}\n\\label{Hades_source&vevs_rho}\n\\begin{array}{llll}\n\\text{Re}\\tilde{z}_{i}(\\rho) & = & \\dfrac{a^{(v)}_{i,0}}{\\rho} + \\dfrac{a^{(s)}_{i,1}}{\\rho^2} + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & , \\\\[4mm]\n\\text{Im}\\tilde{z}_{i}(\\rho) & = & \\dfrac{b^{(s)}_{i,0}}{\\rho} + \\dfrac{b^{(v)}_{i,1}}{\\rho^2} + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & ,\n\\end{array}\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Hades_ab_vevs}\na^{(v)}_{i,0}=\\cosh\\alpha_{i} \\, \\sin\\beta_i\n\\hspace{5mm} , \\hspace{5mm}\nb^{(v\t)}_{i,1} = \\sinh\\alpha_{i} \\,\\, a^{(v)}_{i,0} \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Hades_ab_sources}\nb^{(s)}_{i,0} = - \\cosh\\alpha_{i} \\, \\cos\\beta_i\n\\hspace{5mm} , \\hspace{5mm}\na^{(s)}_{i,1} = - \\sinh\\alpha_{i} \\,\\, b^{(s)}_{i,0} \\ .\n\\end{equation}\nDifferent choices of the Hades parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ translate into different boundary conditions in the expansions (\\ref{Hades_source&vevs_rho}). Note that the boundary theory has sources in (\\ref{Hades_ab_sources}) generically activated except if setting $\\,\\beta_{i}=\\pm\\frac{\\pi}{2}\\,$.\n\n\n\n\\subsubsection{Ridge flows with \\texorpdfstring{$\\,\\alpha_{i}=0\\,$}{alpha=0}}\n\\label{sec:ridge_4D}\n\n\nUnlike for the Janus solutions, setting $\\,\\alpha_{i}=0\\,$ does not recover a regular AdS$_{4}$ vacuum. Instead, the complex scalars in (\\ref{Hades_U1^4_rho_2}) reduce to\n\\begin{equation}\n\\label{Ridge_U1^4_rho_2}\n\\tilde{z}_{i}(\\rho) = \\rho^{-1} \\, e^{i \\left( \\beta_{i} - \\frac{\\pi}{2} \\right)} \\ ,\n\\end{equation}\nand \\textit{ridge flows} of the type investigated in \\cite{Pilch:2015dwa,Pilch:2015vha} appear with constant $\\,\\textrm{arg}\\tilde{z}_{i} = \\beta_{i} - \\frac{\\pi}{2}\\,$ and $\\,k^2=2\\,$ in the singular geometry (\\ref{Hades_U1^4_rho_1}). The $\\,\\rho \\rightarrow \\infty\\,$ expansion in (\\ref{Hades_source&vevs_rho}) and the boundary conditions in (\\ref{Hades_ab_vevs})-(\\ref{Hades_ab_sources}) also simplify drastically\n\n\\begin{equation}\n\\label{Ridge_source&vevs_rho}\n\\begin{array}{llll}\n\\text{Re}\\tilde{z}_{i}(\\rho) & = & \\dfrac{a^{(v)}_{i,0}}{\\rho} + \\dfrac{a^{(s)}_{i,1}}{\\rho^2} + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & , \\\\[4mm]\n\\text{Im}\\tilde{z}_{i}(\\rho) & = & \\dfrac{b^{(s)}_{i,0}}{\\rho} + \\dfrac{b^{(v)}_{i,1}}{\\rho^2} + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & ,\n\\end{array}\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Ridge_ab_definition}\na^{(v)}_{i,0}= \\sin\\beta_i\n\\hspace{8mm} , \\hspace{8mm}\nb^{(s)}_{i,0} = - \\cos\\beta_i\n\\hspace{8mm} , \\hspace{8mm}\na^{(s)}_{i,1} = 0\n\\hspace{8mm} , \\hspace{8mm}\nb^{(v\t)}_{i,1} = 0 \n\\ .\n\\end{equation}\nTwo special cases are immediately identified. Setting $\\,\\beta_{i}=0,\\pi\\,$ renders $\\,\\tilde{z}_{i}(\\rho)\\,$ purely imaginary and the ridge flow from the maximally supersymmetric AdS$_{4}$ vacuum at $\\,\\rho \\rightarrow \\infty\\,$ is triggered by the source modes $\\,b^{(s)}_{i,0}\\,$ of the pseudo-scalars dual to fermion bilinears. On the contrary, setting $\\,\\beta_{i}=\\pm\\frac{\\pi}{2}\\,$ renders $\\,\\tilde{z}_{i}(\\rho)\\,$ purely real and the ridge flow is triggered by the VEV modes $\\,a^{(v)}_{i,0}\\,$ of the proper scalars dual to boson bilinears. As we will see in Section~\\ref{sec:Uplift_11D}, the uplift of these special ridge flows to eleven dimensions will be very different. This is to be contrasted with the situation in four dimensions where the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) is invariant under constant shifts of $\\,\\beta_{i}\\,$. Note also that a shift of the form $\\,\\beta_{i} \\rightarrow \\beta_{i} + \\pi\\,$ amounts to a reflection $\\,\\rho \\rightarrow -\\rho\\,$ in the respective field $\\,\\tilde{z}_{i}\\,$ in (\\ref{Ridge_U1^4_rho_2}) while leaving the Hades metric in (\\ref{Hades_U1^4_rho_1}) invariant. Since the domain of the radial coordinate is fixed to $\\,\\rho \\in [1,\\infty)\\,$, the shift $\\,\\beta_{i} \\rightarrow \\beta_{i} + \\pi\\,$ generically generates a new solution. \n\n\n\n\n\nA fundamental difference between our ridge flows in (\\ref{Hades_U1^4_rho_1}) and (\\ref{Ridge_U1^4_rho_2}) and the ones investigated in \\cite{Pope:2003jp,Pilch:2015dwa,Pilch:2015vha} is that the ones there have a flat-sliced geometry. Therefore they correspond to conventional holographic RG-flows. Our solutions have an AdS$_{3}$-slicing of the geometry, instead. It was further shown in \\cite{Pilch:2015dwa,Pilch:2015vha} that, for the flat-sliced solutions, only a set of discrete values of $\\,\\textrm{arg}\\tilde{z}_{i}\\,$ was compatible with supersymmetry. However, if relaxing supersymmetry, any value of $\\,\\textrm{arg}\\tilde{z}_{i}\\,$ was permitted. In our non-supersymmetric ridge flows, any possible value of $\\,\\beta_{i}\\,$ is permitted too. Generic flows to Hades with $\\,\\alpha_{i} \\neq 0\\,$ and ridge flows with $\\,\\alpha_{i} = 0\\,$ are depicted in Figure~\\ref{fig:Hades_ztilde_U1^4}.\n\n\n\n\n\\subsubsection{Hades with (super) symmetry enhancements}\n\n\n\nAs already discussed for the Janus solutions in Section~\\ref{sec:Janus_sym_enhancement}, imposing identifications between the complex fields $\\,\\tilde{z}_{i}(\\rho)\\,$ translates into different patterns of (super) symmetry enhancements. For example, non-supersymmetric Hades solutions with $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetry are obtained upon identifying the three complex scalars, namely, upon setting $\\,{\\alpha_{1}=\\alpha_{2}=\\alpha_{3}}\\,$ and $\\,{\\beta_{1}=\\beta_{2}=\\beta_{3}}\\,$ in the general Hades solution (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}).\n\n\n\n\nSupersymmetric Hades solutions with an AdS$_{3}$ slicing have previously been constructed in \\cite{Bobev:2013yra} within the $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ invariant sector of the $\\,\\textrm{SO}(8)\\,$ gauged supergravity. As discussed in Section~\\ref{sec:Janus_SO4xSO4}, this sector of the theory is recovered upon setting two of the three complex fields $\\,\\tilde{z}_{i}\\,$ to zero, \\textit{i.e.}, $\\,\\tilde{z}_{2}(\\rho)=\\tilde{z}_{3}(\\rho)=0\\,$. However, it is easy to see that this cannot be achieved by tuning the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ in (\\ref{Hades_U1^4_rho_2}) to any real value. Instead, one must set two complex fields to zero from the start and search for solutions of the field equations. In this manner, one finds Hades solutions of the form \n\\begin{equation}\n\\label{Hades_no_ridge_SO(4)xSO(4)_rho_1}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left( \\frac{d\\rho^{2}}{\\rho^{2}-1} + \n\\frac{\\rho^{2}-1 }{ k^2} \\, d\\Sigma^{2} \\right) \n\\hspace{10mm} \\textrm{ with } \\hspace{10mm}\nk^2= \\sinh^2\\alpha_{1}\n\\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Hades_no_ridge_SO(4)xSO(4)_rho_2}\n\\tilde{z}_{1}(\\rho) = e^{i \\beta_{1}}\\, \\frac{\\cosh\\alpha_{1}}{\\sinh\\alpha_{1} + i \\rho} \n\\hspace{8mm} , \\hspace{8mm} \\tilde{z}_{2}(\\rho)=\\tilde{z}_{3}(\\rho)=0 \\ ,\n\\end{equation}\nwhich turn out to solve the BPS equations in (\\ref{BPS_A})-(\\ref{BPS_scalars}). It is worth emphasising that these supersymmetric Hades with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)$ symmetry do not belong to the same class of solutions as the non-supersymmetric Hades in (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}). Also, they do not admit a ridge flow limit since setting $\\,\\alpha_{1}=0\\,$ implies having a pathological ($k^2=0$) warping of AdS$_{3}$ in the geometry (\\ref{Hades_no_ridge_SO(4)xSO(4)_rho_1}).\n\n\n\n\\section{Uplift to eleven-dimensional supergravity}\n\\label{sec:Uplift_11D}\n\nIn this section we present the uplift to eleven-dimensional supergravity of the Janus and Hades solutions constructed within the four-dimensional SO(8) gauged supergravity. We use the conventions of \\cite{Gauntlett:2002fz} according to which the Lagrangian of eleven-dimensional supergravity \\cite{Cremmer:1978km} takes the form\n\\begin{equation}\n\\mathcal{L}_{11} = \\hat{R} \\, \\text{vol}_{11} - \\tfrac{1}{2} \\, \\hat{F}_{(4)} \\wedge*_{11} \\hat{F}_{(4)} - \\tfrac{1}{6} \\, \\hat{A}_{(3)} \\wedge\\hat{F}_{(4)} \\wedge\\hat{F}_{(4)} \\ .\n\\end{equation}\nA consistent background is then subject to the source-less Bianchi identity\n\\begin{equation}\n\\label{BI_F4}\nd\\hat{F}_{(4)} = 0 \\ ,\n\\end{equation}\nas well as the equations of motion\n\\begin{equation}\n\\label{EOM_11D}\n\\begin{array}\n[c]{rll}%\nd(*_{11} \\hat{F}_{(4)}) + \\frac{1}{2} \\, \\hat{F}_{(4)} \\wedge\\hat{F}_{(4)} & = & 0 \\ ,\\\\[2mm]%\n\\hat{R}_{MN} - \\frac{1}{12} \\left( \\hat{F}_{MPQR} \\, \\hat{F}_{N}{}^{PQR} - \\frac{1}{12} \\, \\hat{F}_{PQRS} \\, \\hat{F}^{PQRS} \\, \\hat{G}_{MN} \\right) & = & 0 \\ .\n\\end{array}\n\\end{equation}\nThe equation of motion for $\\,\\hat{F}_{(4)}\\,$ in (\\ref{EOM_11D}) can be used to introduce the dual flux \n\\begin{equation}\n\\label{F7_definition}\n\\hat{F}_{(7)} \\equiv *_{11} \\hat{F}_{(4)} + \\tfrac{1}{2} \\hat{A}_{(3)} \\wedge \\hat{F}_{(4)} \\ ,\n\\end{equation}\nwhich therefore obeys the Bianchi identity $\\,d\\hat{F}_{(7)}=0\\,$. The flux in (\\ref{F7_definition}) determines the conserved Page charge of M2-branes in the background\\footnote{We have set the string length to unity, \\textit{i.e.}, $\\,\\ell_{s} = 1\\,$.}\n\\begin{equation}\n\\label{M2_brane_charge}\nN_{2} = \\frac{1}{(2 \\pi)^{6}} \\int_{M_{7}} \\hat{F}_{(7)} = \\frac{1}{(2 \\pi)^{6}} \\int_{M_{7}} *_{11} \\hat{F}_{(4)} + \\tfrac{1}{2} \\hat{A}_{(3)} \\wedge \\hat{F}_{(4)} \\ ,\n\\end{equation}\nwhere $\\,M_{7}\\,$ is the internal space. The contribution $\\,*_{11} \\hat{F}_{(4)}\\,$ comes from electric M2-branes and the contribution $\\,{\\tfrac{1}{2} \\hat{A}_{(3)} \\wedge \\hat{F}_{(4)}}\\,$ originates from magnetic M5-branes. \n\n\n\n\\subsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2\\,$}{SU(3) x U(1)2} invariant sector}\n\\label{sec:11D_Janus}\n\nThe eleven-dimensional uplift of the $\\,\\text{SU}(3) \\times\\text{U}(1)^2\\,$ invariant sector of the maximal SO(8) supergravity has been worked out in \\cite{Pilch:2015dwa,Azizi:2016noi} (see also \\cite{Larios:2019kbw}). To describe the internal geometry, we will closely follow the Appendix~B.2 of \\cite{Larios:2019kbw} and use intrinsic coordinates on $\\,\\textrm{S}^{7}\\,$ adapted to its seven-dimensional Sasaki--Einstein structure. In these coordinates, the round metric on $\\,\\textrm{S}^{7}\\,$ takes the form\n\\begin{equation}\n\\label{metric_round_S7}\nds_{7}^{2} = ds_{\\mathbb{CP}_{3}}^{2} + \\left( d\\psi_{-}+ \\sigma_{-} \\right)^2 \\ ,\n\\end{equation}\nwhere $\\,ds_{\\mathbb{CP}_{3}}^{2}\\,$ is the Fubini-Study line element (normalised as in \\cite{Larios:2019kbw})\n\\begin{equation}\n\\label{metric_round_CP3}\nds_{\\mathbb{CP}_{3}}^{2} = d\\tilde{\\alpha}^{2} + \\cos^{2} \\tilde{\\alpha} \\, \\big( \\, ds^{2}_{\\mathbb{CP}_{2}} + \\sin^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma)^{2} \\, \\big)\n\\hspace{6mm} \\textrm{ with } \\hspace{6mm}\n\\sigma_{-} = \\cos^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma) \\ .\n\\end{equation}\nThe ranges of the angles in (\\ref{metric_round_S7})-(\\ref{metric_round_CP3}) are $\\,\\tilde{\\alpha} \\in[0,\\frac\n{\\pi}{2}]\\,$, $\\,\\tau_{-} \\in[0,2 \\pi]\\,$ and $\\,\\psi_{-} \\in[0,2 \\pi]\\,$. Moreover, $\\,\\sigma\\,$ in (\\ref{metric_round_CP3}) is the one-form on $\\,\\mathbb{CP}_{2}\\,$ such that $\\,d\\sigma=2\\boldsymbol{J}\\,$ with $\\,\\boldsymbol{J}\\,$ being the K\\\"ahler form on $\\,\\mathbb{CP}_{2}\\,$. The round metric in (\\ref{metric_round_S7}) occurs when the scalar field in the four-dimensional Lagrangian (\\ref{Lagrangian_model_SU3xU1xU1}) vanishes, \\textit{i.e.}, $\\,\\tilde{z}=0\\,$, and the $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ Freund--Rubin vacuum of eleven-dimensional supergravity is recovered \\cite{Freund:1980xh}. However, whenever non-vanishing, the scalar $\\,\\tilde{z}\\,$ in (\\ref{Janus_solution_SU3xU1xU1}) inflicts a deformation on the Freund--Rubin vacuum so that a new background is generated which displays a smaller $\\,\\text{SU}(3) \\times \\text{U}(1)^{2} \\subset \\textrm{SO}(8)\\,$ isometry group.\n\n\n\nWe are encoding the breaking of isometries caused by $\\,\\tilde{z}\\,$ into a set of metric functions $\\,f\\,$'s and flux functions $\\,h\\,$'s. The eleven-dimensional metric takes the form\n\\begin{equation}\n\\label{11D_metric}\n\\begin{array}{lll}\nd\\hat{s}_{11}^{2} & = & \\frac{1}{2} \\, f_{1} \\, ds_{4}^{2} + 2 \\, g^{-2} \\Big[ f_{2} \\, d\\tilde{\\alpha}^{2} + \\cos^{2} \\tilde{\\alpha} \\, \\big( \\, f_{3} \\, \\, ds^{2}_{\\mathbb{CP}_{2}} + \\sin^{2} \\tilde{\\alpha} \\, f_{4} \\, (d\\tau_{-} + \\sigma)^{2} \\, \\big)\\\\[2mm]\n& + & f_{5} \\, \\big( d\\psi_{-} + \\cos^{2} \\tilde{\\alpha} \\, f_{6} \\, (d\\tau_{-} + \\sigma) \\big)^{2} \\Big] \\ ,\n\\end{array}\n\\end{equation}\nwith $\\,ds_{4}^{2}\\,$ given in (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}). Note that the eleven-dimensional metric (\\ref{11D_metric}) displays an $\\,\\text{SU}(3) \\times \\text{U}(1)_{\\tau_{-}} \\times\\text{U}(1)_{\\psi_{-}}\\,$ symmetry. The $\\,\\textrm{SU}(3)\\,$ factor accounts for the $\\,\\mathbb{CP}_{2}\\,$ isometries and the two $\\,\\textrm{U}(1)\\,$ factors correspond with shifts along the angles $\\,\\tau_{-}\\,$ and $\\,\\psi_{-}\\,$, hence the attached labels. The various metric functions in (\\ref{11D_metric}) depend on the complex scalar $\\tilde{z}$ in\n(\\ref{Janus_solution_SU3xU1xU1}) and on the angle $\\,\\tilde{\\alpha} \\,$ on S$^{7}$. They are given by\n\\begin{equation}\n\\label{f_functions}\n\\begin{array}{c}\nf_{1}^{3} = \\dfrac{ (1+\\tilde{z}) (1+\\tilde{z}^{*})}{(1-|\\tilde{z}|^{2})^{3}} \\, H^{2}\n\\hspace{5mm} , \\hspace{5mm}\nf_{2}^{3\/2} = \\dfrac{H}{(1+\\tilde{z}) (1+\\tilde{z}^{*})}\n\\hspace{5mm} , \\hspace{5mm}\nf_{3}^{3} = \\dfrac{(1+\\tilde{z}) (1+\\tilde{z}^{*})}{H} \\ , \\\\[6mm]\nf_{4}^{3\/2} = \\dfrac{(1-|\\tilde{z}|^{2})^{3}}{ (1+\\tilde{z}) (1+\\tilde{z}^{*})} \\, H \\, K^{-\\frac{3}{2}} \n\\hspace{5mm} , \\hspace{5mm}\nf_{5}^{3\/2} = \\dfrac{1}{ (1+\\tilde{z}) (1+\\tilde{z}^{*})} \\, H^{-2} \\, K^{\\frac{3}{2}} \\ , \\\\[8mm]\nf_{6} = \\Big[ (1+\\tilde{z}) (1+\\tilde{z}^{*}) \\, H + ( \\tilde{z} - \\tilde{z}^{*})^{2} \\cos(2\\tilde{\\alpha}) \\Big] \\, K^{-1} \\ ,\n\\end{array}\n\\end{equation}\nwith\n\\begin{equation}\nH = 1+|\\tilde{z}|^{2} - ( \\tilde{z} +\\tilde{z}^{*}) \\cos(2\\tilde{\\alpha}) \\hspace{8mm} \\text{ and } \\hspace{8mm} K = 1+|\\tilde{z}|^{4} - 2 \\, |\\tilde{z}|^{2} \\, \\cos(4\\tilde{\\alpha}) \\ .\n\\end{equation}\nThe round metric on $\\,\\textrm{S}^{7}\\,$ is recovered from (\\ref{11D_metric}) upon setting $\\,\\tilde{z}=0\\,$, what implies that all the metric functions $\\,H=K=f_{1,\\ldots,6}=1\\,$. The part of the internal geometry in the upper line of (\\ref{11D_metric}) then reconstructs the $\\,\\mathbb{CP}_{3}\\,$ metric in (\\ref{metric_round_CP3}). \n\n\n\nThe eleven-dimensional four-form flux takes a more lengthy expression given in terms of three-, one- and zero-form deformations in four dimensions which we collectively denote $\\,h$'s. Adopting the terminology of \\cite{Pilch:2015dwa}, the four-form flux naturally splits as\n\\begin{equation}\n\\label{11D_F4}\n\\hat{F}_{(4)} = \\hat{F}_{(4)}^{\\textrm{st}} + \\hat{F}_{(4)}^{\\textrm{tr}} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{11D_F4_st}\n\\hat{F}_{(4)}^{\\textrm{st}} =\n-\\frac{1}{2\\sqrt{2}} \\, g \\, h_{1} \\, \\text{vol}_{4} + \\frac{1}{\\sqrt{2}} \\, g^{-1} \\, \\sin(2 \\tilde{\\alpha}) \\,\\, h^{(3)}_{2} \\wedge d\\tilde{\\alpha} \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{11D_F4_tr}\n\\begin{array}{lll}\n\\hat{F}_{(4)}^{\\textrm{tr}} &=& - 2 \\sqrt{2} \\, g^{-3} \\Big[ \\, \\sin(2 \\tilde{\\alpha}) \\, h^{(1)}_{3} \\wedge d\\tilde{\\alpha} \\wedge d\\psi_{-} \\wedge(d\\tau_{-}+\\sigma)\\\\[2mm]\n& + & \\cos^{4} \\tilde{\\alpha} \\,\\, h^{(1)}_{4} \\wedge (d\\tau_{-} + \\sigma) \\wedge\\boldsymbol{J} + \\cos^{2} \\tilde{\\alpha} \\, \\cos(2 \\tilde{\\alpha}) \\,\\, h_{5}^{(1)} \\wedge d\\psi_{-} \\wedge\\boldsymbol{J}\\\\[2mm]\n& + & \\sin(2 \\tilde{\\alpha}) \\, h_{6} \\, d\\tilde{\\alpha} \\wedge d\\psi_{-} \\wedge\\boldsymbol{J} + \\cos^{4} \\tilde{\\alpha} \\,\\, h_{7} \\, \\boldsymbol{J} \\wedge\\boldsymbol{J}\\\\[2mm]\n& + & \\cos^{2}\\tilde{\\alpha} \\, \\sin(2 \\tilde{\\alpha}) \\,\\, h_{8} \\, d\\tilde{\\alpha} \\wedge(d\\tau_{-} + \\sigma) \\wedge\\boldsymbol{J} \\, \\Big] \\ .\n\\end{array}\n\\end{equation}\nFor the space-time part in (\\ref{11D_F4_st}) we have introduced a zero-form\n\\begin{equation}\n\\label{h1_func}\nh_{1} = \\dfrac{1}{(1-|\\tilde{z}|^{2})} \\Big( \\, 3 \\, (1+|\\tilde{z}|^{2}) + ( \\tilde{z} +\\tilde{z}^{*}) \\, (1 - 2 \\cos(2 \\tilde{\\alpha}) ) \\, \\Big) \\ , \n\\end{equation}\nand a three-form\n\\begin{equation}\n\\label{h2_func}\n\\begin{array}{lll}\nh^{(3)}_{2} & = & \\dfrac{1}{(1-|\\tilde{z}|^{2})^{2}} \\Big( (\\tilde{z}^{2}-1) *_{4} d\\tilde{z}^{*} + ((\\tilde{z}^{*})^{2}-1) *_{4} d\\tilde{z} \\Big) \\ ,\n\\end{array}\n\\end{equation}\nwhich has legs along the AdS$_{3}$ factor in the external geometry\\footnote{The Hodge dual $\\,*_{4}\\,$ is defined in four-dimensions using the metric $\\,ds^{2}_{4}\\,$ in (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}).}. For the transverse part in (\\ref{11D_F4_tr}) we have introduced a set of one-forms\n\\begin{equation}\n\\label{11D_F4_one-forms}\n\\begin{array}{lll}\nh^{(1)}_{3} & = & \\dfrac{i}{2} \\left( \\dfrac{d\\tilde{z}^{*} }{(1+\\tilde{z}^{*})^{2}} - \\dfrac{d\\tilde{z} }{(1+\\tilde{z})^{2}} \\right) \\ ,\\\\[4mm]\nh^{(1)}_{4} & = & h^{(1)}_{5} \\,\\, = \\,\\, i \\, H^{-2} \\, \\Big( \\left( 1 - 2 \\cos(2 \\tilde{\\alpha}) \\, \\tilde{z}^{*} + (\\tilde{z}^{*})^{2} \\right) \\, d\\tilde{z} - \\left( 1 - 2 \\cos(2 \\tilde{\\alpha}) \\, \\tilde{z} + \\tilde{z}^{2} \\right) \\, d\\tilde{z}^{*}\n\\Big) \\ ,\n\\end{array}\n\\end{equation}\ntogether with zero-forms\n\\begin{equation}\n\\begin{array}{lll}\nh_{6} & = & i \\, 4 \\, H^{-2} \\, (\\tilde{z}^{*}-\\tilde{z}) \\dfrac{(1+|\\tilde{z}|^{2})}{(1+\\tilde{z})(1+\\tilde{z}^{*})} \\Big( 1+ |\\tilde{z}|^{2} + (\\tilde{z}+\\tilde{z}^{*}) \\sin^{2}\\tilde{\\alpha} \\Big) \\ ,\\\\[4mm]\nh_{7} & = & -i \\, 2 \\, H^{-1} \\, (\\tilde{z}^{*}-\\tilde{z}) \\ ,\\\\[4mm]\nh_{8} & = & i \\, 2 \\, H^{-2} \\, (\\tilde{z}^{*}-\\tilde{z}) \\, \\Big( 1+ |\\tilde{z}|^{2} + (\\tilde{z}+\\tilde{z}^{*}) \\sin^{2}\\tilde{\\alpha} \\Big) \\ .\n\\end{array}\n\\end{equation}\nThe zero-forms $\\,h_{6}\\,$, $\\,h_{7}\\,$ and $\\,h_{8}\\,$ determine the purely internal components in (\\ref{11D_F4_tr}) and vanish if $\\,\\tilde{z}^{*}=\\tilde{z}\\,$. Also the one-form deformations in (\\ref{11D_F4_one-forms}) vanish in this case so that $\\,\\hat{F}_{(4)}^{\\textrm{tr}}=0\\,$. Lastly, the entire eleven-dimensional flux in (\\ref{11D_F4}) preserves an $\\,\\text{SU}(3)\\times\\text{U}(1)_{\\tau_{-}} \\times\\text{U}(1)_{\\psi_{-}}\\,$ symmetry since there is no explicit dependence on the angle $\\,\\psi_{-}\\,$ and, moreover, the two-form $\\,\\boldsymbol{J}\\,$ on $\\,\\mathbb{CP}_{2}\\,$ is not charged under $\\,\\text{U}(1)_{\\tau_{-}}\\,$.\n\n\nTo complete the uplift, the above quantities must be evaluated at the value of the complex scalar $\\,\\tilde{z} \\equiv \\tilde{z}_{1}=\\tilde{z}_{2}=\\tilde{z}_{3}\\,$ both for the Janus (\\ref{Janus_U1^4_rho_2}) and Hades (\\ref{Hades_U1^4_rho_2}) solutions. We have explicitly verified that the resulting eleven-dimensional backgrounds in (\\ref{11D_metric}) and (\\ref{11D_F4}) satisfy the source-less Bianchi identity and equations of motion in (\\ref{BI_F4}) and (\\ref{EOM_11D}), respectively.\n\n\n\\subsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2\\,$}{SU(3) x U(1)2} symmetric Janus}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f1_beta0.pdf} \n\\put(-100,100){$f_{1}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f2_beta0.pdf} \n\\put(-100,100){$f_{2}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f3_beta0.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{3}$}\n\\vspace{5mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f4_beta0.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, \\sin^2\\tilde{\\alpha} \\,f_{4}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f5_beta0.pdf} \n\\put(-100,100){$f_{5}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f6_beta0.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{5}\\, f_{6}$}\n\\end{center}\n\\caption{Regular metric functions in (\\ref{11D_metric}) for the Janus solution with $\\alpha=1$ and $\\beta=0$.}\n\\label{fig:f_functions_beta0}\n\\end{figure}\n\nWe have performed the explicit uplift of the analytic and non-supersymmetric Janus solution in (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}). The resulting eleven-dimensional backgrounds are everywhere regular and depend on the choice of parameters $\\,(\\alpha, \\beta)\\,$ specifying the boundary conditions (\\ref{ab_definition})-(\\ref{ab_algebraic}) of the four-dimensional Janus solution. Plots of the functions entering the metric (\\ref{11D_metric}) for $\\,\\beta=0\\,$ and $\\,\\beta=\\frac{\\pi}{2}\\,$ are depicted in Figure~\\ref{fig:f_functions_beta0} and Figure~\\ref{fig:f_functions_betaPi}. These two choices respectively activate only sources or VEV's in the Janus boundary conditions (\\ref{ab_definition})-(\\ref{ab_algebraic}). In addition, the scalar $\\,\\tilde{z}\\,$ in the $\\text{SU}(3) \\times\\text{U}(1)^2\\,$ symmetric Janus solution of (\\ref{Janus_solution_SU3xU1xU1}) is necessarily complex so that no limit to a real Janus solution exists even in the general case of (\\ref{Janus_solution_U1^4_ztil}). This further implies that all the $\\,h\\,$ functions (and also three- and one-forms) entering $\\,\\hat{F}_{(4)}^{\\textrm{st}}\\,$ in (\\ref{11D_F4_st}) and $\\,\\hat{F}_{(4)}^{\\textrm{tr}}\\,$ in (\\ref{11D_F4_tr}) are generically activated.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f1_betaPi2.pdf} \n\\put(-100,100){$f_{1}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f2_betaPi2.pdf} \n\\put(-100,100){$f_{2}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f3_betaPi2.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{3}$}\n\\vspace{5mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f4_betaPi2.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, \\sin^2\\tilde{\\alpha} \\,f_{4}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f5_betaPi2.pdf} \n\\put(-100,100){$f_{5}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f6_betaPi2.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{5}\\, f_{6}$}\n\\end{center}\n\\caption{Regular metric functions in (\\ref{11D_metric}) for the Janus solution with $\\alpha=1$ and $\\beta=\\frac{\\pi}{2}$.}\n\\label{fig:f_functions_betaPi}%\n\\end{figure}\n\n\n\nIn order to compute the M$2$-brane charge for the $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetric Janus, we first note that the dual seven-form flux can be expressed as\n\\begin{equation}\n\\label{F7_general}\n\\hat{F}_{(7)} = d\\hat{\\alpha} \\wedge h^{(6)} + \\ldots \\ , \n\\end{equation}\nwith $\\,h^{(6)}=\\frac{1}{2} \\boldsymbol{J} \\wedge \\boldsymbol{J} \\wedge d\\tau_{-} \\wedge d\\psi_{-}\\,$ being the volume form of $\\,M_{6}\\,$ spanned by $\\,(\\mathbb{CP}_{2}, \\tau_{-}, \\psi_{-})$, and $\\,\\hat{\\alpha}\\,$ playing the role of an ``adapted'' angular coordinate threaded by the flux. This adapted coordiante is in general a complicated function\n\\begin{equation}\n\\label{hat_alpha}\n\\hat{\\alpha}=\\hat{\\alpha}(\\rho,\\tilde{\\alpha} \\, ; \\,\\alpha,\\beta) \\ ,\n\\end{equation}\nthat depends on the original coordinates $\\,(\\rho,\\tilde{\\alpha})\\,$ as well as on the Janus parameters $\\,(\\alpha,\\beta)\\,$. Lastly, the ellipsis in (\\ref{F7_general}) stand for additional terms with legs on the AdS$_3$ piece of the geometry which do not play a relevant role when computing M$2$-brane charges. Therefore, all the relevant information regarding M$2$-brane charges gets codified into the one-form $\\,d\\hat{\\alpha}\\,$ as it defines an adapted angular direction. It is important to highlight that, when taking the limit $\\,\\rho \\rightarrow \\pm \\infty\\,$, one finds that $\\,d\\hat{\\alpha} \\propto \\sin(2\\tilde{\\alpha}) \\cos^4\\tilde{\\alpha} \\, d\\tilde{\\alpha}\\,$ no longer depends on the Janus parameters $\\,(\\alpha,\\beta)\\,$. In this limit, the dual seven-form flux threads the $\\,\\textrm{S}^7\\,$ as required by the asymptotic $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ geometry of the flow. \n\n\n\nThe computation of the M$2$-brane charge in the Janus solution gives\n\\begin{equation}\n\\label{N2_Janus}\nN_2 = \\frac{1}{(2\\pi)^6}\\int_{\\Gamma \\times \\textrm{M}_{6}} \\hat{F}_{(7)} = \\frac{1}{32 \\pi^2}\\int_{\\partial\\Gamma} \\hat{\\alpha} = \\frac{1}{4\\pi^2 g^6} \\ ,\n\\end{equation}\nwhere the relevant curves $\\,\\Gamma$'s threaded by the purely internal part of the seven-form flux in (\\ref{F7_general}) are specified by their tangent vector field $\\,\\boldsymbol{v} = (\\boldsymbol{v}_{\\mu},\\boldsymbol{v}_{\\tilde{\\alpha}}) = (g \\sqrt{\\rho^2+1} \\, \\partial_{\\rho}\\hat{\\alpha} \\,,\\, \\partial_{\\tilde{\\alpha}}\\hat{\\alpha})\\,$\\footnote{Note that $\\,\\boldsymbol{v}_{\\mu}=\\partial_{\\mu}\\hat{\\alpha}=g \\sqrt{\\rho^2+1} \\, \\partial_{\\rho}\\hat{\\alpha}\\,$ as a consequence of the change of radial coordinate in (\\ref{new_coordinate_Janus}).}. For the Janus, all the curves $\\,\\Gamma\\,$ start at $\\,\\tilde{\\alpha}=0\\,$ and end at $\\,\\tilde{\\alpha}=\\frac{\\pi}{2}\\,$ pointing at the $\\,\\tilde{\\alpha}\\,$ direction on $\\,\\textrm{S}^{7}\\,$ -- see Figure~\\ref{fig:vec_field} for an illustration of such curves in various examples --. Since the $\\,N_{2}\\,$ charge in (\\ref{N2_Janus}) is independent of $\\,\\Gamma\\,$ and also of the Janus parameters $\\,(\\alpha,\\beta)$, it matches the one of the $\\textrm{AdS}_{4} \\times \\textrm{S}^7\\,$ background controlling the asymptotic behaviour of the (regular) Janus solutions at $\\,\\rho \\rightarrow \\pm \\infty\\,$.\n\n\nLastly, it is also interesting to compute the volume of the internal manifold $\\,\\textrm{vol}_7\\,$ along the Janus flow as a function of the radial coordinate $\\,\\rho\\,$ and the Janus parameters $\\,(\\alpha,\\beta)\\,$. The result is a lengthy expression not very illuminating that we have evaluated and plotted in Figure~\\ref{Fig:Janus_S7} for various choices of the Janus parameters. The behaviour is akin a wormhole: the $\\,\\textrm{S}^7\\,$ is a non-contractible seven-manifold whose volume does not vanish anywhere in the flow along the radial direction $\\,\\rho\\,$. Moreover, for a given value of $\\,\\alpha\\,$, there is a range of the parameter $\\,\\beta\\,$ for which the eleven-dimensional Janus features two throats (see right plot in Figure~\\ref{Fig:Janus_S7}).\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Shell_Janus.pdf} \n\\put(-35,30){$\\textrm{Re}\\tilde{z}$}\n\\put(-150,10){$\\textrm{Im}\\tilde{z}$}\n\\hspace{8mm}\n\\includegraphics[width=0.42\\textwidth]{Plots\/Throats_Janus.pdf} \n\\put(0,-5){$\\rho$}\n\\put(-100,135){$\\frac{3 \\, g^7}{ \\, 2^{7\/2} \\pi^{4}} \\, \\textrm{vol}_7$}\n\\end{center}\n\\caption{Left: Volume of the internal seven-sphere as a function of the complex scalar $\\,\\tilde{z}\\,$ (orange dome). Examples of regular Janus flows (loops) are superimposed. Right: Volume of the internal seven-sphere as a function of the radial coordinate $\\,\\rho\\,$ for the regular Janus solutions. The parameters of the curves are: $\\,\\alpha=1\\,$ and $\\,\\beta=-\\frac{\\pi}{2}\\,$ (blue line), $\\,\\beta=\\pi\\,$ (black line) and $\\,\\beta=\\frac{17}{16}\\pi\\,$ (green line). Note the presence of two minima (throats) in the black and green lines.}\n\\label{Fig:Janus_S7}\n\\end{figure}\n\n\n\n\n\n\\subsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2\\,$}{SU3 x U(1)2} symmetric Hades and ridge flows}\n\nSetting $\\,\\alpha \\equiv \\alpha_{1}=\\alpha_{2}=\\alpha_{3}\\,$ and $\\,\\beta \\equiv \\beta_{1}=\\beta_{2}=\\beta_{3}\\,$ enhances the symmetry of the general Hades solution in (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}) from $\\,\\textrm{U}(1)^4\\,$ to $\\,\\text{SU}(3) \\times\\text{U}(1)^2\\,$. Setting $\\,\\alpha \\neq 0\\,$ renders the running of the scalar field (\\ref{Hades_U1^4_rho_2}) along the flow intrinsically complex, as it happened for the Janus case. This again implies that all the $\\,h\\,$ functions (and also three- and one-forms) entering $\\,\\hat{F}_{(4)}^{\\textrm{st}}\\,$ in (\\ref{11D_F4_st}) and $\\,\\hat{F}_{(4)}^{\\textrm{tr}}\\,$ in (\\ref{11D_F4_tr}) are generically activated.\n\n\n\n\n\n\n\n\nThe decomposition of the seven-form flux $\\,\\hat{F}_{(7)} \\,$ in (\\ref{F7_general}) is still at work for the Hades solutions. The computation of the M$2$-brane charge gives\n\\begin{equation}\n\\label{N2_Hades}\nN_2 = \\frac{1}{(2\\pi)^6}\\int_{\\Gamma \\times \\textrm{M}_{6}} \\hat{F}_{(7)} = \\frac{1}{32 \\pi^2}\\int_{\\partial\\Gamma} \\hat{\\alpha} = \\frac{1}{4\\pi^2 g^6} \\ ,\n\\end{equation}\nso that it matches the one of the $\\textrm{AdS}_{4} \\times \\textrm{S}^7\\,$ background controlling the asymptotic behaviour of the Hades solutions at $\\,\\rho \\rightarrow \\infty\\,$. Some examples of Hades flows on the $\\,\\tilde{z}\\,$ complex plane are displayed in Figure~\\ref{fig:Shell_Hades} and superimposed on the volume of the internal seven-sphere.\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.50\\textwidth]{Plots\/Shell_Hades.pdf} \n\\put(-80,20){$\\textrm{Re}\\tilde{z}$}\n\\put(-210,60){$\\textrm{Im}\\tilde{z}$}\n\\end{center}\n\\caption{Volume of the internal seven-sphere (orange dome) as a function of the complex scalar $\\,\\tilde{z}\\,$. Examples of singular Hades flows are superimposed with $\\,(\\alpha,\\beta)=(0,0)\\,$ (green straight line), $\\,{(\\alpha,\\beta)=(0,\\frac{\\pi}{2})}\\,$ (blue straight line), $\\,{(\\alpha,\\beta)=(0,-\\frac{\\pi}{2})}\\,$ (red straight line), $\\,{(\\alpha,\\lambda)=(2,\\frac{\\pi}{2}})\\,$ (blue curved line) and $\\,{(\\alpha,\\lambda)=(2,-\\frac{\\pi}{2}})\\,$ (red curved line).}\n\\label{fig:Shell_Hades}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_janus_beta_0.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\hspace{12mm}\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_janus_beta_pi.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\\\[5mm]\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_ridge_beta_minus_pihalf.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\hspace{12mm}\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_ridge_beta_pihalf.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\end{center}\n\\caption{Plots of the vector field $\\,\\boldsymbol{v} = (\\,g \\sqrt{\\rho^2 \\pm 1} \\, \\partial_{\\rho}\\hat{\\alpha} \\,,\\, \\partial_{\\tilde{\\alpha}}\\hat{\\alpha}\\,)\\,$ on the strip spanned by $\\,(\\rho,\\tilde{\\alpha})\\,$. The $\\,+\\,$ sign must be chosen for the Janus solutions whereas the $\\,-\\,$ sign corresponds to the Hades solutions as a consequence of the change of radial coordinate $\\,d\\rho = g \\, \\sqrt{\\rho^2 \\pm 1} \\, d\\mu \\,$. Top-Left: Janus flow with $\\,\\alpha=1\\,$ and $\\,\\beta=0\\,$. Top-Right: Janus flow with $\\,\\alpha=1\\,$ and $\\,\\beta=\\pi\\,$. Bottom-Left: Ridge flow with $\\,\\beta = -\\frac{\\pi}{2}\\,$. Bottom-Right: Ridge flow with $\\,\\beta = \\frac{\\pi}{2}\\,$.}\n\\label{fig:vec_field}\n\\end{figure}\n\n\n\\subsubsection*{Ridge flows and singularities}\n\n\nIn order to investigate the possible eleven-dimensional resolution of the four-dimensional Hades singularity at $\\,\\rho=1\\,$, we will look at the metric (\\ref{11D_metric}) and analyse the relevant function\n\\begin{equation}\n\\label{Omega_func}\n\\Omega \\equiv f_{1}^{\\frac{1}{2}} \\, e^{A} \\ , \n\\end{equation}\nlying in front of the $\\,\\textrm{AdS}_{3}\\,$ factor of the eleven-dimensional metric describing the world-volume of the (curved) M2-branes in the UV. For simplicity, we will take the limiting case of $\\,\\alpha=0\\,$ and focus on the ridge flows with\n\\begin{equation}\n\\label{Ridge_SU3xU1xU1_rho}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left( \\frac{d\\rho^{2}}{\\rho^{2}-1} + \n\\frac{\\left( \\rho^{2}-1\\right) }{2} \\, d\\Sigma^{2} \\right)\n\\hspace{8mm} \\textrm{ and } \\hspace{8mm}\n\\tilde{z}(\\rho) = \\rho^{-1} \\, e^{i \\left( \\beta - \\frac{\\pi}{2} \\right)} \\ .\n\\end{equation}\nRemarkably, for these flows, the four-dimensional singularity at $\\,\\rho = 1\\,$ gets resolved when uplifting the solutions to eleven dimensions provided $\\,\\beta \\neq \\pm \\frac{\\pi}{2}\\,$. \n\n\n\n\nThe explicit computation of the $\\,\\Omega\\,$ factor in (\\ref{Omega_func}) for the ridge flows yields\n\\begin{equation}\n\\label{Omega_func_ridge}\n\\Omega = (2 g)^{-1} \\left( 1 + \\rho^2 + 2 \\, \\rho \\, \\sin\\beta \\right)^{\\frac{1}{6}} \\left( 1 + \\rho^2 - 2 \\, \\rho \\, \\sin\\beta \\, \\cos(2 \\tilde{\\alpha})\\right)^{\\frac{1}{3}} \\ .\n\\end{equation}\nEvaluating (\\ref{Omega_func_ridge}) at $\\,\\rho=1\\,$ where the four-dimensional singularity is located, one concludes that $\\,\\Omega\\,$ vanishes at $\\,(\\beta,\\tilde{\\alpha})=(\\frac{\\pi}{2},0)\\,$ as well as at $\\,(\\beta,\\tilde{\\alpha})=(-\\frac{\\pi}{2},\\tilde{\\alpha})\\,$ $\\,\\forall \\tilde{\\alpha}\\,$. In other words, the pathology at $\\,\\rho=1\\,$ persists for $\\,\\beta=\\pm \\frac{\\pi}{2}\\,$ and it either localises at $\\,\\tilde{\\alpha}=0\\,$ or gets delocalised along the interval $\\,\\tilde{\\alpha} \\in [0, \\frac{\\pi}{2}]\\,$.\\footnote{A similar class of conventional (flat-sliced) RG-flows with $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetry was constructed in \\cite{Pilch:2015vha}. For the sake of comparison, there is a redefinition of the relevant parameter given by $\\,\\zeta_{\\tiny{\\cite{Pilch:2015vha}}}=\\beta-\\frac{\\pi}{2}\\,$. The singularity of the ridge flows we study here would be similar to that of a (yet to be constructed) non-supersymmetric generalisation of the flows in \\cite{Pilch:2015vha} with $\\,\\cos(3\\zeta)=+1\\,$.} We will look at some limiting examples of ridge flows in order to illustrate their main physical implications.\n\n\n\\subsubsection*{$\\circ\\,$ Singular $\\,\\boldsymbol{\\beta=\\pm\\frac{\\pi}{2}\\,}$ ridge flows:}\n\n\nThe scalar in (\\ref{Ridge_SU3xU1xU1_rho}) becomes real when setting $\\,\\beta=\\frac{\\pi}{2}\\,$. The eleven-dimensional geometry gets simplified to\n\\begin{equation}\n\\label{11D_metric_ridge_beta_pi\/2}\n\\begin{array}{lll}\nds_{11}^2 &=& \\dfrac{f_{-}^{\\frac{2}{3}}}{g^2} \\dfrac{(\\rho+1)^\\frac{2}{3}}{4} \\left[ \\, ds_{\\textrm{AdS}_{3}}^2 +\n\\dfrac{2 \\, d\\rho^2}{(\\rho^2-1)^2} + 8 \\dfrac{d\\tilde{\\alpha}^2}{(\\rho+1)^{2}} \\right. \\\\[6mm]\n& + & \\left. \\dfrac{8}{f_{-}} \\cos^2\\tilde{\\alpha} \\, \\left( ds^2_{\\mathbb{CP}^2} + \\dfrac{(\\rho-1)^2}{f_{+}} \\, \\sin^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma)^{2} \\right) \\right. \\\\[6mm]\n& + & \\left. \n\\dfrac{8}{f_{-}} \\, \\left( \\dfrac{f_{+}^{\\frac{1}{2}}}{\\rho+1} d\\psi_{-} + \\, \\dfrac{\\rho+1}{f_{+}^{\\frac{1}{2}}} \\, \\cos^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma) \\right)^{2} \\, \\right] \\ ,\n\\end{array}\n\\end{equation}\nin terms of the functions\n\\begin{equation}\n\\label{f+-_functions}\nf_{\\pm}= (\\rho \\pm 1)^2 \\mp 4 \\, \\rho \\, \\sin^2 \\tilde{\\alpha} \\ .\n\\end{equation}\nMoreover, since the scalar in (\\ref{Ridge_SU3xU1xU1_rho}) becomes real, one has that\n\\begin{equation}\n\\label{F4tr_ridge_beta_pi\/2}\n\\hat{F}_{(4)}^{\\textrm{tr}}=0 \\ , \n\\end{equation}\nin (\\ref{11D_F4_tr}). The non-vanishing contribution to the three-form gauge potential in this case is given by \n\\begin{equation}\n\\label{Ast_ridge_beta_pi\/2}\n\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{\\rho \\, (3+\\rho+\\rho^2) - 2 \\, (\\rho^2-1) \\cos(2\\tilde{\\alpha})}{8 \\, g^3} \\, \\textrm{vol}_{\\textrm{AdS}_{3}} \\ ,\n\\end{equation}\nproducing a space-time four-form flux in (\\ref{11D_F4_st}) of the form\n\\begin{equation}\n\\label{F4st_ridge_beta_pi\/2}\n\\hat{F}_{(4)}^{\\textrm{st}}=d\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{1}{2 g^3} \\left( \n\\frac{3 + \\rho \\, (2 + 3 \\, \\rho) - 4 \\, \\rho \\, \\cos(2\\tilde{\\alpha})}{4} \\, d\\rho \n+ (\\rho^2-1) \\, \\sin(2\\tilde{\\alpha}) \\, d\\tilde{\\alpha} \\right) \\wedge \\textrm{vol}_{\\textrm{AdS}_{3}}\\ . \n\\end{equation}\nTwo facts suggest an interpretation of this flow as a Coulomb branch type flow very much along the line of \\cite{Cvetic:1999xx}. Firstly, this singular ridge flow lies in the purely proper scalar sector of maximal supergravity as a consequence of $\\,\\beta=\\frac{\\pi}{2}\\,$. Namely, it is triggered from the UV solely by the VEV of the proper scalar dual to the boson bilinears. Secondly, the internal flux in (\\ref{F4tr_ridge_beta_pi\/2}) vanishes all along the flow so there are no magnetic M5-branes sourcing $\\,\\hat{F}_{(7)}\\,$.\n\n\nLet us now investigate the four-dimensional singularity at $\\,\\rho=1\\,$ from a higher-dimensional perspective. To study the eleven-dimensional geometry around $\\,\\rho=1\\,$ it is convenient to look at the Ricci scalar which, in this case, takes the form\n\\begin{equation}\n\\label{Ricci_beta_pi\/2}\n\\hat{R}(\\rho) = g^2 \\, \\frac{ (\\rho -1)^2}{3 \\, (\\rho +1)^{\\frac{2}{3}} f_{-}^{\\frac{8}{3}}} \\,\\, r(\\rho,\\tilde{\\alpha}) \\ ,\n\\end{equation}\nin terms of the negative-definite function\n\\begin{equation}\n\\begin{array}{lll}\nr(\\rho,\\tilde{\\alpha}) &=& \n-(9 \\rho^4 + 12 \\rho^3 + 32 \\rho^2 + 16 \\rho + 11)\n+ 8 \\, \\rho \\, (3 \\rho^2 + 2 \\rho + 3) \\cos (2 \\tilde{\\alpha}) \\\\[2mm]\n& & -2 \\, (\\rho -1) (3 \\rho +1) \\, \\cos(4 \\tilde{\\alpha}) \\ .\n\\end{array}\n\\end{equation}\nThe Ricci scalar in (\\ref{Ricci_beta_pi\/2}) becomes singular at $\\,(\\rho,\\tilde{\\alpha})=(1,0)\\,$. On the other hand, the evaluation of the four-form flux in (\\ref{F4st_ridge_beta_pi\/2}) around the singular value $\\,\\rho=1\\,$ is more subtle. The change of radial coordinate in (\\ref{new_coordinate_Hades}) becomes ill-defined and one must resort to the original coordinate $\\,\\mu\\,$ in (\\ref{metric_ansatz}) using $\\,d\\rho = g \\, \\sqrt{\\rho^2-1} \\, d\\mu \\,$. Then, it becomes clear from (\\ref{F4st_ridge_beta_pi\/2}) that\n\\begin{equation}\n\\left. \\hat{F}_{(4)}^{\\textrm{st}} \\right|_{\\rho = 1} = 0 \\ .\n\\end{equation}\n\n\n\n\nIt is also instructive to look at the flux $\\,\\hat{F}_{(7)} = d\\hat{\\alpha} \\wedge h^{(6)} + \\ldots\\,$ by analysing the expression of the adapted angular variable $\\,\\hat{\\alpha}\\,$. In this case it takes the form\n\\begin{equation}\n\\hat{\\alpha}(\\rho,\\tilde{\\alpha}) = - 8 \\, g^{-6} \\, f_{-}^{-1} (\\rho-1)^2 \\cos^6\\tilde{\\alpha} \\ , \n\\end{equation}\nwith $\\,f_{-}\\,$ given in (\\ref{f+-_functions}). A plot of the curves $\\,\\Gamma\\,$ is presented in Figure~\\ref{fig:vec_field} (bottom-right plot). Note that not all of them start at $\\,\\tilde{\\alpha}=0\\,$ and end at $\\,\\tilde{\\alpha}=\\frac{\\pi}{2}\\,$. There are curves that start at $\\,\\tilde{\\alpha}=0\\,$ but end at some value $\\,0 < \\tilde{\\alpha} < \\frac{\\pi}{2}\\,$ when reaching the singularity at $\\,\\rho = 1\\,$. These curves display a strong singularity bending: the one-form $\\,d\\hat{\\alpha}\\,$ interpolates between being aligned with the $\\,\\textrm{S}^{7}\\,$ direction $\\,d\\tilde{\\alpha}\\,$ at $\\,\\rho \\rightarrow \\infty\\,$ and being aligned with the non-compact direction $\\,d\\rho\\,$ when reaching the singularity at $\\,\\rho =1\\,$.\n\nFinally, recalling the result in Section~\\ref{sec:ridge_4D}, setting $\\,\\beta=-\\frac{\\pi}{2}\\,$ amounts to a reflection of the radial coordinate $\\,\\rho \\rightarrow -\\rho\\,$ (which implies an exchange $\\,f_{+} \\leftrightarrow f_{-}\\,$) while keeping the domain $\\,\\rho \\in [1,\\infty)\\,$. This reflection drastically modifies the eleven-dimensional geometry in (\\ref{11D_metric_ridge_beta_pi\/2}) and (\\ref{f+-_functions}) which becomes singular at $\\,\\rho=1\\,$ for any value of the angular coordinate within the interval $\\,\\tilde{\\alpha} \\in [0,\\frac{\\pi}{2}]\\,$. This can also be viewed in the eleven-dimensional Ricci scalar which reads\n\\begin{equation}\n\\label{Ricci_beta_-pi\/2}\n\\hat{R}(\\rho) = g^2 \\, \\frac{ (\\rho+1)^2}{3 \\, (\\rho -1)^{\\frac{2}{3}} f_{+}^{\\frac{8}{3}}} \\,\\, r(-\\rho,\\tilde{\\alpha}) \\ .\n\\end{equation}\nSince there is no special value of $\\,\\tilde{\\alpha}\\,$ as far as singularities are concerned, the $\\,\\Gamma\\,$ curves constructed from the adapted angular variable \n\\begin{equation}\n\\hat{\\alpha}(\\rho,\\tilde{\\alpha}) = - 8 \\, g^{-6} \\, f_{+}^{-1} (\\rho+1)^2 \\cos^6\\tilde{\\alpha} \\ , \n\\end{equation}\ndo not display any bending when approaching $\\,\\rho=1\\,$. These curves are presented in Figure~\\ref{fig:vec_field} (bottom-left plot). Lastly, the three-form gauge potential at $\\,\\beta=-\\frac{\\pi}{2}\\,$ is given by\n\\begin{equation}\n\\label{Ast_ridge_beta_minus_pi\/2}\n\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{\\rho \\, (3-\\rho+\\rho^2) + 2 \\, (\\rho^2-1) \\cos(2\\tilde{\\alpha})}{8 \\, g^3} \\, \\textrm{vol}_{\\textrm{AdS}_{3}} \\ .\n\\end{equation}\n\n\n\n \n \n \n\n\\subsubsection*{$\\circ\\,$ Regular $\\,\\boldsymbol{\\beta=0,\\pi\\,}$ ridge flows:}\n\nThe scalar in (\\ref{Ridge_SU3xU1xU1_rho}) becomes purely imaginary when setting $\\,\\beta=0\\,$. As a result, this ridge flow is triggered from the UV solely by the source mode of the pseudo-scalar dual to the fermion bilinears. \n\n\n\nThe eleven-dimensional metric reduces in this case to\n\\begin{equation}\n\\label{11D_metric_ridge_beta_0}\n\\begin{array}{lll}\nds_{11}^2 &=& \\dfrac{\\rho^2+1}{4 \\, g^2} \\left[ \\, ds_{\\textrm{AdS}_{3}}^2 +\n\\dfrac{2 \\, d\\rho^2}{(\\rho^2-1)^2} + 8 \\dfrac{d\\tilde{\\alpha}^2}{\\rho^2+1} \\right. \\\\[6mm]\n& + & \\left. 8 \\, \\cos^2\\tilde{\\alpha} \\, \\left( \\dfrac{1}{\\rho^2+1} ds^2_{\\mathbb{CP}^2} + \\dfrac{1}{j_{2}} \\dfrac{(\\rho^2-1)^2}{\\rho^2+1} \\, \\sin^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma)^{2} \\right) \\right. \\\\[6mm]\n& + & \\left. \n\\dfrac{8 \\, j_{1}}{(\\rho^2+1)^3} \\, \\left( \\sqrt{\\dfrac{j_{2}}{j_{1}}} \\, d\\psi_{-} + \\sqrt{\\dfrac{j_{1}}{j_{2}}} \\, \\cos^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma) \\right)^{2} \\, \\right] \\ ,\n\\end{array}\n\\end{equation}\nin terms of the two functions\n\\begin{equation}\n\\label{j1_j2_functions}\nj_{1} = (\\rho^2 + 1)^2 - 4 \\, \\rho^2 \\, \\cos(2 \\tilde{\\alpha}) \n\\hspace{8mm} , \\hspace{8mm}\nj_{2} = (\\rho^2 + 1)^2 - 4 \\, \\rho^2 \\, \\cos^2(2 \\tilde{\\alpha}) \\ .\n\\end{equation}\nThe four-form flux in (\\ref{11D_F4}) comes with both space-time and transverse contributions. The former is given by\n\\begin{equation}\n\\label{Ast_ridge_beta_0}\n\\hat{F}_{(4)}^{\\textrm{st}}=d\\hat{A}_{(3)}^{\\textrm{st}}\n\\hspace{10mm} \\textrm{ with } \\hspace{10mm}\n\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{\\rho \\, (3+\\rho^2)}{8 \\, g^3} \\, \\textrm{vol}_{\\textrm{AdS}_{3}} \\ ,\n\\end{equation}\nwhereas the latter reads\n\\begin{equation}\n\\hat{F}_{(4)}^{\\textrm{tr}}=d\\hat{A}_{(3)}^{\\textrm{tr}} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Atr_ridge_beta_0}\n\\begin{array}{lll}\n\\hat{A}_{(3)}^{\\textrm{tr}} &=& - \\dfrac{4 \\sqrt{2}}{g^{3}} \\dfrac{\\rho}{\\rho^2 +1} \\Big[ \\frac{1}{2} \\sin(2\\tilde{\\alpha}) \\, d\\tilde{\\alpha} \\wedge (d\\tau_{-} + \\sigma) \\wedge d\\psi_{-} \\\\[2mm]\n&& \\qquad\\qquad\\qquad + \\cos^4\\tilde{\\alpha} \\, \\boldsymbol{J} \\wedge (d\\tau_{-} + \\sigma) + \\cos^2\\tilde{\\alpha} \\, \\cos(2\\tilde{\\alpha}) \\, \\boldsymbol{J} \\wedge d\\psi_{-} \\Big] \\ .\n\\end{array}\n\\end{equation}\nThis signals the presence of both electric M2-branes and magnetic M5-branes at a generic point along the flow.\n\n\n\n\nIn order to investigate the four-dimensional singularity at $\\,\\rho=1\\,$ from a higher-dimensional perspective we will look again at the eleven-dimensional Ricci scalar. It reads\n\\begin{equation}\n\\label{Ricci_scalar_Hades_beta=0}\n\\hat{R}(\\rho) = g^2 \\, \\left(1 + \\rho^2\\right)^{-3} \\, \\left(1 + 3 \\, \\rho^2\\right) \\, \\left[ 1 + \\rho^2 \\left( 8 - \\rho^2 \\right) \\right] \\ ,\n\\end{equation}\nand becomes this time independent of the angular variable $\\,\\tilde{\\alpha}\\,$. The Ricci scalar in (\\ref{Ricci_scalar_Hades_beta=0}) features no singularity within the domain $\\,\\rho \\in [1 , \\infty )\\,$. It has a boundary value $\\,\\hat{R}(\\infty) = -3 \\, g^2\\,$ and changes smoothly until reaching the finite value $\\,\\hat{R}(1)=4 \\, g^2\\,$, thus making the eleven-dimensional solution regular. The space-time (\\ref{Ast_ridge_beta_0}) and transverse (\\ref{Atr_ridge_beta_0}) components of the three-form gauge potential are both non-zero when approaching the IR region at $\\,\\rho =1\\,$. However, recalling again the change of radial coordinate $\\,d\\rho = g \\, \\sqrt{\\rho^2-1} \\, d\\mu \\,$, it follows from (\\ref{Ast_ridge_beta_0}) that\n\\begin{equation}\n\\left. \\hat{F}_{(4)}^{\\textrm{st}} \\right|_{\\rho = 1} = 0 \\ .\n\\end{equation}\nTherefore, only magnetic M5-branes source the geometry in the deep IR. The same behaviour was observed for the similar, but flat-sliced, $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ invariant flows constructed in \\cite{Pilch:2015vha}. Such flows were argued to describe how M2-branes in the UV totally dissolve along the flow into magnetic M5-branes, leaving no M2-branes at the core of the regular flows.\\footnote{The same type of behaviour was also observed in the flat-sliced dielectric flows with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetry of \\cite{Pope:2003jp}, although the M2-branes do not totally polarise into M5-branes at the core of these flows.} Moving back to the original radial coordinate\n\\begin{equation}\n\\rho = \\cosh(g \\, \\mu) \n\\hspace{15mm} \\textrm{ with } \\hspace{15mm}\\mu \\in [0,\\infty) \\ ,\n\\end{equation}\nand expanding around $\\,\\mu = 0\\,$ one arrives at\n\\begin{equation}\n\\label{11D_metric_ridge_beta_0_IR}\n\\begin{array}{lll}\n\\left. ds_{11}^2 \\, \\right|_{\\textrm{IR}} & \\approx & \\dfrac{1}{4 \\, g^2} \\left[ \\, \\left(\\dfrac{4}{(g\\mu)^2} + \\dfrac{2}{3} + \\dfrac{4}{15}(g\\mu)^2 + \\ldots \\right) d(g\\mu)^2 + \\left( 2 + (g \\mu)^2 + \\ldots \\right)\\, ds_{\\textrm{AdS}_{3}}^2 \\right. \\\\[6mm]\n& + & \\left. \n\\left( \\dfrac{(g\\mu)^4}{2} - \\dfrac{(g\\mu)^6}{6} + \\ldots \\right) \\, \\Big( (d\\tau_{-} + \\sigma) + 2 \\cos(2\\tilde{\\alpha}) \\, \\left( d\\psi_{-} + \\frac{1}{2} (d\\tau_{-} + \\sigma) \\right) \\Big)^{2} \\, \\right. \\\\[6mm]\n& + &\n\\left. 8 \\, \\left( d\\tilde{\\alpha}^2 + \\cos^2\\tilde{\\alpha} \\, ds^2_{\\mathbb{CP}^2} + \\sin^2(2 \\tilde{\\alpha}) \\, \\left(d\\psi_{-} + \\frac{1}{2}(d\\tau_{-} + \\sigma) \\right)^2 \\right) \\right] \\ .\n\\end{array}\n\\end{equation}\nNote that the $\\,\\mu$-dependent part of the metric only involves the first two lines in (\\ref{11D_metric_ridge_beta_0_IR}). This $\\,\\mu$-dependent part describes a five-dimensional section of the eleven-dimensional geometry that involves the original four coordinates of the ridge flow and an additional $\\,\\textrm{S}^1\\,$ that is non-trivially fibered over a six-dimensional manifold. The latter is described by the last line in (\\ref{11D_metric_ridge_beta_0_IR}). Ignoring this fibration, the five-dimensional section of the geometry verifies $\\,R^{\\textrm{(\\tiny{5D})}}_{\\mu\\nu}= \\frac{1}{5} \\, R^{\\textrm{(\\tiny{5D})}} \\, g^{\\textrm{(\\tiny{5D})}}_{\\mu\\nu}\\,$ with $\\,R^{\\textrm{(\\tiny{5D})}} = -20 \\, g^2 < 0\\,$ at leading order in the radial coordinate $\\,\\mu\\,$. Therefore, up to \nthe non-trivial fibration over the six-dimensional manifold, this ridge flow develops a five-dimensional Einstein geometry in the deep IR.\n\n\n\nThe regularity of the ridge flow at $\\,\\beta=0\\,$ is also reflected in the flux $\\,\\hat{F}_{(7)} = d\\hat{\\alpha} \\wedge h^{(6)} + \\ldots\\,$. The adapted angular variable $\\,\\hat{\\alpha}\\,$ simplifies in this case to\n\\begin{equation}\n\\hat{\\alpha}(\\tilde{\\alpha}) = - 8 \\, g^{-6} \\, \\cos^6\\tilde{\\alpha} \\ , \n\\end{equation}\nso it is independent of $\\,\\rho\\,$. Therefore, all the $\\,\\Gamma\\,$ curves start at $\\,\\tilde{\\alpha}=0\\,$, end at $\\,\\tilde{\\alpha}=\\frac{\\pi}{2}\\,$ and flow parallel to the $\\,\\textrm{S}^7\\,$ angular direction $\\,\\tilde{\\alpha}\\,$ without displaying any bending or pathological behaviour.\n\n\n\n\n\n\n\nFinally, as discussed in Section~\\ref{sec:ridge_4D}, setting $\\,\\beta=\\pi\\,$ amounts to a shift $\\,\\rho \\rightarrow -\\rho\\,$ in the four-dimensional ridge flow solution while keeping the domain $\\,\\rho \\in [1,\\infty)\\,$. This reflection of the radial coordinate leaves the eleven-dimensional metric in (\\ref{11D_metric_ridge_beta_0}) and (\\ref{j1_j2_functions}) invariant. The three-form gauge potential in (\\ref{Ast_ridge_beta_0}) and (\\ref{Atr_ridge_beta_0}) simply flips its sign.\n\n\n\n\n\n\n\n\n\n\\section{Summary and discussion}\n\\label{sec:conclusions}\n\n\nIn this paper we have presented new analytic families of $\\,\\textrm{AdS}_{3} \\times \\mathbb{R}\\,$ Janus and $\\,\\textrm{AdS}_{3} \\times \\mathbb{R}^{+}\\,$ Hades solutions in the $\\,\\mathcal{N}=2\\,$ gauged STU-model in four dimensions \\cite{Cvetic:1999xp}. This supergravity model corresponds to the $\\textrm{U}(1)^{4}$ invariant sector of the maximal SO(8) gauged supergravity that arises upon reduction of eleven-dimensional supergravity on a seven sphere. \n\n\nThe Janus solutions turn out to be surprisingly simple. Using a radial coordinate $\\,\\rho \\in (-\\infty \\, , \\infty)\\,$, the geometry is given by \n\\begin{equation}\n\\label{Janus_metric_conclus}\ng^{2} \\, ds_{4}^{2} = \\frac{d\\rho^{2}}{\\rho^{2}+1} + \n\\frac{ \\rho^{2} + 1 }{ k^2} \\,ds_{\\textrm{AdS}_{3}}^{2} \\ ,\n\\end{equation}\nin terms of the supergravity gauge coupling $\\,g\\,$ and three constant parameters $\\, \\alpha_{i} \\in \\mathbb{R} \\,$. The latter enter (\\ref{Janus_metric_conclus}) through the specific combination\n\\begin{equation}\n\\label{k_factor_Janus_conclus}\nk^2= 1 + \\sum_{i=1}^{3} \\sinh^{2}\\alpha_{i} \\, \\ge \\, 1 \\ .\n\\end{equation}\nThe Janus geometry (\\ref{Janus_metric_conclus}) is supported by $\\rho$-dependent profiles for the three complex scalars in the STU-model. Using the unit-disk parameterisation of the SL(2)\/SO(2) scalar coset, they adopt the form\n\\begin{equation}\n\\label{Janus_scalar_conclus}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}} \\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\rho}\n\\hspace{8mm} \\textrm{ with } \\hspace{8mm} i=1,2,3 \\ ,\n\\end{equation}\nand depend on three additional phases $\\, \\beta_{i} \\in [0,2 \\pi] \\,$. The result is then a six-parameter family $\\,(\\alpha_{i},\\beta_{i})\\,$ of Janus solutions in the STU-model which are everywhere regular for arbitrary choices of the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. These are generically non-supersymmetric solutions (they solve second-order equations of motion) but there is a supersymmetry enhancement when two $\\,\\alpha_{i}\\,$ parameters are set to zero. In this limit the supersymmetric Janus with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetry of \\cite{Bobev:2013yra} is recovered. The very special choice $\\,\\alpha_{i}=0\\,$ $\\forall i \\,$ sets the three scalars to zero. In this limit the maximally supersymmetric AdS$_{4}$ vacuum of the $\\,\\textrm{SO}(8)\\,$ supergravity is recovered which uplifts to the Freund--Rubin $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ vacuum of eleven-dimensional supergravity \\cite{Freund:1980xh}. Note that this vacuum controls the asymptotic behaviour of the Janus solutions at $\\,\\rho \\rightarrow \\pm \\infty\\,$.\\footnote{The Janus solutions in (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) might resemble the ``boomerang RG flows\" studied in \\cite{Donos:2017ljs} within the STU-model. These are flows in supergravity both starting and ending at the maximally supersymmetric AdS$_{4}$ vacuum of the SO(8) gauged supergravity, thus being also relevant for ABJM theory. However the Ansatz for the scalar fields in \\cite{Donos:2017ljs} explicitly breaks translation invariance in the spatial directions of the dual field theory. This is not the case for the Janus solutions (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) which have no dependence on the spatial directions of AdS$_3$.} It is also worth emphasising that the Janus solutions in (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) are everywhere regular and genuinely \\textit{axionic} in nature: $\\,\\textrm{Im}\\tilde{z}_{i}(\\rho) \\neq 0\\,$ for the solution to exist. This fact makes the study of similar solutions in the Euclidean theory (where pseudo-scalars pick up an extra factor of $\\,i\\,$ with respect to proper scalars) interesting in the AdS\/CFT spirit of \\cite{Arkani-Hamed:2007cpn,Bobev:2020pjk}. This could help to understand instanton-like solutions in the context of M-theory, as it has been done for the type IIB non-extremal D-instantons \\cite{Bergshoeff:2004fq,Bergshoeff:2004pg,Bergshoeff:2005zf} (see also \\cite{Hertog:2017owm}), and perhaps to shed new light on axionic wormholes in M-theory. This issue certainly deserves further investigation.\n\n\nThe Hades solutions are closely related to the Janus solutions and turn out to be very simple too. Using this time a radial coordinate $\\,\\rho \\in [1 \\, , \\infty)$, the geometry is given by \n\\begin{equation}\n\\label{Hades_metric_conclus}\ng^{2} \\, ds_{4}^{2} = \\frac{d\\rho^{2}}{\\rho^{2} - 1} + \n\\frac{ \\rho^{2} - 1 }{ k^2} \\,ds_{\\textrm{AdS}_{3}}^{2} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{k_factor_Hades_conclus}\nk^2= -1 + \\sum_{i=1}^3 \\cosh^2\\alpha_{i} \\ ,\n\\end{equation}\nand the scalar profiles read\n\\begin{equation}\n\\label{Hades_scalar_conclus}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}}\\, \\frac{\\cosh\\alpha_{i}}{\\sinh\\alpha_{i} + i \\rho} \n\\hspace{8mm} \\textrm{ with } \\hspace{8mm} i=1,2,3 \\ .\n\\end{equation}\nUnlike the Janus, the Hades solutions are singular at $\\,\\rho=1\\,$ and do not possess a supersymmetric limit upon tuning of the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. Still the maximally supersymmetric AdS$_{4}$ vacuum controls the asymptotic behaviour of the Hades at $\\,\\rho \\rightarrow \\infty\\,$. The special limit $\\,\\alpha_{i}=0\\,$ $\\forall i \\,$ drastically simplifies the Hades solutions giving rise to the so-called ridge flows (see Figure~\\ref{fig:Hades_ztilde_U1^4}).\n\n\nBeing obtained within the $\\textrm{U}(1)^4$ invariant sector of the massless $\\,\\mathcal{N}=8\\,$ supergravity multiplet in four dimensions, the analytic Janus solutions in (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) generalise the supersymmetric ones with $\\,\\text{SO}(4) \\times\\text{SO}(4)\\,$ symmetry constructed in \\cite{Bobev:2013yra}. The non-supersymmetric Hades solutions in (\\ref{Hades_metric_conclus})-(\\ref{Hades_scalar_conclus}) are genuinely new an cannot be continuously connected with the supersymmetric Hades with $\\,\\text{SO}(4) \\times\\text{SO}(4)\\,$ symmetry of \\cite{Bobev:2013yra} upon tuning of $\\,\\alpha_{i}\\,$. In addition, the Janus and Hades solutions presented in this work can be readily uplifted to eleven-dimensional supergravity using the general results for the oxidation of the STU-model worked out in \\cite{Cvetic:1999xp,Azizi:2016noi} and the uplift building blocks collected in the Appendix~\\ref{app:general_uplift}. Instead of uplifting the general $\\textrm{U}(1)^4$ symmetric Janus and Hades solutions, and for the sake of simplicity, we have restricted to the case \n\\begin{equation}\n\\alpha_{1}=\\alpha_{2}=\\alpha_{3}=\\alpha\n\\hspace{10mm} \\textrm{ and } \\hspace{10mm}\n\\beta_{1}=\\beta_{2}=\\beta_{3}=\\beta\n\\end{equation}\nfor which a larger symmetry group $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2 \\subset \\textrm{SO}(8)\\,$ is preserved by the solutions. The Janus solutions are non-supersymmetric and fully regular, both in four and eleven dimensions, for arbitrary values of the parameters $\\,(\\alpha,\\beta)\\,$. The four-dimensional singularity of the Hades may or may not be cured when the solutions are uplifted to eleven-dimensions depending on the choice of parameters $\\,(\\alpha,\\beta)\\,$. For example, in the ridge flow limit $\\,\\alpha=0\\,$, the choice $\\,\\beta=0,\\pi\\,$ eliminates the singularity whereas, if setting $\\,\\beta=\\pm\\frac{\\pi}{2}\\,$, the singularity remains either localised or delocalised in the internal space. It would be interesting to understand the ultimate fate of the singularity in the general Hades solution with $\\,\\textrm{U}(1)^4\\,$ symmetry, as well as to investigate the process of taking the ridge flow limit sequentially on the three scalars $\\,\\tilde{z}_{i}\\,$. Also to further investigate a possible holographic interpretation of these more general flows as interfaces connecting an $\\,\\mathcal{N}=8\\,$ Chern--Simons matter theory to new (non-)conformal phases.\n\n\n\n\n\nSome open questions and follow-up directions regarding the Janus and Hades presented in this work are immediately envisaged. The first one is the issue of the stability, both perturbative and non-perturbative, of the general class of non-supersymmetric Janus and Hades with $\\textrm{U}(1)^4$ symmetry. These solutions can be viewed as AdS$_{3}$ vacua in M-theory, so it would be interesting to investigate their stability in light of the Weak Gravity and Swampland conjectures \\cite{ArkaniHamed:2006dz,Ooguri:2016pdq}. In this respect, and unlike for the Hades, the Janus solutions presented here are continuously connected (in parameter space) to the supersymmetric, and thus stable, Janus solutions with $\\,{\\textrm{SO}(4) \\times \\textrm{SO}(4)}\\,$ symmetry of \\cite{Bobev:2013yra}. This could help in improving the stability properties of the generic non-supersymmetric Janus solution at least within some region in the parameter space $\\,(\\alpha_{i},\\beta_{i})\\,$. Along this line, it would also be interesting to perform a probe brane analysis as a first step towards assessing the non-perturbative stability of the solutions. \n\n\nThe second issue is to understand the higher-dimensional brane picture of the various flows constructed in this work. For a related class of flat-sliced ridge flows, it was shown in \\cite{Pilch:2015vha} (motivated by \\cite{Pope:2003jp}) that the M2-branes in the UV totally polarise into a $\\,(1+3)$-dimensional intersection of M5-branes in the IR generating an AdS$_{5}$ metric at the core of the flow that is non-trivially fibered over a six-dimensional manifold.\\footnote{The appearance of a new strongly-coupled IR phase on the M2-brane involving an extra dimension was argued in \\cite{Pilch:2015vha} to originate from charged solitons that become massless, very much in the spirit of (massless) type IIA string theory and 11D supergravity.} This phenomenon was signaled by the vanishing of the space-time flux component at the IR endpoint of the flow. In our ridge flows with $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^{2}\\,$ symmetry, the expression of the space-time four-form flux (\\ref{11D_F4_st}) at generic $\\,\\beta\\,$ is given by\n\\begin{equation}\n\\label{F4st_ridge_beta_general}\n\\hat{F}_{(4)}^{\\textrm{st}} = \\frac{1}{2 g^3} \\left( \n\\frac{3 \\, (1+\\rho^2) + 2 \\, \\rho \\, \\sin\\beta \\, (1- 2 \\cos(2\\tilde{\\alpha}))}{4} \\, d\\rho \n+ \\sin\\beta \\, (\\rho^2-1) \\, \\sin(2\\tilde{\\alpha}) \\, d\\tilde{\\alpha} \\right) \\wedge \\textrm{vol}_{\\textrm{AdS}_{3}} \\ ,\n\\end{equation}\nso that\n\\begin{equation}\n\\left. \\hat{F}_{(4)}^{\\textrm{st}} \\right|_{\\rho = 1} = 0 \\ ,\n\\end{equation}\nin the deep IR by virtue of the change of radial coordinate $\\,d\\rho = g \\, \\sqrt{\\rho^2-1} \\, d\\mu \\,$. This suggests a possible interpretation in terms of non-supersymmetric dielectric flows with M2-branes being polarised into intersecting M5-branes. Also, in the case of $\\,\\beta=0\\,$, we have shown the appearence of a five-dimensional geometry in the IR non-trivially fibered over a six-dimensional manifold along the lines of \\cite{Pilch:2015vha}. The generalisation to ridge and Hades flows with $\\,\\textrm{U}(1)^4\\,$ symmetry also deserves further investigation.\n\n\nThe third issue has to do with the holographic interpretation of the general Janus and Hades solutions in terms of non-supersymmetric interfaces in the field theory living at the boundary. We have made manifest the strong correlation between the choice of Janus\/Hades parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ (\\textit{i.e.} boundary conditions for the complex scalars $\\,\\tilde{z}_{i}\\,$), the possible emergence of supersymmetry, the source\/VEV and bosonic\/fermionic nature of the dual operators that are turned on in the interface and the (dis)appearance of gravitational singularities. But much work remains to be done to better understand and characterise the physics of the non-supersymmetric interfaces we have presented. \n\n\nFinally, it would also be very interesting to construct charged solutions generalising the Janus and Hades constructed in this work, as well as to investigate the effect of including hypermultiplets in the setup thus going beyond the STU-model. We plan to come back to these and related issues in the future.\n\n\n\n\n\n\\section*{Acknowledgements}\n\nWe are grateful to Ant\\'on Faedo, Carlos Hoyos and Anayeli Ram\\'irez for conversations. The research of AA is supported in part by the Fondecyt Grants 1210635, 1221504 and 1181047 and by the FAPESP\/ANID project 13231-7. The work of AG and MCh-B is partially supported by the AEI through the Spanish grant PGC2018-096894-B-100 and by FICYT through the Asturian grant SV-PA-21-AYUD\/2021\/52177.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}