diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhagu" "b/data_all_eng_slimpj/shuffled/split2/finalzzhagu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhagu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe Weyl anomaly in quantum field theory has a long history (see e.g. \\cite{Duff:1993wm} for a historical overview). It states that the Weyl transformation of the metric in the curved space-time may not be a symmetry of the system even though the quantum field theory under consideration has the conformal symmetry in the flat space-time. Indeed, in most conformal field theories, the Weyl anomaly is non-vanishing, and we say that the Weyl symmetry is quantum mechanically broken in curved space-time.\n\nThe Weyl anomaly has played many important roles in theoretical physics.\nThe existence of the Weyl anomaly gives a constraint on the expectation values of energy-momentum tensor in curved background \\cite{Page:1982fm}, and may be related to the nature of Hawking radiation \\cite{Robinson:2005pd}\\cite{Iso:2006wa}\\cite{Kawai:2017txu}\\cite{Kawai:2014afa}. The fact that vanishing of the Weyl anomaly happens only in a limited class of theories dictates the number of space-time dimensions in critical string theory. \nMore recently, we find that the universal terms in the entanglement entropy of conformal field theories are given by the coefficient of the Weyl anomaly, suggesting a deep relation between geometry and information \\cite{Ryu:2006ef}.\n\nWhat we would like to study in this paper is to find a way to cancel the Weyl anomaly from the other source, e.g. from the position dependent coupling constant.\\footnote{Sorry for the oxymoron. It is no longer constant. Probably it is Dirac who openly advocated this idea in the early days \\cite{Dirac:1938mt}.} Once we put a quantum field theory on a curved manifold, it is natural to assume that coupling constants are position dependent. The position dependent coupling constants then provide an extra contribution to the Weyl anomaly so that we may attempt to cancel the entire Weyl anomaly on the curved manifold. We would like to find under which condition such a cancellation is possible.\n\nThe similar idea of cancelling more general anomalies have been implicitly assumed in many places. For example, if we try to introduce background gauge fields for chiral current operators in the curved background (e.g. in the context of supersymmetric localization), then they may be mutually inconsistent due to the 't Hooft anomaly. One way to avoid this is to cancel the anomaly of the background gauge field from the space-time curvature and vice versa. Similarly, if preserving the Weyl anomaly is the critical issue (e.g. if we try to gauge it), our new way of doing it may be another option to be considered.\n\nThe organization of the paper is as follows. In section 2, we study the cancellation of the Weyl anomaly from position dependent coupling constant in two dimensional conformal field theories. In section 3, we study it in three dimensions and in section 4, we study it in four dimensions. We supplement the holographic viewpoint in section 5 and conclude with some discussions in section 6.\n\n\\section{Two dimensions}\nLet us consider a two-dimensional conformal field theory with an exactly marginal deformation denoted by $g$. For instance, we may take the Gaussian $c=1$ boson with the compactification radius as the exactly marginal deformation here.\nWe put the theory on a curved background with the metric $g_{\\mu\\nu}(x)$ and then vary the coupling constant $g(x)$ over the manifold: schematically we consider the action $S = S_0 + \\int d^2x\\sqrt{g} g(x) O(x)$. Even though the theory is conformal invariant in the Euclidean space $g_{\\mu\\nu} = \\delta_{\\mu\\nu}$ with $g(x) = g$, it is not necessarily so after turning on the background metric and position dependent coupling constant.\\footnote{We define the scale transformation by the change of the difference of the coordinate as in \\cite{Osborn:1991gm} rather than the coordinate itself \\cite{Dong:2012ua}.} \nThis obstruction is known as the Weyl anomaly under the infinitesimal Weyl rescaling: $\\delta g_{\\mu\\nu}(x) = 2\\delta \\sigma g_{\\mu\\nu}$. \n\nIn terms of the free energy functional $e^{-F[g_{\\mu\\nu}(x),g(x)]} = \\int \\mathcal{D} \\Phi e^{-S[\\Phi]}$, the Weyl anomaly for a two-dimensional conformal field theory is given by (e.g. see \\cite{Osborn:1991gm})\\footnote{In this paper, we always assume that the conformal field theories under consideration preserve the CP symmetry.}\n\\begin{align}\n\\delta F_{\\sigma} = \\int d^2x\\sqrt{g} \\delta \\sigma(x) (cR -\\frac{1}{2} \\partial^\\mu g(x) \\partial_\\mu g(x)) \\ \\label{twoa}\n\\end{align}\nin a certain renormalization scheme so that the exactly marginal deformation has a flat line metric. Otherwise, we can always redefine the coupling constant or the renormalization scheme so that it is flat. It is clear that when $g(x) = g$, the only way to cancel the Weyl anomaly is to require $c=0$, which is typically what we demand in critical string theory.\n\nHowever, we see that this is not the only available option. Now, given a positive curvature $R(x)\\ge 0$, we may try to cancel the curvature term in the Weyl anomaly \\eqref{twoa} against the second term originating from the position dependent coupling constant by solving the equation\n\\begin{align}\nc R = \\frac{1}{2} \\partial^\\mu g(x) \\partial_\\mu g(x) \\ . \\label{twoc}\n\\end{align}\nThis is possible for positive curvature $R(x) \\ge 0$ (assuming $c>0$ in unitary conformal field theories). \n\n\nFor example, if we take the Fubini-Study metric on the sphere with the complex coordinate $z$ and $\\bar{z}$:\n\\begin{align}\nds^2 = \\frac{dzd\\bar{z}}{(1+|z|^2)^2} \\ , \n\\end{align}\nthe solution of \\eqref{twoc} is \n\\begin{align}\ng(x) = \\sqrt{c} \\cdot \\mathrm{arctan}(|z|) + \\mathrm{const} \\ .\n\\end{align}\nIn this way, one can cancel the Weyl anomaly on the sphere by introducing the position dependent coupling constant. Note, however, that the position dependence of the coupling constant reduces the symmetry of the sphere from $SO(3)$ down to $SO(2)$. The idea here is we gained extra ``Weyl symmetry\" at the sacrifice of the rotational symmetry.\\footnote{This is not necessarily a bad idea: for examle, in the supersymmetric localization, we often do not keep the full isometry of the sphere but only the $U(1)$ subgroup of it.}\n\nThe above cancellation works both for infinitesimal generic Weyl transformation $\\delta g_{\\mu\\nu}(x) = 2\\delta \\sigma(x) g_{\\mu\\nu}(x)$ or finite but constant Weyl transformation $g_{\\mu\\nu}(x) \\to e^{2\\bar{\\sigma}} g_{\\mu\\nu}(x)$, where $\\bar{\\sigma}$ is a finite constant. The latter is because the equation to be solved in \\eqref{twoc} trivially scales under the constant Weyl transformation, so once it is solved then it is also solved after finite but constant Weyl transformation.\nFor finite generic Weyl transformation, however, the cancellation may not persist. The point is that the curvature term in the Weyl anomaly is non-trivially transforms under the Weyl transformation:\n\\begin{align}\nR \\to e^{-2\\sigma(x)} (R -2D^2 \\sigma) \\ , \n\\end{align}\nwhere $D^2$ is the Laplacian, \nwhile the Weyl transformation of the second term from the position dependent coupling constant $\\partial^\\mu g(x) \\partial_\\mu g(x)$ is trivial:\n\\begin{align}\n\\partial^\\mu g(x) \\partial_\\mu g(x) \\to e^{-2\\sigma(x)} \\partial^\\mu g(x) \\partial_\\mu g(x)\n\\end{align}\nThus, even though one may solve the cancellation condition for a given $g_{\\mu\\nu}(x)$ with a certain position dependent coupling constant $g(x)$, the cancellation does not persist for the Weyl transformed geometry. \n\n Nevertheless, we realize that the cancellation is still intact if we restrict\\footnote{A similar restriction on the Weyl transformation has been studied in \\cite{Edery:2014nha}.} ourselves to the harmonic Weyl transformation, which satisfies $D^2 \\sigma = 0$. Thus, we may construct a quantum field theory which is exactly invariant under the harmonic Weyl transformation by cancelling the Weyl anomaly from the position dependent coupling constant.\n\n\nMore generically, one may consider theories with several exactly marginal deformations. The Weyl anomaly has the generalized form\n\\begin{align}\n\\delta F_{\\sigma} = \\int d^2x\\sqrt{g} \\delta\\sigma(x) (cR - \\chi_{ij}(g) \\partial^\\mu g^i(x) \\partial_\\mu g^j(x)) + \\partial_\\mu \\delta\\sigma(x) w_i(g) \\partial^\\mu g^i(x) \\ , \\label{twog}\n\\end{align}\nwhere $\\chi_{ij}(g)$ and $w_i(g)$ may depend on the exactly marginal deformations $g^i(x)$. \nFor a constant Weyl transformation, the condition for the cancellation is essentially the same as before since the last term in \\eqref{twog} drops out.\n\nFor infinitesimal generic Weyl transformation, however, we have to think about the cancellation of the third term proportional to $\\partial_\\mu \\delta \\sigma(x)$. We did not talk about it in the single coupling case because we were able to remove it from the local counterterm $\\int d^2x\\sqrt{g} b(g) R$, but we have to discuss it now with several coupling constants when it has the non-trivial curvature $\\partial_i w_j - \\partial_j w_i$.\nWhile the Wess-Zumino consistency condition does not say anything about the (non-)existence of this term \\cite{Osborn:1991gm}, the recent analysis in \\cite{Gomis:2015yaa} tells that on the conformal manifold spanned by the exactly marginal deformations, the curvature is trivial (i.e. $\\partial_i w_j - \\partial_j w_i = 0$) and can be removed by the local counterterm $\\int d^2x\\sqrt{g} b(g^i) R$,\n so we actually do not have to worry about its cancellation. The non-existence of the curvature $\\partial_i w_j -\\partial_j w_i$ is related to the gradientness of the beta functions and it may have a deep implication in renormalization group flows \\cite{Osborn:1991gm}\\cite{Friedan:2009ik}\\cite{Gukov:2016tnp}.\n\n\n\n\\section{Three dimensions}\nThere is no curvature dependent Weyl anomaly in three dimensions. The position dependent exactly marginal deformations do not introduce the additional Weyl anomaly either under the assumption of the CP symmetry \\cite{Nakayama:2013wda}. Thus there is no interesting scenario we can imagine in three dimensions.\n\n\\section{Four dimensions}\nLet us consider a four-dimensional conformal field theory with an exactly marginal deformation denoted by $g$. We put the theory on a curved background with the metric $g_{\\mu\\nu}(x)$ and then vary the coupling constant over the manifold $g(x)$. \nIn a certain renormalizaiton scheme, the first order Weyl transformation (i.e. the Weyl anomaly) is given by \\cite{Osborn:1991gm} (See also \\cite{Nakayama:2013is}\\cite{Jack:2013sha}.)\n\\begin{align}\n\\delta F_{\\sigma} = \\int d^4x \\sqrt{g} \\delta\\sigma(x) &\\left( c(g) \\mathrm{Weyl}^2 - a \\mathrm{Euler} + (D^2 g D^2g - 2G_{\\mu\\nu} \\partial^\\mu g \\partial^\\nu g -\\frac{R}{3}\\partial^\\mu g\\partial_\\mu g) \\right. \\cr\n& \\left. + \\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g \\right) \\ . \\label{foura}\n\\end{align}\nHere $G_{\\mu\\nu} = R_{\\mu\\nu} - \\frac{R g_{\\mu\\nu}}{2}$ is the Einstein tensor and $D^\\mu$ is the covariant derivative. In addition, we have introduced $\\mathrm{Weyl}^2 = R_{\\mu\\nu\\rho\\sigma}^2 -2R_{\\mu\\nu}^2 + \\frac{1}{3}R^2$ and $\\mathrm{Euler} = R_{\\mu\\nu\\rho\\sigma}^2 - 4R_{\\mu\\nu}^2 + R^2$. \nIn principle $c(g)$ can depend on $g$, but the only such theories known are constructed in somewhat artificial holographic realization \\cite{Nakayama:2017oye}.\n\n \nTo simplify the analysis, let us focus on the regime in which the last quartic term in \\eqref{foura} i.e. $\\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g$ can be neglected (e.g. in the small coupling regime). \nNeglecting the quartic term, we try to solve the equation\n\\begin{align}\n-c\\mathrm{Weyl}^2 + a \\mathrm{Euler} = (D^2 g D^2g - 2G_{\\mu\\nu} \\partial^\\mu g \\partial^\\nu g -\\frac{R}{3}\\partial^\\mu g\\partial_\\mu g) \\ . \\label{fourc}\n\\end{align}\nIn particular, suppose that the metric $g_{\\mu\\nu}(x)$ is Ricci flat. Then the equation \\eqref{fourc} becomes\n\\begin{align}\n\\sqrt{(a-c) R_{\\mu\\nu\\rho\\sigma}^2} = D^2 g \\ , \n\\end{align}\nwhich may be solved by using Green's function for the Laplacian \n\\begin{align}\ng(x) = \\int d^4x' G(x,x') \\sqrt{(a-c) R_{\\mu\\nu\\rho\\sigma}^2(x')}\n\\end{align}\nwhen the manifold is non-compact (otherwise the regular solution does not exist).\n\n\n\nAs in two-dimensions, the above argument works both for infinitesimal Weyl transformation or finite but constant Weyl transformation.\nFor finite generic Weyl transformation, one may define the analogue of harmonic Weyl transformation. For this purpose, it is more convenient to choose a different renormalization scheme so that the Weyl anomaly takes the form (e.g. \\cite{Gomis:2015yaa})\n\\begin{align}\n\\delta F_{\\sigma} = \\int d^4x \\sqrt{g} \\delta\\sigma(x) &\\left( (c(g)-a) \\mathrm{Weyl}^2 - 4a Q + g \\Delta_4 g \\right. \\cr\n& \\left. + \\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g \\right) \\ , \\label{alt} \n\\end{align}\nwhere $\\Delta_4$ is the Fradkin-Tseytlin-Riegert-Paneitz conformal operator \\cite{Fradkin:1982xc}\\cite{Fradkin:1981jc}\\cite{Riegert:1984kt}\\cite{PA}\n\\begin{align}\n\\Delta_4 = (D^2)^2 + 2G_{\\mu\\nu}D^\\mu D^\\nu + \\frac{1}{3}(D^\\mu R) D_\\mu + \\frac{1}{3}R D^2, \n\\end{align}\nwhich is Weyl covariant $\\Delta_4 \\to e^{-4\\sigma} \\Delta_4$, \nand $Q$ is what is called the Q-curvature \\cite{Q}:\n\\begin{align}\nQ = \\frac{-1}{6} D^2 R -\\frac{1}{2}R^{\\mu\\nu}R_{\\mu\\nu} + \\frac{1}{6}R^2 \\ \n\\end{align}\nwhich has a nice mathematical property under the Weyl transformation\n\\begin{align}\nQ \\to e^{-4\\sigma} (Q + \\Delta_4 \\sigma)\n\\end{align}\n\nThe advantage of this rewriting or a choice of the particular local counterterm is as follows. Suppose we cancelled the Weyl anomaly at a particular background by demanding\n\\begin{align}\n0 = (c(g)-a) \\mathrm{Weyl}^2 - 4a Q + g \\Delta_4 g + \\chi_4(g) \\partial_\\mu g \\partial^\\mu g \\partial_\\nu g \\partial^\\nu g \n\\end{align}\nThen, we are still able to cancel the Weyl anomaly on the Weyl transformed manifold whenever the Weyl rescaling is annihilated by the Fradkin-Tseytlin-Riegert-Paneitz operator:\n\\begin{align}\n\\Delta_4 \\sigma = 0 \\ . \\label{ann}\n\\end{align}\nThis is because all the terms in \\eqref{alt} except for the Q-curvature transform covariantly under the finite Weyl transformation.\nIf the Weyl scaling factor satisfies \\eqref{ann}, the cancellation of the Weyl anomaly therefore persists even for finite Weyl transformation. This is precisely analogous to the special role of harmonic Weyl transformation in two dimensions.\n\n\nLet us move on to the most generic cases with multiple coupling constants. The Weyl transformation is given by\n\\begin{align}\n\\delta F_{\\sigma} = \\int d^4x \\sqrt{g} \\delta \\sigma(x) &( c(g) \\mathrm{Weyl}^2 - a \\mathrm{Euler} + \\chi_{ij}(g) (D^2 g^i D^2g^j - 2G_{\\mu\\nu} \\partial^\\mu g^i \\partial^\\nu g^j -\\frac{R}{3}\\partial^\\mu g^i \\partial_\\nu g^j) \\cr\n& \\left. + \\chi_{ijkl}(g) \\partial_\\mu g^i \\partial^\\mu g^j \\partial_\\nu g^k \\partial^\\nu g^l \\right) \\cr\n&+ \\partial_\\mu \\delta\\sigma G^{\\mu\\nu} w_i(g) \\partial_\\nu g^j \\ . \\label{fourm}\n\\end{align}\nFor finite but constant Weyl transformation, we only have to cancel the first two lines in \\eqref{fourm}, which is essentially equivalent to what we have done in the above. On the other hand, for infinitesimal but generic Weyl transformation, we have to cancel the third line as well, which requires either $\\partial_i w_j - \\partial_j w_i =0$ or $G_{\\mu\\nu} = 0$ in the background.\n\nTo conclude the analysis, we would like to mention the other obstructions to the Weyl transformation if the dimension two operator $O(x)$ exist in the theory. If this is the case, there is a further operator Weyl anomaly such as \n\\begin{align}\n\\int d^4x \\sqrt{g} \\delta\\sigma(x) (\\eta(g) R O(x) + \\epsilon(g) \\partial_\\mu g \\partial^\\mu g O + \\tau(g) D^2 O + \\delta(g) D^2 g O) + \\partial_\\mu \\delta \\sigma(x) \\theta(g) \\partial^\\mu g O \\ .\n\\end{align}\nIt has been shown that such Weyl anomaly can be removed when $g(x) = g$ \\cite{Jack:2013sha} (see also \\cite{Nakayama:2013wda}\\cite{Farnsworth:2017tbz} for similar analysis), but with the space-time dependent coupling, we need the extra cancellation to get the consistent picture. Schematically, the Wess-Zumino consistency condition demands $\\eta = 0$ and one can always remove $\\theta$ and $\\tau$ by local counterterms. Then we need to cancel $\\epsilon$ term against the $\\delta$ term. Since the existence of dimension two operator is non-generic, we will not pursue the cancellation in further details.\n\n\n\\section{Holographic models}\nWe revisit the cancellation mechanism we have studied in previous sections from the holographic perspective. For definiteness we consider the case of four dimensional conformal field theories with the five dimensional bulk. Let us study the Einstein gravity coupled with a scalar field $\\phi$ given by the minimal action\n\\begin{align}\nS = \\int d^5x \\sqrt{g} \\left( R + \\Lambda + \\frac{1}{2}\\partial^M \\phi \\partial_M \\phi \\right) .\n\\end{align}\n\nIn the AdS\/CFT correspondence, we compute the on-shell action for a given boundary condition at $\\rho= \\epsilon$ (i.e. $\\phi_{(0)}(x)$ and $g_{(0)\\mu\\nu}(x)$ below) with the expansion \n\\begin{align}\n\\phi &= \\phi_{(0)}(x) + \\rho \\phi_{(1)}(x) + \\rho^2 \\phi_{(2)}(x) + \\cdots \\cr\ng_{\\mu\\nu} &= g_{(0)\\mu\\nu}(x) + \\rho g_{(1)\\mu\\nu}(x) + \\rho^2 g_{(2) \\mu\\nu}(x) + \\cdots\n\\end{align}\nin the Graham-Fefferman gauge \n\\begin{align}\nds^2 = G_{MN} dx^M dx^N = \\frac{d\\rho^2}{\\rho^2} + \\frac{g_{\\mu\\nu} dx^\\mu dx^\\nu}{\\rho} \\ .\n\\end{align}\n\nThe resulting on-shell action is generically divergent in the limit $\\epsilon \\to 0$ from the $\\rho$ integration as $\\int_{\\epsilon} d\\rho \\rho^{-1} S_{\\log} = \\log \\epsilon S_{\\log}$, leading to the holographic Weyl anomaly \\cite{Henningson:1998gx}. Explicitly \\cite{Nojiri:1998dh}, we have\n\\begin{align}\nS = \\log \\epsilon \\int d^4x &\\left( \\frac{1}{8} R_{\\mu\\nu (0)}^2 -\\frac{1}{24} R_{(0)}^2 + \\frac{1}{4} (D^2 \\phi_{(0)})^2 \\right. \\cr \n& \\left. - \\frac{1}{2} R_{(0)}^{\\mu\\nu} \\partial_\\mu \\phi \\partial_\\nu \\phi + \\frac{1}{6} R_{(0)} \\partial^\\mu \\phi_{(0)} \\partial_\\nu \\phi_{(0)} + \\frac{1}{3}(\\partial_\\mu \\phi_{(0)} \\partial^\\mu\\phi_{(0)})^2 \\right) \\ \n\\end{align}\nwhich is exactly what we had in section 4 for the constant Weyl transformation.\nThus cancelling the Weyl anomaly from the position dependent coupling constant corresponds to the choice of the boundary values of $\\phi(x)$ such that the on-shell gravity action is finite without the logarithmic divergence. The choice of such boundary conditions make the AdS\/CFT correlation functions better behaved, so classifying such supergravity background may be of theoretical interest.\n\n\\section{Discussions}\nIn this paper, we have studied a novel way to cancel the Weyl anomaly from the position dependent coupling constant. Here we would like to mention further possibilities to cancel the Weyl anomaly.\n\nFirst of all, if the theory under consideration possesses a conserved current $J_\\mu$, one may introduce the background field strength by the coupling $\\int d^4x \\sqrt{g} A^\\mu J_\\mu$. This gives another contribution to the Weyl anomaly as $\\int d^4x \\delta\\sigma(x) b_0 F_{\\mu\\nu}(x) F^{\\mu\\nu}(x)$, where $b_0$ is the coefficient of the one-loop beta function determined from the current two-point function (which is positive in unitary conformal field theories) and $F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$. Then we may try to cancel the Weyl anomaly from this contribution.\nActually, simultaneous use of the gauge field and the position dependent coupling constant may not be a good idea because of the existence of the vector beta functions \\cite{Nakayama:2013ssa}. Again we have to think about the cancellation of the extra operator Weyl anomaly such as $\\int d^4x \\delta \\sigma \\rho(g) \\partial^\\mu g J_\\mu$. To avoid the appearance of the vector beta functions, we may only introduce the position dependent coupling constant which is neutral under the symmetry generated by $J_\\mu$.\n\nIt could have been extremely interesting if we were able to find a novel class of Weyl gauging without demanding $c=0$ in two dimensions, and $c=a=0$ in four dimensions.\\footnote{See e.g. \\cite{Fradkin:1983tg} for a possibility in the context of supersymmetric Weyl gravity.} \nCurrently, the closest way to do this is to demand all the non-trivial Weyl anomalies vanish, say $a=0$ in four dimensions, and then try to cancel the $c \\mathrm{Weyl}^2$ term against the space-time dependent coupling constants. Here, we should further employ the non-unitariness of the model (since $a=0$ from the beginning suggests it must be so) to obtain the cancellation in the Weyl anomalies. This is because unitarity demands the positivity of the both terms and the cancellation only happens by using the non-unitary property. Whether this is better than just demanding $c=a=0$ is yet to be seen in the context of quantum Weyl gravity in which we would like to gauge the Weyl symmetry exactly.\\footnote{Note, however, that the introduction of the position dependent coupling constant modifies the conservation of the energy-momentum tensor, and the consistecy with the dynamical gravity must be discussed more carefully. We stress that in the main part of this paper, we have focused on the fixed gravitational background, so there is no inconsistency. The author would like to thank S.~Deser for the discussions.}\n\nIn two dimensions, we did not obtain any new possibilities to gauge the entire Weyl symmetry than demanding $c=0$ from the beginning. We are still able to gauge the harmonic Weyl symmetry, but the physical interest in such gauging (e.g. whether it defines new class of quantum gravity in two dimensions) should be discussed more in detail.\n\nFinally, we point out that there is an alternative option. Once we know how to solve $g(x)$ to cancel the Weyl anomaly in a given metric $g_{\\mu\\nu}(x)$, one may introduce the extra transformation on $g(x)$ so that the Weyl anomaly is always cancelled (irrespective of the obstructions we have discussed above). This possibility requires further investigation if such transformation can be defined systematically and then whether such a generalized notion of the Weyl transformation is useful or not.\n\n\n\\section*{Acknowledgements}\nThis work is in part supported by JSPS KAKENHI Grant Number 17K14301. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Effects of the code improvements on the wind}\nIn this section, we evaluate the effect on the wind of the three improvements that we presented to the treatment of the radiation field: (i) radial dependence of $f_\\text{\\tiny UV}$, (ii) improved radiation transport, and (iii) relativistic corrections.\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/comparisons.pdf}\n \\caption{Effects of our improvements to the treatment of the radiation field on the trajectories of gas blobs. All simulations have been done with $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, and $R_\\mathrm{in}=50R_\\mathrm{g}$.}\n \\label{fig:comparisons}\n\\end{figure*}\nThe impact of each improvement is shown in \\autoref{fig:comparisons}. In the first panel (blue), we calculate the optical depth $\\tau_\\text{\\tiny UV}$ from the centre, rather than attenuating each individual light ray coming from the disc. We also take a constant $f_\\text{\\tiny UV}=0.85$ and we do not include relativistic corrections. Calculating $\\tau_\\text{\\tiny UV}$ from the centre overestimates the attenuation of the UV radiation field in the outer parts of the wind, since it assumes that the radiation is crossing all the inner gas. This produces a failed wind in the outer regions of the disc (which is on too small a scale to be seen in \\autoref{fig:comparisons}), since the inner gas is optically thick to the UV radiation. In the next panel (green), we calculate $\\tau_\\text{\\tiny UV}$ by attenuating each individual light ray as explained in \\autoref{sec:radiation_transport}, consequently, most of the wind escapes from the disc since the UV attenuation is less strong, with the average wind velocity being slightly lower because we are now averaging over the slower trajectories in the outer part of the wind. The next improvement we evaluate is the inclusion of the radial dependence of $f_\\text{\\tiny UV}$ (third panel, orange). The change in wind geometry can be explained by considering the distribution of the UV emissivity, which is skewed towards the centre of the disc thus pushing the outer wind along the equator. This also leads to a more powerful wind, due to the increase in UV luminosity at small radii, where the fastest and most massive streamlines originate. Lastly, the effect of including relativistic corrections is shown in the 4th panel (red). As expected, the velocity of the wind is considerably lower, and hence also its kinetic luminosity. Furthermore, the wind flows at a slightly higher polar angle, as the vertical component of the radiation force gets weaker where the wind has a high vertical velocity. \n\\section{Conclusions and future work}\n\nIn this paper, we have presented \\textsc{Qwind3}, a model that builds upon the non-hydrodynamical code \\textsc{Qwind} first introduced in \\cite{risaliti_non-hydrodynamical_2010} to model UV line-driven winds in the context of AGNs. In this new version, we generalise the CAK formalism for stellar winds to AGNs, which we use to calculate the initial density and velocity with which the wind is launched at the surface of the accretion disc. We have highlighted the importance of correctly accounting for the fraction of luminosity emitted in the UV by each disc annulus, which in other numerical codes is assumed to be a constant over the whole disk. We have also introduced an algorithm to do ray-tracing calculations throughout the wind, allowing us to compute the radiation transfer of the system taking into account the full geometry of the wind and the spatial distribution of the X-ray and UV radiation sources. Furthermore, we have also included special relativistic corrections to the calculation of the radiation flux, which are important for ensuring that the wind does not achieve superluminal speeds. We note two important assumptions that still remain: the simplified dependence of the X-ray opacity on the ionisation parameter and the X-ray luminosity fraction, which we here assume is constant at $f_X=0.15$, independent of black hole mass and mass accretion rate. We will address these issues in the next {\\sc Qwind} release. \n\nWe have used the new code to explore under what conditions UV line-driven winds are successfully launched from accretion discs, studying how the normalised mass loss rate, kinetic luminosity, momentum outflow rate, and final velocity change as a function of black hole mass, mass accretion rate, and initial wind launching radius. We find that winds can carry a mass loss rate up to 30\\% of the mass accretion rate,\nso this can have a moderate impact on the mass accretion rate at radii below the wind launching radius. The next {\\sc Qwind} release will address this by reducing the mass accretion rate through the disc, and hence reducing the UV flux produced at small radii, to make the model self-consistent (see also \\citealt{nomura_line-driven_2020}).\nThe current code produces winds where the \nkinetic power and especially the momentum outflow rate can \nbe comparable to that of the radiation power at high $\\dot{m}$, and even exceed it which is unphysical. This shows that the effect of line blanketing must be important in reducing the UV flux driving the wind below that predicted by electron scattering opacity alone. We will address this in the next {\\sc Qwind} release, but \noverall, the wind clearly carries sufficient power to meet the criteria for an efficient feedback mechanism in galaxy formation \\citep{hopkins_stellar_2016}.\n\nOur results here show that the outflow velocity is mildly relativistic across a broad parameter space, with velocity $0.1-0.3$~c even excluding the unphysically efficient winds. Thus UV line-driven winds can reach UFO velocities even when special relativistic corrections (radiation drag) are included. This contrasts with \\cite{luminari_speed_2021}, who conclude that UV line-driving is not capable of generating such high velocity winds. \nWe caution that the two codes make different assumptions about the initial conditions and ray tracing (both of which \\textsc{Qwind} does more accurately and self-consistently) as well as the opacity (which their code does better), so it is premature to rule out UV line-driving in favour of other mechanisms such as magnetic driving \\citep{blandford_hydromagnetic_1982, fukumura_magnetic_2017} as the origin of UFOs until these factors are all incorporated together. We will explore this more fully in the next \\textsc{Qwind} release.\n\nThe normalised wind mass loss rate and normalised kinetic luminosity vary substantially as a function of black hole mass and accretion rate, the latter being the most significant factor. The ratio of wind kinetic power to mass accretion rate scales steeply with $\\dot{m}$, in contrast to the constant ratio normally assumed in the AGN feedback models currently implemented in cosmological simulations of galaxy formation. Implementing this new more physically-based AGN feedback prescription in simulations will therefore change how galaxies are predicted to evolve across cosmic time.\n\n\n\\section{Calculating the gas trajectories}\n\\label{sec:gas_trajectories}\n\\subsection{Initial radii of gas trajectories}\n\nThe first thing to consider when calculating the gas trajectories is the initial location of the gas blobs. The innermost initial position of the trajectories is taken as a free parameter $R_\\mathrm{in}$, and the outermost initial position, $R_\\mathrm{out}$, is assumed to be the self-gravity radius of the disc, where the disc is expected to end \\citep{laor_massive_1989}, which for our reference BH corresponds to 1580 $R_g$. We initialise the first trajectory at $R_\\mathrm{in}$, the next trajectory starts at $R=R_\\mathrm{in} + \\Delta R$, where $\\Delta R$ is the distance between adjacent trajectories. We determine $\\Delta R$ by considering two quantities: (i) the change in optical depth between two adjacent trajectories along the base of the wind and (ii) the mass loss rate along a trajectory starting at $R$ and at a distance $\\Delta R$ to the next one. Regarding (i), the change in optical depth $\\Delta\\tau$ between two trajectories initially separated by $\\Delta R$ is given by\n\\begin{equation}\n \\Delta\\tau = \\int_R^{R+\\Delta R_1} n(R')\\, \\sigma_{\\text{\\tiny T}} \\,\\mathrm{d} R'.\n\\end{equation}\nWe consider\n\\begin{equation}\n \\Delta\\tau = \\begin{cases} \n 0.05 & \\mathrm{ if } \\;\\tau(R) < 5,\\\\\n 0.5 & \\mathrm{ if } \\;\\tau(R) < 10,\\\\\n 5 & \\mathrm{ if } \\;\\tau(R) < 100,\\\\\n 20 & \\mathrm{ if } \\;\\tau(R) > 100,\\\\\n \\end{cases}\n\\end{equation}\nwhere $\\tau(R) = \\sum_i \\Delta\\tau_i$. This guarantees that the spacing between trajectories resolves the transition from optically thin to optically thick for both the UV and X-ray radiation. The quantity (ii) is given by\n\\begin{equation}\n \\Delta \\dot M_\\mathrm{wind} = \\int_R^{R+\\Delta R_2}2\\pi R'\\rho(R')v(R') \\mathrm{d} R',\n\\end{equation}\nwhere we consider $\\Delta \\dot M_\\mathrm{wind}=0.01\\dot M$, so that no streamline represents more than $1\\%$ of the accreted mass rate. The $\\Delta R$ step is thus given by \n\\begin{equation}\n \\Delta R = \\min (\\Delta R_1, \\Delta R_2)\n\\end{equation}\nand we repeat this process until $R=R_\\mathrm{out}$\n\n\\subsection{Solving the equation of motion}\n\nThe equation of motion of the wind trajectories is the same as in \\citetalias{quera-bofarull_qwind_2020}, and we solve it analogously by using the Sundials IDA integrator \\citep{hindmarsh_sundials_2005}. The wind trajectories are calculated until they either exceed a distance from the centre of $10^4 R_g$, fall back to the disc, or self intersect. To detect when a trajectory self intersects we use the algorithm detailed in Appendix \\ref{app:intersections}.\n\nBy considering the full wind structure for the radiation ray tracing, we run into an added difficulty: the interdependence of the equation of motion with the density field of the wind. To circumvent this, we adopt an iterative procedure in which the density field of the previous iteration is used to compute the optical depth factors for the current iteration. We first start assuming that the disc's atmosphere is void, with a vacuum density of $n_\\text{vac} = 10^2$ cm$^{-3}$. Under no shielding of the X-ray radiation, all the trajectories fall back to the disc in a parabolic motion. After this first iteration, we calculate the density field of the resulting failed wind and use it to calculate the wind trajectories again, only that this time the disc atmosphere is not void. We keep iterating until the mass loss rate and kinetic luminosity do not significantly change between iterations. It is convenient to average the density field in logarithmic space between iterations, that is, the density field considered for iteration $k$, $n_k$, is given by\n\\begin{equation}\n \\log_{10}(n_{k}(R, z)) = \\frac{1}{2}\\, \\left(\\log_{10}(n_{k-1}(R,z)) + \\log_{10}(n_{k-2}(R,z))\\right).\n\\end{equation}\nWe do not take the average for the first two iterations. The number of iterations required depends on the initial radius of the wind, but we typically find convergence after $\\sim 20$ iterations, although we run several more to ensure that the standard deviation of the density field between iterations is small. The normalised mass loss rate and kinetic luminosity for our fiducial model at each iteration are shown in \\autoref{fig:iterations}.\n\n\\begin{figure}\n \\centering\n \\includegraphics{figures\/iterations.pdf}\n \\caption{Normalised mass loss rate and kinetic luminosity for each iteration for our fiducial case.}\n \\label{fig:iterations}\n\\end{figure}\n\nFor a given iteration, solving the equation of motion for the different gas trajectories is an embarrassingly parallel problem, and so our code's performance scales very well upon using multiple CPUs, allowing us to quickly run multiple iterations, and scan the relevant parameter spaces. The computational cost of running one iteration is $\\sim 5$ CPU hours, so one is able to obtain a fully defined wind simulation after $\\sim 100$ CPU hours.\n\n\\section{Critical point derivation}\n\\label{app:initial_conditions}\n\nIn this appendix we aim to give a detailed and self-contained derivation of the initial conditions for launching the wind from the surface of the accretion disk. Let us first start with the 1D wind equation (now including gas pressure forces),\n\\begin{equation}\n \\label{eq:ic_1d}\n \\rho \\frac{\\mathrm{d} v_z}{\\mathrm{d} t} = \\rho (a_\\mathrm{rad}^z - a_\\mathrm{grav}^z) - \\frac{\\partial P}{\\partial z}.\n\\end{equation}\nThe left-hand side of the equation can be expanded to\n\\begin{equation}\n \\rho \\frac{\\mathrm{d} v_z}{\\mathrm{d} t} = \\rho \\frac{\\partial v_z}{\\partial t} + \\rho v_z \\frac{\\partial v_z}{\\partial z} ,\n\\end{equation}\nwhere we note that $\\partial v_z \/ \\partial t$ = 0, since we are only interested in steady solutions that do not depend explicitly on time. For simplicity, we drop the partial derivative notation going forward, since only derivatives along the $z$ direction are involved. We assume that the wind is isothermal, with an equation of state given by\n\\begin{equation}\n P = c_s^2 \\rho\n\\end{equation}\nwhere $c_s$ is the isothermal sound speed, which is assumed constant throughout the wind. Using the mass conservation equation (\\autoref{eq:mass_conservation}), $\\dot M = A\\, \\rho\\, v_z$, where $A = 2\\pi r \\Delta r$, we can write\n\\begin{equation}\n \\frac{\\mathrm{d} P}{\\mathrm{d} z} =c_s^2\\frac{\\mathrm{d} \\rho}{\\mathrm{d} z} = -\\rho c_s^2 \\left(\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A} + \\frac{\\mathrm{d} v_z \/ \\mathrm{d} z}{v_z}\\right),\n\\end{equation}\nso we can rewrite \\autoref{eq:ic_1d} as\n\\begin{equation}\n v_z \\frac{\\mathrm{d} v_z}{\\mathrm{d} z}\\left(1-\\frac{c_s^2}{v_z^2}\\right) = a_\\mathrm{rad}^z - a_\\mathrm{grav}^z + c_s^2\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A}.\n\\end{equation}\nWe now focus on the radiation force term, which is the sum of the electron scattering and line-driving components,\n\\begin{equation}\n a_\\mathrm{rad}^z = a_\\mathrm{rad}^{\\mathrm{es}, z} + \\mathcal M \\, a_\\mathrm{rad}^{\\mathrm{es}, z}.\n\\end{equation}\nWhen the wind is rapidly accelerating, as it is the case at low heights, the force multiplier is well approximated by its simpler form (see section 2.2.3 of \\citetalias{quera-bofarull_qwind_2020}),\n\\begin{equation}\n \\label{eq:fm_simple}\n \\mathcal M(t) = k \\, t^{-\\alpha},\n\\end{equation}\nwhere $\\alpha$ is fixed to $0.6$, $k$ depends on the ionisation level of the gas (in the \\citetalias{stevens_x-ray_1990} parameterisation), and $t = \\kappa_{\\text{\\tiny e}} \\, \\rho \\, v_\\text{th} \\left | \\mathrm{d} v\/\\mathrm{d} z \\right | ^{-1}$. As discussed in section 3.1.3 of \\citetalias{quera-bofarull_qwind_2020}, the thermal velocity, $v_\\mathrm{th}$ (\\autoref{eq:thermal_velocity}), is computed at a fixed temperature of $T=2.5 \\times 10^4$ K. Once again making use of the mass conservation equation we can write\n\\begin{equation}\n \\mathcal M = k\\, t^{-\\alpha} = \\frac{k}{(\\kappa_{\\text{\\tiny e}}\\, v_\\text{th}\\, \\rho)^\\alpha}\\left |\\frac{dv}{dz}\\right|^\\alpha = \\frac{k}{\\left(\\kappa_{\\text{\\tiny e}} v_\\text{th} \\, \\right)^\\alpha}\\left(\\frac{A}{\\dot M} v_z \\left|\\frac{dv_z}{dz}\\right|\\right)^\\alpha\n\\end{equation}\nand so the full equation to solve with all derivatives explicit is\n\\begin{equation}\n \\begin{split}\n v_z \\frac{\\mathrm{d} v_z}{\\mathrm{d} z}\\left( 1- \\frac{c_s^2}{v_z^2}\\right) &= a_\\mathrm{rad}^{\\mathrm{es}, z} (z) -a_\\mathrm{grav}^z (z) \\; \\\\\n &+ a_\\mathrm{rad}^{\\mathrm{es}, z}(z)\\frac{k}{\\left(\\kappa_{\\text{\\tiny e}} v_\\text{th} \\, \\right)^\\alpha}\\left(\\frac{A(z)}{\\dot M} v_z \\left|\\frac{dv_z}{dz}\\right|\\right)^\\alpha \\\\\n &+ c_s^2\\frac{\\mathrm{d} A(z) \/ \\mathrm{d} z}{A(z)}.\n \\end{split}\n\\end{equation}\nWe assume that $\\mathrm{d} v_z \/ \\mathrm{d} z >0$ always, so the absolute value can be dropped. It is convenient to introduce new variables to simplify this last equation. The first change we introduce is $W=v_z^2\/2$, so that $v_z \\, \\mathrm{d} v_z \/ \\mathrm{d} z = \\mathrm{d} W \/ \\mathrm{d} z$. Next, we aim to make the equation dimensionless, so we consider $R$ as the characteristic length, $B_0 = GM_\\mathrm{BH}\/R^2$ as the characteristic gravitational acceleration value, $W_0=GM_\\mathrm{BH}\/R = B_0 R$ as the characteristic $W$ value, $A_0 = 2\\pi R \\Delta R$ as the characteristic $A$ value, and finally we take the characteristic radiation force value to be \n\\begin{equation}\n\\label{eq:gamma_0}\n \\gamma_0 = \\frac{k}{(\\kappa_{\\text{\\tiny e}} v_\\text{th})^\\alpha} \\; a_\\mathrm{rad, 0}^{\\mathrm{es}, z},\n\\end{equation}\nwhere $a_\\mathrm{rad, 0}^{\\mathrm{es}, z}$ is \\autoref{eq:radiation_acceleration_approx} without considering the $\\tau_\\text{\\tiny UV}$ factor, since we do not include attenuation in this derivation. With all this taken into account, we can write\n\\begin{equation}\n \\label{eq:half_way}\n \\begin{split}\n B_0 \\frac{dw}{dx} \\left(1 - \\frac{s}{w}\\right) & = -a_\\mathrm{grav}^z + a_\\mathrm{rad}^{\\mathrm{es}, z} \\left(1+ \\frac{k}{(\\kappa_{\\text{\\tiny e}} v_\\text{th} )^\\alpha}\\left(\\frac{A}{\\dot M} B_0\\frac{dw}{dx}\\right)^\\alpha\\right) \\\\\n & + c_s^2\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A},\n\\end{split}\n\\end{equation}\nwhere we have defined $x=z\/R$, $w=W\/W_0$, and $s=c_s^2 \/ (2 W_0)$. We also define $a=A\/A_0$, and $\\varepsilon = \\dot M \/ \\dot M_0$ with\n\\begin{equation}\n \\dot M_0 = \\alpha (1-\\alpha)^{(1-\\alpha) \/ \\alpha} \\frac{(\\gamma_0 A_0)^{1 \/ \\alpha}}{(B_0 A_0)^{(1-\\alpha) \/ \\alpha}}.\n \\label{eq:Mdot0_def}\n\\end{equation}\nThis last definition may seem a bit arbitrary, but it is taken such that $\\varepsilon = 1$ corresponds to the classical \\citetalias{castor_radiation-driven_1975} $\\dot M$ value for O-stars. \\autoref{eq:half_way} then becomes\n\\begin{equation}\n \\begin{split}\n \\frac{\\mathrm{d} w}{\\mathrm{d} x} \\left(1-\\frac{s}{w}\\right) &= \\frac{(-a_\\mathrm{grav}^z + a_\\mathrm{rad}^{\\mathrm{es}, z})}{B_0} + \\frac{1}{\\alpha^\\alpha(1-\\alpha)^{1-\\alpha}}\\frac{a_\\mathrm{rad}^{\\mathrm{es}, z}}{a_\\mathrm{rad,0}^{\\mathrm{es}, z}} \\left(\\frac{a}{\\varepsilon}\\frac{\\mathrm{d} w}{\\mathrm{d} x}\\right)^\\alpha \\\\\n &+ \\frac{4sx}{a},\n \\end{split}\n\\end{equation}\nwhere we have used\n\\begin{equation}\n a = \\frac{A}{A_0} = \\frac{2\\pi r \\Delta r}{2\\pi r_0 \\Delta r_0} = \\frac{r^2}{r_0^2},\n\\end{equation}\nso that\n\\begin{equation}\n \\frac{c_s^2}{B_0}\\frac{\\mathrm{d} A \/ \\mathrm{d} z}{A} = \\frac{4sx}{a}.\n\\end{equation}\nWe note that we have assumed that the area $A$ changes with $z$ like the 2D solution, despite the fact that we are considering here a 1D wind. This small correction guarantees that we find critical-point like solutions for all initial radii, since it guarantees that the \\citetalias{castor_radiation-driven_1975} conditions for the existence of a critical point are satisfied.\nFinally, we define\n\\begin{equation}\n\\label{eq:nozzle_f}\n f = \\frac{1}{\\alpha^\\alpha (1-\\alpha)^{1-\\alpha}}\\frac{a_\\mathrm{rad}^{\\mathrm{es}, z}}{a_\\mathrm{rad, 0}^{\\mathrm{es}, z}},\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:nozzle_h}\n h = \\frac{(a_\\mathrm{grav}^z - a_\\mathrm{rad}^{\\mathrm{es}, z})}{B_0} - 4sxa,\n\\end{equation}\nso that\n\\begin{equation}\n \\frac{\\mathrm{d} w}{\\mathrm{d} x} \\left(1-\\frac{s}{w}\\right) = -h(x) + f(x) \\left(\\frac{a}{\\varepsilon}\\frac{\\mathrm{d} w}{\\mathrm{d} x}\\right)^\\alpha,\n\\end{equation}\nwhich is the dimensionless wind equation. The choice of sign in $h(x)$ is to make the further steps clearer.\nIt is useful to interpret this equation as an algebraic equation for $w'=\\mathrm{d} w \/ \\mathrm{d} x$,\n\\begin{equation}\n F(x,w,w') = w' \\left(1-\\frac{s}{w}\\right) + h(x) - f(x) \\left(\\frac{a}{\\varepsilon}w'\\right)^\\alpha = 0.\n \\label{eq:CAK_F_wprime}\n\\end{equation}\nThis has the same general form as Equation~26 in \\citetalias{castor_radiation-driven_1975}, which applies to a spherical wind from a star, but the functions $h(x)$ and $f(x)$ are different. We note that $f(x)>0$ and $0<\\alpha<1$, but $h(x)$ can have either sign. (Also note that we have chosen the opposite sign for $h(x)$ to CAK, for later convenience.)\nGiven values of $x$ and $\\varepsilon$, we can distinguish 5 different regions of solutions for $w'$ (we follow the original enumeration of regions by \\citetalias{castor_radiation-driven_1975}):\n\n\\begin{itemize}\n \\item Subsonic stage ($w0$, $F(w'=0)$ is positive and as $w'$ increases, $F$ goes to negative values crossing $F=0$ once, so there is one solution for $w'$.\n \\end{itemize}\n \\item Supersonic stage ($w >s$): We have three regions\n \\begin{itemize}\n \\item Region III: If $h(x)<0$, $F(w'=0)$ is negative and as $w'$ increases, $F$ goes to positive values crossing $F=0$ once, so there is one solution for $w'$.\n \\item If $h(x) > 0$, $F(w'=0)$ is positive, then $F$ initially decreases until its minimum, $w_\\mathrm{min}'$, and then increases again. If $F(w_\\mathrm{min}') >0$, we have no solution for $w'$ and if $F(w_\\mathrm{min}') \\leq 0$ we have two solutions, which are the same if $F(w_\\mathrm{min}') = 0$.\n The minimum can be found by solving\n \\begin{equation}\n \\frac{\\partial F}{\\partial w'} = 0,\n \\end{equation}\n which gives\n \\begin{equation}\n \\label{eq:w_min}\n w_\\mathrm{min}' = \\left( \\frac{1-s\/w}{\\alpha f (a\/\\varepsilon)^\\alpha}\\right)^\\frac{1}{\\alpha -1},\n \\end{equation}\n so we have\n \\begin{itemize}\n \\item Region IV: If $F(w_\\mathrm{min}') > 0$, there is no solution.\n \\item Region II: If $F(w_\\mathrm{min}') \\leq 0$, there are two solutions, one with $w' < w_\\mathrm{min}'$ and the other with $w' > w_\\mathrm{min}'$\n \\end{itemize}\n \\end{itemize}\n\\end{itemize}\n\nWe assume that the wind starts subsonic ($ws) and extends to $x\\to\\infty$, this means that the wind must end at Region~III. However, because $h(x)>0$ in Region~I and $h(x)<0$ in Region~III, these two regions must be connected by Region~II in between. At the boundary between Regions~I and II, $w=s$ and $h(x)>0$, while at the boundary between Regions~II and III, $w>s$ and $h(x)=0$.\n\nConsidering first the boundary between Regions~I and II, setting $w=s$ in \\autoref{eq:CAK_F_wprime} gives\n\\begin{equation}\n h - f \\left(\\frac{a}{\\varepsilon} w'\\right)^\\alpha = 0,\n\\end{equation}\nwith $h>0$. Considering \\autoref{eq:w_min} for $w_\\mathrm{min}'$ as $w \\to s^{+}$, we find that the wind solution at this boundary must lie on the branch with $w' < w_\\mathrm{min}'$.\n\nConsidering next the boundary between Regions~II and III, setting $h=0$ in \\autoref{eq:CAK_F_wprime} (neglecting the $w'=0$ solution) gives\n\\begin{equation}\n 1-\\frac{s}{w} - f \\left(\\frac{a}{\\varepsilon}\\right)^\\alpha w'^{\\alpha-1} = 0,\n\\end{equation}\nAgain using \\autoref{eq:w_min} and $w>s$, we find that the wind solution at this boundary must lie on the branch with $w' > w_\\mathrm{min}'$.\n\n\nHowever, we assume that $w'$ must be continuous throughout the wind, which would not be the case if the two branches of region II were not connected. Therefore, both branches of region II must coincide at some point, so the condition\n\\begin{equation}\n F(w_\\mathrm{min}') = 0\n\\end{equation}\nmust hold at that point, with $w' = w_\\mathrm{min}'$ for both branches. This point is in fact the critical point of the solution, and $\\partial F\/\\partial w' = 0$ there. Upon substitution of $w_\\mathrm{min}'$, this condition is equivalent to\n\\begin{equation}\n \\label{eq:nozzle_function_sonic}\n \\varepsilon \\, \\left(1 - \\frac{s}{w} \\right) = \\alpha (1-\\alpha)^\\frac{1-\\alpha}{\\alpha} \\, \\frac{f^{1\/\\alpha} a}{h^\\frac{1-\\alpha}{\\alpha}}.\n\\end{equation}\nThe right-hand side of the equation is usually referred to as the Nozzle function $\\mathcal N$,\n\\begin{equation}\n\\label{eq:nozzle_function}\n \\mathcal N(x) = \\alpha (1-\\alpha)^\\frac{1-\\alpha}{\\alpha} \\frac{f^{1\/\\alpha}a}{h^\\frac{1-\\alpha}{\\alpha}}.\n\\end{equation}\nWe define $x=x_c$ to be the position of the critical point. We now assume that at the critical point the wind is highly supersonic ($w \\gg s$). We verify this assumption in \\autoref{subsection:verify_critical_point}. Then $1-s\/w \\approx 1$, and so the normalised mass loss rate is given by\n\\begin{equation}\n \\varepsilon = \\mathcal N(x_c).\n \\label{eq:eps_n_xc}\n\\end{equation}\n\nWe now show that the location of the critical point $x_c$ is at the minimum of $\\mathcal N(x)$. Let us consider the total derivative of $F$ with respect to $x$ taken along a wind solution. Since $F=0$ at all points along the solution,\n\\begin{equation}\n \\frac{\\mathrm{d} F}{\\mathrm{d} x} = \\frac{\\partial F}{\\partial x} + \\frac{\\partial F}{\\partial w}w' + \\frac{\\partial F}{\\partial w'}\\frac{\\mathrm{d} w'}{\\mathrm{d} x} = 0.\n\\end{equation}\nBecause of our assumption $w\\gg s$, we have $\\frac{\\partial F}{\\partial w} = 0$, and, at the critical point, $\\frac{\\partial F}{\\partial w'} = 0$, so that\n\\begin{equation}\n \\left .\\frac{\\mathrm{d} F}{\\mathrm{d} x}\\right |_{w'_\\mathrm{min}, x_c} = \\left .\\frac{\\partial F}{\\partial x}\\right |_{w'_\\mathrm{min}, x_c} = 0,\n\\end{equation}\nwhich gives (a prime denotes $\\mathrm{d} \/ \\mathrm{d} x$)\n\\begin{equation}\n\\label{eq:dF_dx}\n h' - \\left(\\frac{w'_\\mathrm{min}}{\\varepsilon}\\right)^\\alpha \n \\frac{\\mathrm{d}}{\\mathrm{d} x} \\left( f a^{\\alpha} \\right) = 0.\n\\end{equation}\nNow using \\autoref{eq:w_min} for $w'_\\mathrm{min}$ along with \\autoref{eq:eps_n_xc} and \\autoref{eq:nozzle_function}, we obtain $w'_\\mathrm{min} = \\frac{\\alpha}{(1-\\alpha)} h$. Substituting in \\autoref{eq:dF_dx} and using \\autoref{eq:eps_n_xc} and \\autoref{eq:nozzle_function} again, we obtain \n\\begin{equation}\n(1-\\alpha) \\frac{h'}{h} - \\frac{1}{f a^{\\alpha}} \\frac{\\mathrm{d}}{\\mathrm{d} x} \\left( f a^{\\alpha} \\right) = 0.\n\\end{equation}\nComparing to \\autoref{eq:nozzle_function}, we see that this is equivalent to the condition $\\mathrm{d} \\mathcal{N}\/\\mathrm{d} x =0$. Therefore the critical point happens at an extremum of $\\mathcal{N}(x)$, but since the nozzle function is not upper bounded, the extremum has to be a minimum. We have therefore shown that the critical point $x_c$ corresponds to the minimum of the nozzle function $\\mathcal{N}(x)$.\n\n\n\nIn \\autoref{fig:nozzle_function}, we plot the nozzle function for $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, $k=0.03$, and $R=20R_\\mathrm{g}$ and we highlight the position of the critical point. The value of the nozzle function at the critical point determines the mass-loss rate along the streamline, through $\\dot M = \\varepsilon \\dot M_0$, so that the surface mass loss rate is given by\n\\begin{equation}\n \\dot \\Sigma = \\frac{\\dot M}{A}.\n\\end{equation}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/nozzle_function.pdf}\n \\caption{Nozzle function for $\\protectM_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$, $\\protect\\dot m =0.5$, $k=0.03$, and $R=20 R_\\mathrm{g}$} \n \\label{fig:nozzle_function}\n\\end{figure}{}\n\nIt is interesting to point out that the position of the critical point and the value of $\\epsilon$ do not depend on the chosen value of $k$ in the force multiplier parametrisation (\\autoref{eq:fm_simple}), however, \nthe mass loss rate does depend on $k$ through the value of $\\dot{M}_0$.\nIn the \\citetalias{stevens_x-ray_1990} parametrisation, $k$ is a function of the ionisation state of the gas, $k=k(\\xi)$, so the mass loss rate directly depends on the ionisation conditions at the critical point location. Since this would make our results dependent on the modelling of the vertical structure of the disc, which is out of scope for the purpose of this work, we assume that the gas is always ionised when it reaches the critical point, so we take $k=0.03$, corresponding to the minimum value of $k$ in \\citetalias{stevens_x-ray_1990}. Similarly, we set the initial height of the wind to $z=0$ to avoid dependencies on the disc vertical structure.\n\n\n\\subsection{Scaling of the initial conditions with BH properties}\n\\label{subsection:ic_scaling}\n\n\\begin{figure}\n \\centering\n \\includegraphics{figures\/nozzle_scaling.pdf}\n \\caption{Value of the nozzle function at the critical point ($\\varepsilon = \\mathcal N(x_c)$) as a function of radius for varying $M_\\mathrm{BH}$ (left panel) with $\\dot m =0.5$ and varying $\\dot m$ (right panel) with $M_\\mathrm{BH}=10^8 \\mathrm{M}_\\odot$}\n \\label{fig:nozzle_scaling}\n\\end{figure}\n\nSince we explore the BH parameter space in \\autoref{sec:results_bh}, it is useful to assess how the initial number density and velocity of the wind scale with $M_\\mathrm{BH}$ and $\\dot m$. To this end, we need to determine how the values of $\\varepsilon$ and $\\dot \\Sigma_0$ change with $M_\\mathrm{BH}$ and $\\dot m$. For this analysis, we ignore the dependence of $f_\\text{\\tiny UV}(R)$ on $M$ and $\\dot m$. We also have $v_\\text{th}(R) \\propto T^{1\/2} \\propto (M_\\mathrm{BH} \\dot{M}\/R^3)^{1\/8}$ (using \\autoref{eq:radiation_flux}). Accounting for the fact that the disk size scales as $R \\propto R_\\mathrm{g} \\propto M_\\mathrm{BH}$, this gives $v_0 = v_\\text{th} \\propto (\\dot{m}\/M_\\mathrm{BH})^{1\/8}$, where $v_0$ is the initial velocity. This is a weak dependence, so we ignore it here. Looking at \\autoref{eq:gamma_0}, and using \\autoref{eq:radiation_acceleration_approx}, we get $\\gamma_0 \\propto M_\\mathrm{BH} \\dot{M}\/R^3 \\propto \\dot m \/ M_\\mathrm{BH}$. Similarly, $B_0 \\propto M_\\mathrm{BH}\/R^2 \\propto 1 \/ M_\\mathrm{BH}$. Using \\autoref{eq:Mdot0_def}, we then have $\\dot \\Sigma_0 \\propto \\dot{M}_0\/A_0 \\propto \\gamma_0^{1\/\\alpha}\/B_0^{(1-\\alpha)\/\\alpha} \\propto \\dot m^{1\/\\alpha} \/ M_\\mathrm{BH}$.\n\nThe scaling of $\\varepsilon = \\mathcal{N}(x_c)$ is a bit more complicated, since it depends on the exact position of the critical point for each value of $M_\\mathrm{BH}$, $\\dot m$, and $R$. In \\autoref{fig:nozzle_scaling}, we plot the values of $\\varepsilon$ as a function of radius for varying $M_\\mathrm{BH}$ (left panel) and $\\dot m$ (right panel), ignoring the dependence of $f_\\text{\\tiny UV}$ with $M_\\mathrm{BH}$ and $\\dot m$ (we set $f_\\text{\\tiny UV} = 1$). We note that $\\varepsilon$ does not scale with $M_\\mathrm{BH}$, and changes very little with $\\dot m$. Including the dependence of $f_\\text{\\tiny UV}$ with $M_\\mathrm{BH}$ and $\\dot m$, effectively reduces the value of $\\varepsilon$ at the radii where $f_\\text{\\tiny UV}$ is small, but it does not change substantially in the radii that we would expect to launch an escaping wind. We can then conclude that $\\varepsilon$ has a very weak scaling with $M_\\mathrm{BH}$ and $\\dot m$ so that\n\\begin{equation}\n \\label{eq:sigma_scaling}\n \\dot\\Sigma \\propto \\dot\\Sigma_0 \\propto \\frac{\\dot m^{\\frac{1}{\\alpha}}}{M_\\mathrm{BH}},\n\\end{equation}\nwhich is the same result that \\cite{pereyra_steady_2006} found in applying the \\citetalias{castor_radiation-driven_1975} formalism to cataclysmic variables.\n\n\\subsection{Verification of the critical point conditions}\n\\label{subsection:verify_critical_point}\n\nFor our fiducial case (\\autoref{sec:results}), we plot the critical point location compared to the wind trajectories in \\autoref{fig:verify_critical_point}. All escaping trajectories are vertical at the critical point, so our treatment of the wind as a 1D flow for the initial conditions derivation is justified. Nonetheless, we emphasise again that this treatment does not hold for the inner failed wind. The wind is highly supersonic (~$10^3$ times the sound speed) at the critical point, as shown in the bottom panel of \\autoref{fig:verify_critical_point}, so our assumption $w\\gg s$ is validated.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/verify_critical_point.pdf}\n \\caption{Results for fiducial case $\\protectM_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$ and $\\protect\\dot m =0.5$. Top panel: wind trajectories compared to the critical point position, plotted on a linear scale. Bottom panel: Velocity at the critical point plotted on a log scale}\n \\label{fig:verify_critical_point}\n\\end{figure}\n\\section{Initial conditions}\n\\label{sec:initial_conditions}\n\nAs initial conditions, we determine the density and velocity at the base of the wind following the \\citetalias{castor_radiation-driven_1975} formalism. Let us consider a wind originating from the top of an accretion disc. At low heights, a gas blob is mostly irradiated by the local region of the disc that is just below it. Since this local disc area can be considered to be at a uniform temperature, the direction of the radiation force is mostly upwards, and thus the wind flows initially vertically and can be considered a 1D wind. The corresponding equation describing the vertical motion is\n\n\\begin{equation}\n \\rho \\frac{\\mathrm{d} v_z}{\\mathrm{d} t} = \\rho\\, (a_\\mathrm{grav}^z + a_\\mathrm{rad}^z) - \\frac{\\partial P}{\\partial z},\n\\end{equation}\nEven though the \\textsc{Qwind} model does not include hydrodynamic forces when solving the 2D trajectories of gas parcels, we do include the force term due to gas pressure here, since it is necessary for deriving critical point like solutions (see Appendix \\ref{app:initial_conditions}). \n\nThe study of 1D line-driven winds was pioneered by \\citetalias{castor_radiation-driven_1975}, who defined a framework to find steady state solutions of the 1D wind equation. Their methodology can be extended to any particular geometry of the gravitational and radiation fields, in particular, \\cite{pereyra_steady_2006} (hereafter \\citetalias{pereyra_steady_2006}) apply the \\citetalias{castor_radiation-driven_1975} formalism to the study of cataclysmic variables (CVs). We here aim to further extend this approach to our case, by using the \\citetalias{castor_radiation-driven_1975} formalism to calculate the properties of the 1D wind solutions from an accretion disk as initial conditions for the global 2D wind solution.\n\nThe core result of the \\citetalias{castor_radiation-driven_1975} approach is that if a steady state solution of the 1D wind equation satisfies the following conditions:\n\n\\begin{itemize}\n \\item the velocity increases monotonically with height,\n \\item the wind starts subsonic,\n \\item the wind extends towards arbitrarily large heights,\n \\item the wind becomes supersonic at some height,\n \\item the velocity gradient is a continuous function of position,\n\\end{itemize}\nthen the wind must pass through a special point called the critical point $z_c$, which can be derived without solving the wind differential equation, and thus the global properties of the wind such as its mass loss rate can be determined without resolving the full wind trajectory. To keep the main text concise, we refer the reader to Appendix \\ref{app:initial_conditions} for a detailed derivation.\n\nThe previously specified conditions for the existence of a critical point solution may not be satisfied for all of the wind trajectories that we aim to simulate. For instance, a wind trajectory that starts in an upward direction and falls back to the disc because it failed to achieve the escape velocity does not have a velocity that increases monotonically with height. Furthermore, eventually the wind trajectory is no longer vertical, and the 1D approach breaks down. Having considered these possibilities, and only for the purpose of deriving the initial conditions of the wind, we assume that these conditions hold, so that we can derive the wind mass loss rate at the critical point, which we in turn use to determine the initial conditions of the wind. The full 2D solution of the wind may then not satisfy these conditions. \n\nThe location of the critical point as a function of radius is plotted in \\autoref{fig:critical_points}, where we also plot the height of the disc,\n\\begin{equation}\n z_\\mathrm{h}(R) = \\frac{\\kappa_{\\text{\\tiny e}} \\mu_e \\mathcal F(R) R^3}{GM_\\mathrm{BH} c},\n\\end{equation}\ndefined as the point of equality between the vertical gravitational and radiation force. Overall, we notice that the critical point height increases slowly with radius, except for a bump at $R\\sim 20R_\\mathrm{g}$ which is caused by the UV fraction dependence with radius (see \\autoref{fig:uv_fractions}). For radii $R \\lesssim 50\\, R_\\mathrm{g}$ the critical point height is comparable to the disc radius, so our approximation that streamlines are vertical at that point may not be applicable. We assess the validity of this assumption in \\autoref{subsection:verify_critical_point}. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/critical_points.pdf}\n \\caption{Position of the critical point $z_c$ as a function of radius, for $M_\\mathrm{BH} = 10^8 \\mathrm{M}_\\odot$ and $\\dot m =0.5$.} \n \\label{fig:critical_points}\n\\end{figure}{}\n\nWe assume that the wind originates from the disc surface with an initial velocity $v_0$ equal to the thermal velocity (or isothermal sound speed) at the local disc temperature, \n\\begin{equation}\n \\label{eq:thermal_velocity}\n v_\\text{th}(R) = \\sqrt{\\frac{k_\\mathrm{B} T(R)}{\\mu \\, m_p}}.\n\\end{equation}\nGiven that the critical point is close to the disc surface, and that the wind is supersonic at the critical point (see Appendix \\ref{app:initial_conditions}), this is a good starting point. Since mass conservation holds, the initial number density of the wind can then be calculated as\n\\begin{equation}\n n_0 (R) = \\frac{\\dot \\Sigma (R)}{v_\\text{th}(T(R)) \\, \\mu \\, m_\\mathrm{p}},\n\\end{equation}\nwhere $\\dot \\Sigma(R)$ is the mass loss rate per unit area at the critical point. In \\autoref{fig:initial_conditions}, we plot the initial number density and velocity for $M_\\mathrm{BH} =10^8\\, \\mathrm{M}_\\odot$, $\\dot m =0.5$. We note that the initial velocity stays relatively constant, only varying by a factor of $\\sim 3$ across the radius range. However, the initial number density varies by more than 5 orders of magnitude, showing that the assumption of a constant density at the base of the wind used in the previous versions of the model was poor. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/initial_conditions.pdf}\n \\caption{Initial conditions at the base of the wind as a function of radius. The original \\textsc{Qwind} code assumed \n constant initial density and velocity across the disk, and these were free parameters. We now calculate these from first principles. The left panel shows that the initial density changes by a factor of $10^6$, very different to the original assumption, while the right panel (note change in y axis scale) shows that the initial velocity changes by less than a factor 10. } \n \\label{fig:initial_conditions}\n\\end{figure}{}\n\n\\subsection{Comparison to other models}\n\nAs a partial test of this new section of the code, we can compare our findings with \\cite{nomura_modeling_2013} (hereafter \\citetalias{nomura_modeling_2013}), in which the authors also use the sonic velocity as the initial velocity of the wind, and derive the initial density profile by assuming the same functional form for the mass loss rate as in \\citetalias{castor_radiation-driven_1975}, but using the AGN $M_\\mathrm{BH}$ and $\\dot m$ instead. We note that the use of the CAK formula directly for accretion discs may not be appropriate because the geometry of the system is very different from stellar winds. Additionally, in \\citetalias{nomura_modeling_2013} the dependence of $f_\\text{\\tiny UV}$ on the disc radius is not considered. Hence we first hardwire $f_\\text{\\tiny UV}$ for the comparison. This comparison is shown in the upper panels of \\autoref{fig:nomura_comparison} for different $M_\\mathrm{BH}$ (left, all at $\\dot{m}=0.5$), and (right) for $M_\\mathrm{BH}$ fixed at $10^8\\,\\mathrm{M}_\\odot$ with different $\\dot{m}$. It is clear that the initial density now derived in \\textsc{Qwind3} (dashed lines) has a steeper decrease with radius than in \\citetalias{nomura_modeling_2013}. This is probably due to their use of the direct CAK formula, which assumes a spherical geometry rather than an accretion disc. However, the inferred densities are within an order of magnitude of each other, and both approaches give a linear scaling of the initial number density profile with $M_\\mathrm{BH}$, but \\textsc{Qwind3} gives an almost quadratic scaling with $\\dot m$, compared to a linear one for \\citetalias{nomura_modeling_2013} (see \\autoref{subsection:ic_scaling}).\n\nThe lower panels of \\autoref{fig:nomura_comparison} show instead the comparison of the \\textsc{Qwind} models using the self consistent $f_\\text{\\tiny UV}$ with its radial dependence (solid lines) instead of assuming $f_\\text{\\tiny UV}=1$ (dashed lines). There is a very strong drop in the initial density at radii where $f_\\text{\\tiny UV}$ drops (see Fig. \\ref{fig:uv_fractions}). This shows the importance of including the self-consistent calculation of $f_\\text{\\tiny UV}$ in \\textsc{Qwind3}. \n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/nomura_comparison.pdf}\n \\caption{Initial density as a function of radius. In the top panels we compare with the results of \\protect\\citetalias{nomura_modeling_2013} (solid lines), the only other diskwind code which used UV line driving to calculate the initial density. That code assumes constant $f_{UV}$, so we fix $f_\\text{\\tiny UV}=1$ in \\textsc{Qwind} to compare results (dashed lines). In the bottom panels we compare the \n \\textsc{Qwind} results with $f_\\text{\\tiny UV}=1$ (dashed lines) with the full \\textsc{Qwind} results for $f_\\text{\\tiny UV}(R_\\mathrm{d})$. Left and right panels show results for fixed $\\dot m = 0.5$, and vary $\\protectM_\\mathrm{BH} \\in (10^7, 10^8, 10^9) \\, \\mathrm{M}_\\odot$. Right panels: We fix $M_\\mathrm{BH} = 10^8\\,\\mathrm{M}_\\odot$ and vary $\\protect \\dot m \\in(0.05, 0.1, 0.5)$.}\n \\label{fig:nomura_comparison}\n\\end{figure}{}\n\n\n\n\n\n\\section{Introduction}\n\nAGN feedback is a very important process in shaping the growth of galaxies, but the prescriptions for it that are included in current cosmological simulations are generally highly simplified and \nnot based on any deeper understanding of the physical processes involved. Jets from AGN are poorly understood, but some types of AGN winds can be calculated ab initio from the fundamental parameters of black hole mass, mass accretion rate and spin. Observations show the existence of ultra-fast outflows (UFOs) in AGN, likely originating from the accretion disc close to the central supermassive black hole (BH). These outflows can reach velocities of $v$ $\\sim (0.03-0.3)\\,c$ \\citep{weymann_comparisons_1991, pounds_evidence_2003, pounds_high-velocity_2003, reeves_compton-thick_2009, crenshaw_feedback_2012, tombesi_evidence_2010, fiore_agn_2017}. UV line driving is a mechanism which is especially likely to be present in luminous AGN, with their accretion disc spectrum peaking in the UV, where there are multiple strong atomic transitions in low ionisation material. These transitions absorb the photon momentum, producing the strong winds seen from similar temperature material in O star photospheres \\citep{howarth_stellar_1989}, which were first extensively studied by \\cite{castor_radiation-driven_1975} (hereafter \\citetalias{castor_radiation-driven_1975}) and \\cite{abbott_theory_1980}. In the context of AGN, the study of UV line-driven winds started with analytical studies \\citep{murray_accretion_1995}, and continued with the use of radiation-hydrodynamic simulations \\citep{proga_radiation-driven_1998, nomura_radiation_2016}. However, the computational complexity of the radiation-hydrodynamics codes prevents us from efficiently exploring the input parameter space. Even more importantly, the complexity of these codes can obscure the effect of some of the underlying assumptions, e.g. the lack of scattered emission, on setting the radiation environment \\citep{higginbottom_line-driven_2014}, or the effect of wind mass loss on the net accretion rate and hence the disc emission \\citep{nomura_line-driven_2020}. \n\nTo circumvent this, we build on the pioneering approach of \\textsc{Qwind} \\citep{risaliti_non-hydrodynamical_2010} in developing a non-hydrodynamic code. This calculates ballistic trajectories, ignoring pressure forces (which should be negligible in a supersonic flow) but including gravity and radiation forces, to obtain the streamlines, making the computer code much faster, and simpler, so that it can be used to explore the parameter space much more fully. In \\cite{quera-bofarull_qwind_2020} (hereafter \\citetalias{quera-bofarull_qwind_2020}) we released a modern version of this code (\\textsc{Qwind2}), but this was still based on some underlying, arbitrary parameter choices, including for the launching of the wind from the accretion disk, and used simplified radiation transport. \nHere, we aim to significantly improve the predictive power of the \\textsc{Qwind} code. We present a model to derive the initial conditions of the wind, a radiative transfer algorithm that takes into account the wind geometry and density structure, and we include special relativistic corrections to our calculations of the radiation force on the wind. We then use this new model, which we refer as \\textsc{Qwind3}, to study the dependence of the mass loss rate and kinetic power of the wind on the BH mass and mass accretion rate. These results can form the basis for a physical prescription for AGN feedback that can be used in cosmological simulations to explore the coeval growth of galaxies and their central black holes across cosmic time. \n\n\\section*{Acknowledgements}\n\nAQB acknowledges the support of STFC studentship (ST\/P006744\/1) and the JSPS London Pre\/Postdoctoral Fellowship for Foreign Researchers. CD and CGL acknowledge support from STFC consolidated grant ST\/T000244\/1. CD acknowledges support for vists to Japan from Kavli Institute for the Physics\nand Mathematics of the Universe (IPMU) funding from the National\nScience Foundation (No. NSF PHY17-48958).\nThis work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST\/K00042X\/1, ST\/P002293\/1, ST\/R002371\/1 and ST\/S002502\/1, Durham University and STFC operations grant ST\/R000832\/1. DiRAC is part of the National e-Infrastructure. This work was supported in part by JSPS Grant-in-Aid for Scientific Research (A) JP21H04488 (KO), Scientific Research (C) JP18K03710 (KO), Early-Career Scientists JP20K14525 (MN). This work was also supported by MEXT as \"Program for Promoting Researches on the Supercomputer Fugaku\" (Toward a unified view of the universe: from large scale structures to planets, JPMXP1020200109) (KO), and by Joint Institute for Computational Fundamental Science (JICFuS, KO). This paper made use of the Matplotlib \\citep{hunter_matplotlib_2007} and SciencePlots \\citep{garrett_scienceplots_2021} software packages.\n\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Review of Qwind}\n\nThe \\textsc{Qwind} code of Q20 is based on the approach of \\cite{risaliti_non-hydrodynamical_2010}, which calculates ballistic trajectories of gas blobs launched from the accretion disc. These blobs are subject to two forces: the gravitational pull of the BH, and the outwards pushing radiation force, which can be decomposed into an X-ray and a UV component, with the later being dominant. On the one hand, the X-ray photons couple to the accretion disc material via bound-free transitions with outer electrons, ionising the gas. On the other hand, the UV opacity is greatly enhanced when the material is not over-ionised, since the UV photons can then excite electrons to higher states while transferring their momentum to the gas in the process. If sufficient momentum is transferred from the radiation field to the gas, the latter may eventually reach the gravitational escape velocity, creating an outgoing flow. This mechanism for creating a wind is known as UV line-driving. The conditions under which the wind can escape depend on the density and velocity structure of the flow, where part of the material can be shielded from the X-ray radiation while being illuminated by the UV emitting part of the accretion disc. \n\n\\subsection{Radiation force}\n\nUsing a cylindrical coordinate system $(R, \\phi, z)$, with $r^2 = R^2 + z^2$, let us consider a BH of mass $M_\\mathrm{BH}$, located at $r=0$, accreting mass at a rate $\\dot M$, and an accretion disc located at the $z=0$ plane. We use the gravitational radius, $R_g = G M_\\mathrm{BH} \/ c^2$, as our natural unit of length. The total luminosity of the system is related to the accreted mass through\n\\begin{equation}\n L_\\text{bol} = \\eta \\dot M c^2\n\\end{equation}\nwhere $\\eta$ is the radiation efficiency. We set $\\eta = 0.057$ throughout this work, as we only consider non-rotating BHs \\citep{thorne_disk-accretion_1974}. We frequently refer to the Eddington fraction $\\dot m = \\dot M \/ \\dot M_\\text{Edd}$, where $\\dot M_\\text{Edd}$ is the mass accretion rate corresponding to the Eddington luminosity,\n\\begin{equation}\n \\dot M_\\text{Edd} = \\frac{L_\\text{Edd}}{\\eta c^2} = \\frac{4\\pi G M_\\mathrm{BH}}{\\eta c \\kappa_{\\text{\\tiny e}}},\n\\end{equation}\nwhere $\\kappa_{\\text{\\tiny e}}$ is the electron scattering opacity, related to the electron scattering cross section $\\sigma_{\\text{\\tiny T}}$ through\n\\begin{equation}\n \\kappa_{\\text{\\tiny e}} = \\frac{\\sigma_{\\text{\\tiny T}}}{m_p \\, \\mu_e},\n\\end{equation}\nwhere $m_p$ is the proton mass, and $\\mu_e$ is the mean molecular weight per electron. We set $\\mu_e = 1.17$ corresponding to a fully ionised gas with solar chemical abundance \\citep{asplund_chemical_2009}.\n\nThe emitted UV radiated power per unit area by a disc patch located at $(R_\\mathrm{d}, \\phi_\\mathrm{d}, 0)$ is given by \\citep{shakura_black_1973}\n\\begin{equation}\n \\label{eq:radiation_flux}\n \\mathcal F_{\\rm UV} = \\frac{3 G M_\\mathrm{BH} \\dot M}{8\\pi R_\\mathrm{d}^3} f_\\text{\\tiny UV}(R_\\mathrm{d}) f_\\text{\\tiny NT}(R_\\mathrm{d}, R_\\mathrm{isco}),\n\\end{equation}\nwhere $f_\\text{\\tiny NT}$ are the Novikov-Thorne relativistic factors \\citep{novikov_astrophysics_1973}, and $f_\\text{\\tiny UV}$ is the fraction of power in the UV band, which we consider to be (200--3200) \\AA. (The total power radiated per unit area $\\mathcal F$ is given by setting $f_\\text{\\tiny UV}=1$ in the above equation.) The force per unit mass exerted on a gas blob at a position $(R, 0, z)$ due to electron scattering is (see \\citetalias{quera-bofarull_qwind_2020})\n\\begin{equation}\n \\label{eq:radiation_acceleration}\n \\bmath{a}_\\mathrm{rad}^\\mathrm{es}(R,z) = \\mathcal C \\,z\\int\\int\\frac{f_\\text{\\tiny UV} f_\\text{\\tiny NT}}{R_\\mathrm{d}^2 \\, \\Delta^4} e^{-\\tau_\\text{\\tiny UV}} \\begin{pmatrix}R-R_\\mathrm{d}\\cos\\phi_\\mathrm{d}\\\\ -R_\\mathrm{d} \\sin \\phi_\\mathrm{d}\\\\ z \\end{pmatrix} \\, \\mathrm{d} R_\\mathrm{d} \\mathrm{d} \\phi_\\mathrm{d},\n\\end{equation}\nwhere \n\\begin{equation}\n \\mathcal C = \\frac{3 G M_\\mathrm{BH} \\dot M \\kappa_{\\text{\\tiny e}}}{8 \\pi^2 c},\n\\end{equation}\n$\\Delta^2 = R^2 + z^2 - 2 R R_\\mathrm{d} \\cos\\phi_\\mathrm{d}$, and $\\tau_\\text{\\tiny UV}$ is the UV optical depth measured from the disc patch to the gas blob. We note that it is enough to consider the case $\\phi=0$ due to the axisymmetry of the system, and, furthermore, the $\\phi$ component of the force vanishes upon integration. The total radiation force can be greatly amplified by the contribution from the line opacity, which we parameterise as $\\kappa_\\mathrm{line} = \\mathcal{M} \\,\\kappa_{\\text{\\tiny e}}$, such that the total radiation opacity is $(1 + \\mathcal{M}) \\kappa_{\\text{\\tiny e}}$, implying that\n\\begin{equation}\n \\bmath{a}_\\mathrm{rad} = (1 + \\mathcal M) \\; \\bmath{a}_\\mathrm{rad}^\\mathrm{es}.\n\\end{equation}\nThe parameter $\\mathcal M$ is known as the force multiplier, and we use the same parametrisation as \\citetalias{quera-bofarull_qwind_2020} \\citep{stevens_x-ray_1990} (hereafter \\citetalias{stevens_x-ray_1990}). A limitation of our assumed parametrisation is that we do not take into account the dependence of the force multiplier on the particular spectral energy distribution (SED) of the accretion disc \\citep{dannen_photoionization_2019}. Furthermore, the force multiplier is also expected to depend on the metallicity of the gas \\citep{nomura_radiation_2021}. A self consistent treatment of the force multiplier with relation to the accretion disc and its chemical composition is left to future work. \n\nIt is useful to consider that, close to the disc's surface, the radiation force is well approximated by considering the radiation force produced by an infinite plane at a temperature equal to the local disc temperature,\n\\begin{equation}\n \\label{eq:radiation_acceleration_approx}\n \\bmath{a}_\\mathrm{rad, 0}^\\mathrm{es}(R) = \\frac{3 GM_\\mathrm{BH} \\dot M \\kappa_{\\text{\\tiny e}}}{8\\pi^2 R^3 c} f_\\text{\\tiny UV} f_\\text{\\tiny NT}\\, e^{-\\tau_\\text{\\tiny UV}}\\; \\bmath{\\hat{z}},\n\\end{equation}\nwhere $\\tau_\\text{\\tiny UV}$ is calculated along a vertical path. The radiation force is then vertical and almost constant at small heights \n($z \\lesssim 0.1 R_\\mathrm{g}$).\nTo speed up calculations and minimise numerical errors, we use this expression when $z < 0.01 R_\\mathrm{g}$.\n\n\n\\subsection{Equations of motion}\n\nThe equations of motion of the gas blob trajectories are\n\n\\begin{equation}\n\\label{eq:trajectory_ode}\n \\begin{split}\n &\\frac{\\mathrm{d} R}{\\mathrm{d} t} &=& \\; v_R,\\\\\n \n &\\frac{\\mathrm{d} z}{\\mathrm{d} t} &= &\\; v_z,\\\\\n &\\frac{\\mathrm{d} v_R}{\\mathrm{d} t} &= &\\; a^\\mathrm{grav}_R + a^\\mathrm{rad}_R + \\frac{\\ell^2}{R^3},\\\\\n \n &\\frac{\\mathrm{d} v_z}{\\mathrm{d} t} &= &\\; a^\\mathrm{grav}_z + a^\\mathrm{rad}_z,\n \\end{split}\n\\end{equation}\nwhere $\\ell$ is the specific angular momentum (assumed constant for a given blob), and $\\bmath{a}_\\mathrm{grav}$ is the gravitational acceleration,\n\\begin{equation}\n\\label{eq:gravity}\n\\bmath{a}_\\mathrm{grav}\\,(R,z) = -\\frac{GM_\\mathrm{BH}}{r^2}\\,\\begin{pmatrix}R\/r \\\\ 0 \\\\ z \/ r\\end{pmatrix}.\n\\end{equation}\nWe assume that initially the gas blobs are in circular orbits around the BH, so that $\\ell = \\sqrt{G M_\\mathrm{BH} R_0}$, where $R_0$ is the launch radius, and thus the azimuthal velocity component at any point is $v_\\phi = \\ell \/ r$.\n\nAs in \\citetalias{quera-bofarull_qwind_2020}, we ignore contributions from gas pressure (except when calculating the launch velocity and density) as these are negligible compared to the radiation force, especially since we focus our study on the supersonic region of the wind. We assume that the distance between two nearby trajectories at any point, $\\Delta r$, is proportional to the distance to the centre $\\Delta r \\propto r$, so that the surface mass loss rate along a particular streamline is $\\dot \\Sigma = \\dot M_\\mathrm{streamline} \/ (2 \\pi \\, r_0 \\, \\Delta r_0)$. This both captures the fact that streamlines are mostly vertical close to the disc, and diverge in a cone-like shape at large distances. The gas blob satisfies the approximate mass conservation equation along its trajectory (\\citetalias{quera-bofarull_qwind_2020}),\n\\begin{equation}\n \\label{eq:mass_conservation}\n \\dot M_\\mathrm{streamline} = 2 \\, \\pi \\, r \\, \\Delta r \\, \\rho \\, v,\n\\end{equation}\nwhere $\\rho$ is the density of the wind, related to the number density $n$ through $\\rho = \\mu \\, m_p \\, n$, and $\\Delta r = (r \/ r_0) \\Delta R_0$. We set the mean molecular weight $\\mu$ to $\\mu = 0.61$, corresponding to a fully ionised gas with solar abundance \\citep{asplund_chemical_2009}. We use \\autoref{eq:mass_conservation} to determine the gas density at each point along a trajectory.\n\n\\subsection{Improvements to \\textsc{Qwind}}\n\nIn \\citetalias{quera-bofarull_qwind_2020}, a series of assumptions are made to facilitate the numerical solution of the presented equations of motion. Furthermore, the initial conditions of the wind are left as free parameters to explore, limiting the predictive power of the model. In this work, we present a series of important improvements to the \\textsc{Qwind} code. \nFirstly we calculate $f_\\text{\\tiny UV}$ from the disc spectrum rather than have this as a free parameter (\\autoref{sec:fuv}).\nSecondly, we derive the initial conditions of the wind in \\autoref{sec:initial_conditions}, based on the methodology introduced in \\citetalias{castor_radiation-driven_1975}, and further developed in \\cite{abbott_theory_1982, pereyra_steady_2004, pereyra_further_2005, pereyra_steady_2006}. This removes the wind's initial velocity and density as degrees of freedom of the system. Thirdly, we vastly enhance the treatment of the radiative transfer in the code, reconstructing the wind density and velocity field from the calculated gas trajectories. This allows us to individually trace the light rays coming from the accretion disc and the central X-ray source, correctly accounting for their attenuation. This is explained in detail in \\autoref{sec:radiation_transport}. Lastly, we include the relativistic corrections from \\cite{luminari_importance_2020} in the calculation of the radiation force, solving the issue of superluminal winds, and we later compare our findings to \\cite{luminari_speed_2021}.\nReaders interested only in the results should skip to Section \\ref{sec:results}. \n\nThe improvement in the modelling of the physical processes comes at the expense of added computational cost. We have ported the \\textsc{Qwind} code to the Julia programming language \\citep{bezanson_julia_2017}, which is an excellent framework for scientific computing given its state of the art performance, and ease of use. The new code is made available to the community under the GPLv3 license on GitHub\\footnote{https:\/\/github.com\/arnauqb\/Qwind.jl}.\n\n\\section{Updates to the radiation transport}\n\\label{sec:radiation_transport}\n\nIn \\citetalias{quera-bofarull_qwind_2020}, the radiation transfer is treated in a very simple way. The disc atmosphere (i.e. the wind) is assumed to have constant density, and so the line of sight absorption does not take into consideration the full geometry and density structure of the wind (see section 2 of \\citetalias{quera-bofarull_qwind_2020}). Furthermore, the UV optical depth is measured from the centre of the disc, and assumed to be the same for radiation from all disc patches, regardless of the position and angle relative to the gas parcel. In this section, we improve \\textsc{Qwind}'s radiative transfer model, by reconstructing the wind density from the gas blob trajectories, thus accounting fully for the wind geometry. The disc is assumed to be flat and thin, with constant height $\\bar z_\\mathrm{h} = 0$, thus we do not model the effect of the disc itself on the radiation transfer. To illustrate the improvements, we consider our fiducial model with $M_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$, and $\\dot m=0.5$, and present the ray tracing engine of the code in an arbitrary wind solution. We discuss particular physical implications and results in \\autoref{sec:results}.\n\n\\begin{figure}\n \\includestandalone[width=\\columnwidth]{diagram}\n \\caption{Diagram of the disc-wind geometry. The blue line corresponds to the light path an X-ray photon takes, while the violet line corresponds to an example of a UV light ray from the accretion disc.}\n \\label{fig:geometry_diagram}\n\\end{figure}\n\n\\subsection{Constructing the density interpolation grid}\n\\label{sec:interp_grid}\n\nGiven a collection of trajectories, we aim to obtain the wind density field at every point in space. The first step is to delimit where the wind is spatially located by computing the concave hull that contains all the points of all wind trajectories. We use the algorithm described in \\cite{moreira_concave_2007} and implemented in \\cite{stagner_lstagnerconcavehulljl_2021}. The resulting concave set is illustrated in \\autoref{fig:wind_hull}. Outside the concave hull, the density is set to the vacuum density which is defined to be $n_\\text{vac} = 10^2$ cm$^{-3}$. \nSince the density varies by orders of magnitude within the wind, we compute the density at a point by linearly interpolating $\\log_{10} n$ in logarithmic space ($\\log R$ - $\\log z$) from the simulated wind trajectories. We use the interpolation algorithm \\textsc{LinearNDInterpolator} from \\textsc{SciPy} \\citep{virtanen_scipy_2020}. The resulting density map is shown in \\autoref{fig:density_grid}. We note that using a concave hull envelope is important, since the interpolation algorithm we use restricts the interpolation space to the convex hull of the input points, which, due to non-convexity of the wind geometry, would otherwise lead us to overestimate the obscuration in certain regions.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/wind_hull.pdf}\n \\caption{Gas parcel trajectories encapsulated by the concave hull containing the points. Left panel on linear scale and right panel on logarithmic scale. Note that the non-convexity of the wind prevents us from using a simpler convex hull envelope.}\n \\label{fig:wind_hull}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/density_grid.pdf}\n \\caption{Interpolated density grid from the simulated wind positions and densities. Left panel: position in linear scale. Right panel: position in logarithmic scale.}\n \\label{fig:density_grid}\n\\end{figure}\n\nOnce we have built the interpolator, we construct a rectilinear grid with the interpolated values. This allows us to implement an efficient ray tracing algorithm to compute the UV and X-ray optical depths. The vertical coordinates of the grid nodes are logarithmically spaced from $10^{-6}$ $R_g$ to the wind's maximum height. The horizontal coordinates are taken at the initial positions of the wind's trajectories, plus an additional range logarithmically spaced from the initial position of the last streamline to the highest simulated $R$ coordinate value.\n\n\n\\subsection{Ray tracing}\n\nTo compute the optical depth along different lines of sight, we need to calculate an integral along a straight path starting at a disc point $(R_\\mathrm{d}, \\phi_\\mathrm{d}, 0)$ to a point $(R, \\phi, z)$. Due to the axisymmetry of the system, the radiation acceleration is independent of $\\phi$ so we can set $\\phi = 0$. We can parametrise the curve in the Cartesian coordinate system with a single parameter $t\\in [0,1]$, (see \\autoref{fig:geometry_diagram}),\n\\begin{equation}\n \\begin{split}\n x(t) &= R_\\mathrm{d} \\cos\\phi_\\mathrm{d} \\, (1-t) + t \\, R , \\\\\n y(t) &= R_\\mathrm{d} \\sin\\phi_\\mathrm{d} \\, (1-t),\\\\\n z(t) &= t \\, z\\\\\n \\end{split}\n\\end{equation}\nso that the cylindrical radius varies along the path as\n\\begin{equation}\n R_t^2(t) = R_\\mathrm{d}^2(1-t)^2 + t^2 R^2 + 2R_\\mathrm{d} R \\cos\\phi_\\mathrm{d} \\,t (1-t) ,\n\\end{equation}\nand $\\phi(t) = \\arctan{(y(t) \/ x(t))}$. In this parametrisation, $t=0$ points to the disc plane point, and $t=1$ to the illuminated wind element. The integral to compute is thus\n\\begin{equation}\n \\tau = \\Delta \\, \\int_0^1 \\sigma(t)\\; n(t)\\, \\mathrm{d} t,\n\\end{equation}\nwhere $\\Delta$ is the total path length defined earlier.\nGiven a rectilinear density grid in the R-z plane, we compute the intersections of the light ray with the grid lines, $\\{ R_i, z_i\\}$, such that we can discretise the integral as,\n\\begin{equation}\n \\tau \\approx \\sum_i \\sigma(R_i, z_i) \\, n(R_i, z_i)\\Delta d_i,\n\\end{equation}\nwhere $\\Delta d_i$ is the 3D distance between the $i$-th intersection point and the $(i-1)$-th,\n\\begin{equation}\n \\Delta d_i = \\sqrt{R_i^2 + R_{i-1}^2 + (z_i - z_{i-1})^2 -2 R_i R_{i-1} \\cos(\\phi_i-\\phi_{i-1})},\n\\end{equation}\nTo find the intersections, we need to calculate whether the light ray crosses an $R_i$ grid line or a $z_i$ grid line. We start at the initial point $(R_\\mathrm{d}, \\phi_\\mathrm{d}, 0)$, and compute the path parameter $t_R$ to hit the next $R$ grid line $R_i$ by solving the second degree equation $R(t_R) = R_i$, and similarly for the next $z_i$ line, $t_z = z_i \/ z$. The next intersection is thus given by the values of $R(t_m)$ and $z(t_m)$ where $t_m = \\min(t_R, t_z)$. Geometrically, the projection of the straight path onto the $(R-z)$ grid is in general a parabola as we can see in \\autoref{fig:tikz:ray_tracing}. \n\n\\begin{figure}\n \\includestandalone[width=\\columnwidth]{ray_tracing_diagram}\n \\caption{Projections of two typical light rays onto the $R-z$ interpolation grid. The X-ray radiation (path with blue dots) is assumed to come from the centre of the grid $(0,0)$, so the projection of the light curve onto the $R-z$ grid is always a straight line. However, for the UV case (purple dots), the light ray can originate from any $\\phi_\\mathrm{d}$, so the path on the interpolation grid is, in general, a parabola.}\n \\label{fig:tikz:ray_tracing}\n\\end{figure}\n\n\\subsection{X-ray optical depth}\n\nThe X-ray opacity depends on the ionisation level of the gas and is assumed to have the same functional form as \\citetalias{quera-bofarull_qwind_2020},\n\\begin{equation}\n \\label{eq:xray_opacity}\n \\sigma_\\text{\\tiny X}(\\xi) = \\begin{cases}\\sigma_{\\text{\\tiny T}} & \\text{ if } \\xi > 10^5 \\text{erg cm s}^{-1}\\\\ 100 \\sigma_{\\text{\\tiny T}} & \\text{ if }\\xi \\leq 10^5 \\text{erg cm s}^{-1}\\end{cases},\n\\end{equation}\nwhere $\\sigma_{\\text{\\tiny T}}$ is the Thomson scattering cross section, $\\xi$ is the ionisation parameter,\n\\begin{equation}\n \\xi = \\frac{4\\pi F_\\text{\\tiny X}}{n},\n\\end{equation}\nand $F_\\text{\\tiny X}$ is the X-ray radiation flux, $F_\\text{\\tiny X} = L_\\text{\\tiny X} \\exp(-\\tau_\\text{\\tiny X}) \/ (4\\pi r^2)$. We notice that to compute the value of $\\tau_\\text{\\tiny X}$ we need to solve an implicit equation, since the optical depth depends on the ionisation state of the gas which in turn depends on the optical depth. One thus needs to compute the distance $d_\\text{\\tiny X}$ at which the ionisation parameter drops below $\\xi_0 = 10^5 \\text{ erg cm s}^{-1}$,\n\\begin{equation}\n \\xi_0 - \\frac{L_\\text{\\tiny X}}{n d_\\text{\\tiny X}^2} \\exp{(-\\tau_\\text{\\tiny X})} = 0.\n\\end{equation}\nThis equation needs to be tested for each grid cell along the line of sight as it depends on the local density value $n$. Therefore for a cell at a distance $d_{0_i}$ from the centre, density $n_i$, and intersection length $\\Delta d_i$ with the light ray (see \\autoref{fig:tikz:ray_tracing}), the contribution $\\Delta \\tau_\\text{\\tiny X}$ to the optical depth is\n\\begin{equation}\n \\Delta \\tau_\\text{\\tiny X} = \\sigma_{\\text{\\tiny T}} \\, n_i \\cdot \\left[\\max(0, d_\\text{\\tiny X}-d_{0_i}) + 100 \\cdot \\max(0, \\Delta d_i - (d_\\text{\\tiny X} - d_{0_i}))\\right],\n\\end{equation}\nwhere $d_\\text{\\tiny X}$ is calculated from\n\\begin{equation}\n \\xi_0 - \\frac{L_\\text{\\tiny X}}{n_i d_\\text{\\tiny X}^2} \\exp{\\left(-\\tau_{0_i} - \\sigma_{\\text{\\tiny T}} \\cdot n_i \\cdot (d_\\text{\\tiny X} - d_{0_i})\\right)} = 0,\n\\end{equation}\nwhere $\\tau_{0_i}$ is the accumulated optical depth from the centre to the current position. We solve the equation numerically using the bisection method. In \\autoref{fig:xray_grid} we plot the X-ray optical depth grid for our example wind. We observe that there is a region where $\\tau_\\text{\\tiny X} \\gg 1$ at very low heights. This shadow is caused by the shielding from the inner wind, which makes the ionisation parameter drop below $\\xi_0$, substantially increasing the X-ray opacity. As we will later discuss in the results section, the shadow defines the acceleration region of the wind, where the force multiplier is very high.\n\n\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/xray_tau_grid.pdf}\n \\caption{X-ray optical depth as a function of position, measured from $R=0$ and $z=0$. Left panel: position in linear scale. Right panel: position in logarithmic scale.}\n \\label{fig:xray_grid}\n\\end{figure}\n\n\\subsection{UV optical depth}\n\nThe UV opacity calculation is significantly simpler than that for the X-rays, since we assume that the line shift due to the Doppler effect in an accelerating wind is sufficient to always reveal fresh, unabsorbed continuum, so that the opacity is constant at the Thomson (electron scattering) value, $\\sigma(R_i, z_i) = \\sigma_{\\text{\\tiny T}}$. \nIn \\autoref{fig:uv_grid}, we plot the UV optical depth as a function of $R$ and $z$ for light rays originating at the disc position $R_\\mathrm{d} = 500$, $\\phi_\\mathrm{d}=0$. Nevertheless, there are many more sight-lines to consider as the UV emission is distributed over the disc, making this ray tracing calculation the highest contributor to the computational cost of the model. The total UV flux and its resultant direction at any given position in the wind have to be calculated as the sum over each disc element (see \\autoref{eq:radiation_flux}) where now $\\tau_\\text{\\tiny UV} = \\tau_\\text{\\tiny UV}(R_d, \\phi_d, R, z)$. \n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/uv_tau_grid.pdf}\n \\caption{UV optical depth as a function of position, measured from $R=500R_\\mathrm{g}$ and $z=0$. Left panel: position in linear scale. Right panel: position in logarithmic scale.}\n \\label{fig:uv_grid}\n\\end{figure}\n\nAs already noted in \\citetalias{quera-bofarull_qwind_2020}, the integral in \\autoref{eq:radiation_acceleration} is challenging to calculate numerically, and in this case the computational cost is further increased by the refined UV ray tracing. In spite of that, by using the adaptive integration scheme presented in \\cite{berntsen_adaptive_1991} and implemented in \\cite{johnson_juliamathcubaturejl_2021}, the integration typically converges after $\\mathcal O(10^4)$ integrand evaluations, which results in a computation time of a few milliseconds, keeping the simulation tractable. At low heights, $z\\sim 0$, the trajectory solver requires several evaluations of the radiation force to correctly compute the adaptive time-step, which makes the computation particularly slow. Fortunately, the approximation in \\autoref{eq:radiation_acceleration_approx} comes in very handy at reducing the overall computational cost at low heights.\n\n\n\\section{Special relativity effects}\n\\label{sec:relativistic}\nWhen the gas trajectory approaches the speed of light, one should consider special relativistic effects such as relativistic beaming and Doppler shifting \\citep[see eg][chapter 4]{rybicki_radiative_1986}. The importance of taking these effects into account is highlighted in \\cite{luminari_importance_2020}. We include a correction to the radiation flux seen by the gas (\\autoref{eq:radiation_flux}),\n\\begin{equation}\n \\mathrm{d} F_\\text{relativistic}= \\Psi(R_d, \\phi_d, R, z, v_R, v_z) \\; \\mathrm{d} F,\n\\end{equation}\nwhere $v_R$ and $v_z$ are the radial and vertical velocity components of the gas at the position $(R, 0, z)$. We ignore the contribution from the angular velocity component, $v_{\\phi}$ for simplicity, as its inclusion would break our assumption that angular momentum is conserved along a gas blob trajectory. The correction $\\Psi$ is given by \\citep{luminari_speed_2021},\n\\begin{equation}\n \\Psi = \\frac{1}{\\gamma^4 (1+\\beta\\cos\\theta)^4},\n\\end{equation}\nwhere $\\gamma$ is the Lorentz factor, $\\beta = \\sqrt{v_R^2 + v_z^2} \/ c$, and $\\theta$ is the angle between the incoming light ray and the gas trajectory,\n\\begin{equation}\n \\cos\\theta = \\frac{(R - R_d\\cos\\phi_d) v_R + z v_z}{\\beta \\Delta}.\n\\end{equation}\nIntuitively, when the incoming light ray is parallel to the gas trajectory, $\\cos\\theta=1$, so the correction reduces to $\\Psi = \\left(\\frac{1-\\beta}{1+\\beta}\\right)^2$, which is 0 when $\\beta=1$ and 1 when $\\beta=0$, as expected. \n\nIt is worth noting that this is a local correction which needs to be integrated along all the UV sight-lines (see \\autoref{eq:radiation_acceleration}). Nonetheless, the computational cost of calculating the radiation force is heavily dominated by ray tracing and the corresponding UV optical depth calculation, so this relativistic correction does not significantly increase the computation time. \n\nThe X-ray flux, which determines the ionisation state of the gas, is also likewise corrected for these special relativistic effects, but there is only one such sight-line to integrate along for any position in the wind, as the X-ray source is assumed to be point-like. \n\n\n\\section{Results}\n\\label{sec:results}\n\nHere, we evaluate the dependence of the wind properties on the initial wind radius, BH mass, and mass accretion rate. We also study the impact of the relativistic corrections on the wind velocity and structure. All of the parameters that are not varied are specified in \\autoref{table:fixed_parameters}.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ c c c }\n Parameter & Value \\\\ \n \\hline\\hline \n $f_\\mathrm{\\tiny X}$ & 0.15 \\\\\n $z_0$ & 0 \\\\\n $R_\\text{out}\/ R_\\mathrm{g}$ & 1580 \\\\\n $\\mu$ & 0.61 \\\\ \n $\\mu_e$ & 1.17 \\\\ \n $\\alpha$ & 0.6 \\\\\n $k_\\text{ic}$ & 0.03 \\\\\n\\end{tabular}\n\\caption{\\label{table:fixed_parameters} Fixed parameters for the results section. Note that $k_\\text{ic}$ refers to the value of $k$ used to compute the initial conditions of the wind (\\autoref{eq:fm_simple}), but we use the SK90 parametrisation $k=k(\\xi)$ elsewhere.}\n\\end{table}\n\n\\subsection{The fiducial case}\n\nTo gain some intuition about the structure of the wind trajectories solutions, we first have a close look at our fiducial simulation with $M_\\mathrm{BH} = 10^8\\, \\mathrm{M}_\\odot$, $\\dot m =0.5$, and $R_\\mathrm{in} = 50\\,R_\\mathrm{g}$. We run the simulation iterating 50 times through the density field, to make sure that our density grid has converged (see \\autoref{sec:interp_grid}). \n\nIn \\autoref{fig:failed_wind}, we plot the wind streamline shapes, zooming in on the innermost region where we also show the ionisation state of the gas. \\autoref{fig:initial_conditions} shows that the initial density should be $\\sim 3\\times 10^{12}$~cm$^{-3}$. Hence the initial ionisation parameter at $R_\\mathrm{in}$ is $\\xi=f_\\text{\\tiny X}\\, 0.5\\, L_\\mathrm{Edd}\/(n_\\mathrm{in} R_\\mathrm{in}^2)\\sim 800$ so the base of the wind is already in the regime where the X-ray opacity is high. \nThe wind starts, but the drop in density as the material accelerates means it reaches a high ionisation parameter where the force multiplier is low before it reaches escape velocity. Hence the material falls back to the disc as a failed wind region. \n\nThe failed wind region has a size characterised by $\\tau_x \\lesssim 5$, acting as a shield to the outer wind from the central X-ray source. The X-ray obscuration is especially large in the failed wind shadow, due to the jump in X-ray opacity at $\\xi_0 = 10^5$ erg cm s$^{-1}$ (see contour shown by turquoise line in the left panel of \\autoref{fig:failed_wind}), but its opening angle may be small (\\autoref{fig:xray_grid}). The shadow region defines the acceleration region of the wind, where the force multiplier is greatly enhanced and the wind gets almost all of its acceleration. Eventually this acceleration is enough that the material reaches the escape velocity before it emerges from the shadow, and is overionised by the X-ray radiation. The left panel of \nFig. \\ref{fig:failed_wind} shows these first escaping streamlines (blue) which are close to $R_\\mathrm{in}$. \n\n\\autoref{fig:ufo_pros} shows the resulting wind parameters at \na distance $r=5000 \\, R_\\mathrm{g}$. We plot the wind column density, and density-weighted mean velocity and ionisation parameter as a function of the polar angle $\\theta = \\arctan(R\/z)$. The column is almost constant at $N_H\\sim 2\\times 10^{23}$~cm$^{-2}$ (optical depth of $\\sim 0.1$ to electron scattering) across the range $25^\\circ<\\theta<85^\\circ$. For $\\theta > 85^\\circ$, the sight-line intercepts the inner failed wind and the column density increases to $N_H\\sim 10^{24}$~cm$^{-2}$. The\ntypical wind velocity at this point is $\\simeq (0.1-0.4)\\,c$\nbut it is always very ionised ($\\xi > 10^5$ erg cm s $^{-1}$) at these large distances. This is too ionised to allow even H- and He-like iron to give visible atomic features in this high velocity gas, although these species may exist at smaller radii where the material is denser. We will explore the observational impact of this in a future work, specifically assessing whether UV line-driving can be the origin of the ultra-fast outflows seen in some AGN\n(see also \\citealt{mizumoto_uv_2021}). \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/failed_wind.pdf}\n \\caption{Simulated wind trajectories for our fiducial system ($M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, and $R_\\mathrm{in} = 50 R_\\mathrm{g}$) zooming in on the failed wind region, where we also colour-plot the ionisation parameter $\\xi$, showing the contour at $\\xi = 10^5$ erg cm s$^{-1}$ as the light turquoise line. We plot the failed trajectories in green and the escaping trajectories in blue.}\n \\label{fig:failed_wind}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/ufo_pros.pdf}\n \\caption{Wind properties for a system with $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$, $\\dot m =0.5$, $R_\\mathrm{in}=50R_\\mathrm{g}$ measured along a sight-line at angle $\\theta$ at a distance $r=5000R_\\mathrm{g}$ from the centre. First panel: column density. Second panel: outward mean velocity weighted by density. Third panel: mean ionisation parameter weighted by density. Fourth panel: kinetic luminosity per unit angle. Fifth panel: wind momentum rate per unit angle.}\n \\label{fig:ufo_pros}\n\\end{figure}\n\nThe escaping wind carries a mass loss rate of $\\dot M_\\mathrm{wind} \\simeq 0.26$ $\\mathrm{M}_\\odot \/$ yr, corresponding to $\\dot M_\\mathrm{wind} \/ \\dot M \\simeq 11\\%$ of the mass accretion rate, and a kinetic luminosity of $L_\\mathrm{kin} \\approx 7 \\times 10^{44}$ erg \/ s, which is equal to 10\\% of the bolometric luminosity. As the two bottom panels of \\autoref{fig:ufo_pros} show, most of the energy and momentum of the wind is located at small polar angles ($\\sim 20^\\circ$), which is consistent with the initial density profile since the innermost streamlines carry the largest amount of mass.\n\n\\subsection{Dependence on the initial radius \\texorpdfstring{$R_\\mathrm{in}$}{Rin}}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/rin_scan.pdf}\n \\caption{Mass loss rate (first panel), kinetic luminosity (second panel), momentum loss rate (third panel) and average velocity (fourth panel) for different values of $R_\\mathrm{in}$, at a fixed $M_\\mathrm{BH}=10^8\\mathrm{M}_\\odot$ and $\\dot m =0.5$. The mass loss rate is normalised to the system's mass accretion rate, while we normalise the luminosity to the bolometric luminosity. The average velocity is taken as \\protect$v_\\mathrm{r} = \\sqrt{2 L_\\mathrm{kin} \/ \\dot M_\\mathrm{wind}}$.}\n \\label{fig:rin_scan}\n\\end{figure*}\n\nAs we already mentioned in \\autoref{sec:gas_trajectories}, the initial radius of the innermost trajectory ($R_\\mathrm{in}$) is left as a free parameter to explore. This parameter is likely dependent on the structure of the accretion flow, which we do not aim to model here. As we increase $R_\\mathrm{in}$, the amount of mass that can potentially be lifted from the disc decreases, both because of the reduction in the extent of the launching region and the decrease in initial density with radius (\\autoref{fig:initial_conditions}). Furthermore, increasing $R_\\mathrm{in}$ also narrows the failed wind shadow, since it reduces its subtended angle, thus reducing the accelerating region of the wind. We therefore expect the wind to flow at higher polar angles and smaller velocities when increasing $R_\\mathrm{in}$. In \\autoref{fig:rin_scan}, we plot the predicted normalised mass loss rate, kinetic luminosity, momentum loss rate, and average velocity of the wind as a function of $R_\\mathrm{in}$. As we expected, both the mass loss rate and the kinetic luminosity decrease with $R_\\mathrm{in}$, with the latter decreasing much faster. This difference in scaling is not surprising, since the fastest part of the wind originates from the innermost part of the disc, where most of the UV radiation is emitted. To further illustrate this, we plot the average velocity of the wind for each simulation in the rightmost panel of \\autoref{fig:rin_scan}, observing that the maximum velocity decreases with initial radius. We note that the wind successfully escapes for $R_\\mathrm{in} \\gtrsim 175 R_\\mathrm{g}$, which is a consequence of the initial number density profile (\\autoref{fig:initial_conditions}) sharply declining after $R \\gtrsim 100 R_\\mathrm{g}$. There is a physically interesting situation happening at $R_\\mathrm{in} \\sim 12R_\\mathrm{g}$, where the average velocity of the wind drops. This is due to the failed wind being located where most of the UV emission is, thus making the wind optically thick to UV radiation (with respect to electron scattering opacity). Lastly, we note that the wind consistently reaches velocities $\\gtrapprox 0.3$ $c$ for $R_\\mathrm{in} \\lesssim 50\\, R_\\mathrm{g}$.\n\n\n\\subsection{Dependence on BH mass and mass accretion rate}\n\\label{sec:results_bh}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{figures\/bh_scan.pdf}\n \\caption{Wind mass loss rate normalised by the mass accretion rate (first panel column), kinetic luminosity normalised by bolometric luminosity (second panel column), momentum loss rate normalised by $L_\\mathrm{bol} \/ c$ (third panel column), and average velocity (fourth panel column) as functions of the Eddington fraction $\\dot m$ for different $M_\\mathrm{BH}$.The average velocity is taken as \\protect $v_\\mathrm{r} = \\sqrt{2 L_\\mathrm{kin} \/ \\dot M_\\mathrm{wind}}$.}\n \\label{fig:bh_scan}\n\\end{figure*}\n\nWe now investigate the dependence of the wind mass loss rate, kinetic luminosity, momentum loss rate, and average velocity on $M_\\mathrm{BH}$, $\\dot m$, and $R_\\mathrm{in}$. We scan the BH parameter range for $M_\\mathrm{BH}\\in(10^6 - 10^{10})$, $\\dot m \\in(0.01-0.5)$, and $R_\\mathrm{in} = (10, 25, 50)R_\\mathrm{g}$. We fix $f_\\mathrm{X} = 0.15$ and recognise that keeping it constant throughout the parameter scan is a limitation, since in reality it depends on $M_\\mathrm{BH}$ and $\\dot m$. The results are shown in \\autoref{fig:bh_scan}, where all of the quantities have been normalised to their characteristic scales. The red dashed line denotes the limit when quantities become unphysical since the wind is carrying more mass, energy, or momentum than the disc can provide. The first thing to note is that we do not obtain any wind for $\\dot m \\lesssim 0.06$, regardless of $M_\\mathrm{BH}$. This is initially surprising, as the force multiplier is of order $1000$ for cool material, apparently allowing a wind to escape for $\\dot{m}>0.001$. However, the initial density drops as $\\dot{m}^{1\/\\alpha}$ (see \\autoref{subsection:ic_scaling}) so the X-ray shielding drops dramatically, strongly suppressing the wind. Overall, the weakest winds are seen from the highest ($M_\\mathrm{BH}=10^{10}\\,\\mathrm{M}_\\odot$) and lowest ($M_\\mathrm{BH}=10^6\\,\\mathrm{M}_\\odot$) black hole masses. This can be explained by the behaviour of the UV fraction (\\autoref{fig:uv_fractions}). For the $M_\\mathrm{BH}=10^6\\,\\mathrm{M}_\\odot$ case, the UV bright disc annuli are located at large radii, where the disc luminosity is lower; for the $M=10^{10}\\mathrm{M}_\\odot$ case, $f_\\text{\\tiny UV}$ is only high at very small radii, and overall small in the wind launching region. Furthermore, the high disc temperatures expected for the lowest mass systems ($M_\\mathrm{BH} = 10^6\\,\\mathrm{M}_\\odot$), especially at high $\\dot{m}$, mean that the disc contributes to the ionising X-ray flux. This effect is not considered in our work here, but makes it likely that even our rather weak UV line-driven wind is an overestimate for these systems.\n\nFor the values of $M_\\mathrm{BH}$ where the wind is robustly generated across the rest of the parameter space, $M_\\mathrm{BH} \\in (10^7, 10^8, 10^9)\\mathrm{M}_\\odot$, we find a weak dependence of the normalised wind properties on $M_\\mathrm{BH}$. This is expected, since the initial density profile scales with $M_\\mathrm{BH}$ and $\\dot m$ as (\\autoref{subsection:ic_scaling})\n\\begin{equation}\n n_0 \\propto \\frac{\\dot m^{\\frac{1}{\\alpha}}}{M_\\mathrm{BH}},\n\\end{equation}\nwhere we ignore the dependence of $f_\\text{\\tiny UV}$ on $M_\\mathrm{BH}$. The initial wind velocity $v_0 = v_\\text{th} \\propto T^{1\/2}$, hence\n\\begin{equation}\n v_0 \\propto \\left( \\frac{\\dot m}{M_\\mathrm{BH}} \\right)^{1\/8} \n\\end{equation}\n(see \\autoref{eq:radiation_flux}.)\nThis then implies\n\\begin{equation}\n \\dot M_\\mathrm{wind} \\propto n_0 \\; v_0 \\; R^2 \\propto \\left(\\frac{\\dot m^{\\frac{1}{\\alpha}}}{M_\\mathrm{BH}}\\right) \\left(\\frac{\\dot m}{M_\\mathrm{BH}}\\right)^{1\/8}\\left( M_\\mathrm{BH}^2\\right)\n\\end{equation}\nSince scaling due to the dependence on $v_0$ is particularly weak (it depends on the 1\/8-th power), we choose to ignore it so that we can write\n\\begin{equation}\n \\frac{\\dot M_\\mathrm{wind}}{\\dot M_\\mathrm{acc}} \\propto \\frac{\\dot m^{\\frac{1}{\\alpha}} \\, M_\\mathrm{BH}}{\\dot m M_\\mathrm{BH}}\\propto \\dot m^{\\frac{1}{\\alpha} -1},\n\\end{equation}\nwhere we have used the fact that $\\dot M_\\mathrm{acc} = \\dot m \\dot M_\\mathrm{Edd} \\propto \\dot m M_\\mathrm{BH}$. Similarly,\n\\begin{equation}\n \\frac{L_\\mathrm{kin}}{L_\\mathrm{bol}} \\propto \\frac{\\dot M_\\mathrm{wind} v_f^2}{\\dot M_\\mathrm{acc}} = \\dot m^{1 + \\frac{1}{\\alpha}},\n\\end{equation}\nwhere $v_f$ is the final wind velocity and we have assumed $v_f \\propto \\dot m$. This last assumption is verified (for $0.1 \\lesssim \\dot m \\lesssim 0.5$) in the rightmost panel of \\autoref{fig:bh_scan}. Consequently, ignoring the scaling of $f_\\text{\\tiny UV}$, both the normalised mass loss rate and normalised kinetic luminosity are independent of $M_\\mathrm{BH}$. For our value of $\\alpha=0.6$, we find $L_\\mathrm{kin} \\propto \\dot m^{2.7} L_\\mathrm{bol}$. This scaling is significantly different from the one often assumed in models of AGN feedback used in cosmological simulations of galaxy formation, where the energy injection rate is assumed to be proportional to the mass accretion rate and hence to the bolometric luminosity \\citep{schaye_eagle_2015, weinberger_simulating_2017, dave_simba_2019}. However, it is consistent with the results found in the hydrodynamical simulations of \\cite{nomura_line-driven_2017} and with current observational constraints \\citep{gofford_suzaku_2015, chartas_multiphase_2021}.\n\nFor the $R_\\mathrm{in} = 10 R_\\mathrm{g}$ case, many parameter configurations of $M_\\mathrm{BH}$ and $\\dot m$ give rise to winds that are unphysical, since they carry more momentum than the radiation field. This is caused by us not considering the impact that the wind mass loss would have on the disc SED, and an underestimation of the UV opacity in our ray tracing calculation of the UV radiation field, in which we only include the Thomson opacity. We also observe that for the lower and upper ends of our $M_\\mathrm{BH}$ range the existence of a wind for different $\\dot m$ values depends on the value of $R_\\mathrm{in}$. This is a consequence of a complex dependence of the failed wind shadow on the initial density and $R_\\mathrm{in}$.\n\nWe now investigate the geometry of the wind: where it originates and at which angles it flows outwards. Despite the strong dependence of the kinetic luminosity on $\\dot m$, the wind launching region does not vary significantly with $\\dot m$, as is shown in \\autoref{fig:bh_scan_radii}, where we plot the average launch radius, weighted by mass loss rate or kinetic luminosity. The exception is the case $R_\\mathrm{in} =10R_\\mathrm{g}$, where the lower inner density at $R_\\mathrm{in}=10R_\\mathrm{g}$ (see \\autoref{fig:initial_conditions}), and its decrease with $\\dot m$ produce a larger failed wind region for $\\dot m \\lesssim 0.15$. The dependence of the mass weighted average radius with $M_\\mathrm{BH}$ is a consequence of the dependence of $f_\\text{\\tiny UV}$ on $M_\\mathrm{BH}$. Larger $M_\\mathrm{BH}$ black holes have the $f_\\text{\\tiny UV}$ peak closer in, so the wind carries more mass at smaller radii.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/bh_scan_avg_radius.pdf}\n \\caption{Average launch radius weighted by the trajectories' mass loss rate (left panels) and kinetic luminosity (right panels) as a function of $M_\\mathrm{BH}$ and $\\dot m$ for the 3 values of $R_\\mathrm{in}$.}\n \\label{fig:bh_scan_radii}\n\\end{figure}\n\nFinally, we plot the wind opening angle for the scanned parameter space in \\autoref{fig:wind_angle}. We again find a small dependence on $M_\\mathrm{BH}$, except for near the boundaries of our $M_\\mathrm{BH}$ range, where the angle is quite sensitive to $\\dot m$. The wind flows closer to the equator as we decrease $\\dot m$, hence whether the wind is more polar or equatorial depends on the mass accretion rate of the system. This can be explained by considering that lower disc luminosities do not push the wind as strongly from below, and thus the wind escapes flowing closer to the equator.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/wind_angle.pdf}\n \\caption{Wind opening angle, measured as the smallest polar angle, $\\theta_\\mathrm{min}$, of all escaping streamlines as a function of $M_\\mathrm{BH}$ and $\\dot m$, for the 3 studied $R_\\mathrm{in}$ values.}\n \\label{fig:wind_angle}\n\\end{figure}\n\n\\subsection{Can UV line-driven winds be UFOs?}\n\nIn \\citetalias{quera-bofarull_qwind_2020}, wind trajectories could achieve arbitrarily large velocities, often surpassing the speed of light, due to the neglect of relativistic effects. With the introduction of relativistic corrections (\\autoref{sec:relativistic}), the wind is always sub-luminal, as we show in \\autoref{fig:rin_scan} and \\autoref{fig:bh_scan}. Nonetheless, throughout the simulated parameter space, outflows consistently achieve speeds of $(0.1 - 0.8)$ $c$, scaling approximately as $\\dot{m}$\nwith little dependence on \n$M_\\mathrm{BH}$. If we limit ourselves to the simulations that conserve the overall momentum and energy of the system, then the simulated wind still achieve speeds in the range $(0.1 - 0.4)$c. This implies that UV line-driving is a feasible mechanism to produce UFOs even when relativistic corrections are included. The final velocity of the wind depends on how much a gas blob can be accelerated while it is shadowed from the X-ray radiation. In \\autoref{fig:velocity_dependence}, we plot the velocity profile for a trajectory starting at $R=100R_\\mathrm{g}$ in our fiducial simulation density grid for different values of the initial velocity. We find that the final velocity of the trajectory is independent of the initial velocity and that the wind is able to drastically accelerate (up to 6 orders of magnitude in velocity) over a very small distance ($\\lesssim 1 R_\\mathrm{g}$), which suggests that line-driving can be very effective even when the X-ray shadowed region is very small.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/velocity_dependence.pdf}\n \\caption{Velocity as a function of $z$ for a trajectory starting at $R=100R_\\mathrm{g}$ for our fiducial model. Different colours correspond to different initial velocities. }\n \\label{fig:velocity_dependence}\n\\end{figure}\n\nWe thus find that UV line-driving is sufficient to reproduce the range of observed UFO velocities, as opposed to the findings of \\cite{luminari_speed_2021}. \nWe note that their code assumes an initial density (similar to \\citetalias{quera-bofarull_qwind_2020}) rather than calculating it from first principles, and does not include the full ray tracing of both UV and X-rays that are considered here. On the other hand, our treatment of the force multiplier is simplified compared to their calculation, where they use the radiative transfer code XSTAR \\citep{kallman_photoionization_2001} to compute the radiation flux absorbed by the wind. Nonetheless, our results show that \na small shaded region can produce a very fast wind (see \n\\autoref{fig:velocity_dependence}), \nand the range of $R_\\mathrm{in}$ that gives rise to velocities $\\geq 0.3 c$ is quite wide (see \\autoref{fig:bh_scan_radii}). It then seems\nquite likely that UV line-driven winds can indeed reach these velocities and hence be the origin of the majority of UFOs seen. \nThere are even higher velocities claimed for a few absorption features in the literature, but these are generally low signal-to-noise detections. \n\\section{Intersection of trajectories}\n\\label{app:intersections}\n\nOnce we have solved the equations of motion of the different gas blobs, it is common for the resulting trajectories to cross each other. Trajectories of gas elements computed using our ballistic model should not be confused with the streamlines of the actual wind fluid, since the latter cannot cross each other as it would imply the presence of singular points where the density and the velocity fields are not well defined.\n\nNonetheless, if we aim to construct a density and velocity field of the wind, we need to define the density and velocity at the crossing points. To circumvent this, once two trajectories intersect, we terminate the one that has the lowest momentum density at the intersection point.\n\nTo determine at which, if any, point two trajectories cross, we consider a trajectory as a collection of line segments $\\{s_i\\}$. Two trajectories $\\{s_i\\}$, and $\\{t_j\\}$ cross each other if it exists $i, j$ such that $s_i \\cap t_j \\neq 0$.\n\nSuppose the line segment $s_i$ is bounded by the points $\\bmath{p_1}$ and $\\bmath{p_2}$ such that $\\bmath{p_2} = \\bmath{p_1} + \\alpha' (\\bmath{p_2} - \\bmath{p_1})$ with $\\alpha' \\in [0,1]$. Similarly, $t_j$ is limited by $\\bmath{q_1}$ and $\\bmath{q_2}$ such that $\\bmath{q_2} = \\beta' (\\bmath{q_2} - \\bmath{q_1})$ with $\\beta' \\in [0,1]$. The condition that $s_i$ intersects $t_j$ is equivalent to finding $\\alpha$, $\\beta$ $\\in [0,1] \\times [0,1]$ such that\n\\begin{equation}\n \\bmath{p_1} + \\alpha' (\\bmath{p_2} - \\bmath{p_1}) = \\bmath{q_1} + \\beta' (\\bmath{q_2} - \\bmath{q_1}),\n\\end{equation}\nwhich corresponds to the linear system $\\mathbfss{A}\\bmath{x} = \\bmath{b}$ with\n\\begin{equation}\n \\mathbfss{A} = \\left(\\bmath{p_2}-\\bmath{p_1}, \\bmath{q_1}-\\bmath{q_2}\\right),\n\\end{equation}\n$\\bmath{x} = \\left(\\alpha', \\beta'\\right)^\\intercal$, and $\\bmath{b} = \\bmath{q_1} - \\bmath{p_1}$. The segments intersect if this linear system is determined with solution inside the unit square.\n\n\\section{Radial dependence of \\texorpdfstring{$f_\\text{\\tiny UV}$}{fuv}}\n\\label{sec:fuv}\n\nWe first address the validity of assuming a constant emitted UV fraction with radius. We can calculate its radial dependence using\n\\begin{equation}\n f_\\text{\\tiny UV}(R_\\mathrm{d}) = \\frac{\\int_{E_1}^{E_2} B(E, T(R_\\mathrm{d}))\\, \\mathrm{d} E}{\\int_{0}^{\\infty} B(E, T(R_\\mathrm{d}))\\, \\mathrm{d} E},\n\\end{equation}\nwhere $B(E,T)$ is the blackbody spectral radiance, $E_1 = 0.0038$ keV and $E_2=0.06$ keV (the standard definition of the UV transition band: (3200-200) \\AA), and $T^4(R_\\mathrm{d})=\\mathcal{F} \/ \\sigma_\\text{\\tiny SB}$ (where $\\mathcal F$ is defined in \\autoref{eq:radiation_flux}). In \\autoref{fig:uv_fractions}, we plot the UV fraction as a function of radius for different $M_\\mathrm{BH}$ and $\\dot m$. The disc temperature is related to the total flux (given by setting $f_\\text{\\tiny UV}=1$ in \\autoref{eq:radiation_flux}) so $T^4\\propto (M\\dot{M}f_\\text{\\tiny NT}\/R^3)\\propto \\dot{m}\/M (R\/R_\\mathrm{g})^3$.\nThus the disc temperature increases with decreasing $R\/R_\\mathrm{g}$, and for the fiducial case of $M_\\mathrm{BH}=10^8 \\mathrm{M}_\\odot$ this leads to the majority of the disc emission in the UV coming from $R\/R_\\mathrm{g}\\le 100$ (left panel of \\autoref{fig:uv_fractions}: orange line).\nHowever, the increase in disc temperature at a given $R\/R_\\mathrm{g}$ for decreasing mass means that the same $\\dot{m}$ for $M_\\mathrm{BH}=10^6 \\mathrm{M}_\\odot$ gives a UV flux which peaks at $R\/R_\\mathrm{g}>100$, as the inner regions are too hot to emit within the defined UV bandpass (left panel of \\autoref{fig:uv_fractions}: blue line). Conversely, for the highest BH masses of $M_\\mathrm{BH}=10^{10}\\mathrm{M}_\\odot$ the disc is so cool that it emits UV only very close to the innermost stable circular orbit (left panel of \\autoref{fig:uv_fractions}: purple line). The universal upturn at $R=10R_\\mathrm{g}$ is caused by the the temperature sharply decreasing at the inner edge of the accretion disc due to the viscuous torque dropping to zero there.\n\nSimilarly, the right panel of \\autoref{fig:uv_fractions} shows the impact of changing $\\dot{m}$ for the fiducial mass of $10^8\\mathrm{M}_\\odot$. The dashed orange line shows the case $\\dot{m}=0.5$, as before, \nand the disc temperature decreases with decreasing $\\dot{m}$ to $0.1$ (dashed green line) and $0.05$ (dashed blue line), reducing the radial extent of the UV-emitting zone. \n\nWe note that the assumption used in \\citetalias{quera-bofarull_qwind_2020} (and many other UV line driven disc wind codes) of $f_\\text{\\tiny UV}$ being a constant value is poor overall, highlighting the importance of including the radial dependence of the UV flux.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/uv_fractions.pdf}\n \\caption{UV fraction as a function of disc radius. Left panel: dependence on $M_\\mathrm{BH}$ for fixed $\\dot m =0.5$. Right panel: dependence on $\\dot m$ for fixed $M_\\mathrm{BH} = 10^8\\mathrm{M}_\\odot$.}\n \\label{fig:uv_fractions}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nCryptocurrencies are notoriously known to attract financial speculators who often seek to multiply their potential monetary upside and financial gains through leverage. Leverage is realized by borrowing assets to perform trades~---~commonly referred to as margin trading. It is apparent that margin trading, speculating with borrowed assets in general, is an incredibly risky endeavor. Yet, the borrowing and lending markets on blockchains are thriving and have reached a collective~$39.88$B USD of {\\em total value locked (TVL)} at the time of writing\\footnote{\\url{https:\/\/defipulse.com\/}}.\n\nLoans on a blockchain typically operate as follows. Lenders with a surplus of money provide assets to a lending smart contract. Borrowers then provide a security deposit, known as collateral, to borrow cryptocurrency. Because the lending and borrowing on blockchains lacks compulsory means on defaults,\nthe amount of debt borrowers can take on is typically inferior to the security deposit in value~---~resulting in \\emph{over-collateralized loans}. Over-collateralized loans are interesting from a financial perspective, as they enable borrowers to take on leverage.\n\nIf the collateral value decreases under a specific threshold (e.g., below $150$\\% of the debt value~\\cite{makerdao}), the associated debt can be recovered through three means: \\emph{(1)} a loan can be made available for liquidation by the smart contract. Liquidators then pay back the debt in exchange for receiving the collateral at a discount (i.e., \\emph{liquidation spread}), or the collateral is liquidated through an auction. \\emph{(2)} Debt can also be rescued by ``topping up'' the collateral, such that the loan is sufficiently collateralized. \\emph{(3)} Finally, the borrower can repay parts of their debt. While users can repay their debts manually, this appears impractical for the average user, as it requires infrastructure to constantly monitor the blockchain, collateral price, and transaction fee fluctuations. For example, even professional liquidation bots from MakerDAO failed to monitor and act upon price variations during blockchain congestion~\\cite{maker-fail}. \n\nIn this paper we make the following contributions. \n\n\\begin{enumerate}\n \\item {\\bf Liquidation Models and Insights:} We provide the first longitudinal study of the four major lending platforms MakerDAO, Aave, Compound, and dYdX, capturing collectively over $85$\\% of the borrowing\/lending market on the Ethereum blockchain. By focusing on the protocol's liquidation mechanisms, we systematize their liquidation designs. MakerDAO, for instance, follows an auction-based liquidation process, while Aave, Compound, and dYdX operate under a fixed spread liquidation model.\n \n \\item {\\bf Data Analytics:} We provide on-chain data analytics covering the entire existence of the four protocols ($2$ years). Our findings show how the accumulative liquidation proceeds amount to~\\empirical{$347.62$M}~USD\\xspace, we identify~\\empirical{$1,600$}\\xspace unique liquidator addresses and~\\empirical{$21,792$}\\xspace liquidation events, of which~\\empirical{$265$}\\xspace} \\newcommand{\\empirical{$\\numprint{467.44}$K}~USD\\xspace}{\\empirical{$12.20$k}~USD\\xspace auction liquidations are not profitable for the liquidators. We show how~\\empirical{$72.85\\%$}\\xspace of the liquidations pay an above average transaction fee, indicating competitive behavior. We find the existence of bad debts, the borrowing positions that do not financially incentivize borrowers to perform a close. Notably, Aave V2 has accumulated up to~$87.4$K~USD of bad debts by the end of April,~2021.\n \n We quantify how sensitive debt behaves to collateral price declines and find that, for example, a~$43$\\% reduction of the ETH price (analogous to the ETH price decline on the~13th of March,~2020) would engender liquidatable collateral volume of~$1.07$B USD on MakerDAO.\n \\item {\\bf Objective Liquidation Mechanism Comparison:} We provide a methodology to compare quantitatively whether a liquidation mechanism favors a borrower or a liquidator. We find evidence that fixed spread liquidation mechanisms favor liquidators over borrowers. That is, because existing DeFi systems are parameterized to allow more collateral than necessary to be liquidated.\n \n \\item {\\bf Optimal Fixed Spread Liquidation Strategy:} We propose an optimal fixed spread liquidation strategy. This strategy allows liquidators to lift the restrictions of the close factor (the upper limit of repaid debts in a single liquidation, cf.\\ Section~\\ref{sec:terminology}) within two successive liquidations. We provide a case study of a past liquidation transaction and show that the optimal strategy could have increased the liquidation profit by \\empirical{$53.96$K}~USD\\xspace (\\empirical{$1.36\\%$}\\xspace), validated through concrete execution on the real blockchain state. This optimal strategy can further aggravate the loss of borrowers.\n\\end{enumerate}\n\nThe remainder of the paper is organized as follows. Section~\\ref{sec:background} outlines the background on blockchain and lending, while we systematize existing liquidation mechanisms in Section~\\ref{sec:existing-protocols}. Section~\\ref{sec:insights} provides liquidation data insights from empirical data.\nWe discuss how to objectively compare liquidation mechanisms and the optimal liquidation strategy in Section~\\ref{sec:better-liquidation}.\nWe outline related work in Section~\\ref{sec:related-work} and conclude the paper in Section~\\ref{sec:conclusion}.\n\n\\section{Lending on the Blockchain}\\label{sec:background}\nWe proceed by outlining the required background on blockchain and DeFi for the remainder of the paper.\n\n\\subsection{Blockchain \\& Smart Contract}\nBlockchains are distributed ledgers that enable peers to transact without the need to entrust third-party intermediaries. There exist two categories of blockchains: \\textit{(i)} permissionless blockchains, where any entity is able to join and leave without permission; \\textit{(ii)} permissioned blockchains, which are typically composed of a group of authenticated participants. In this work, we only focus on permissionless blockchains, on top of which DeFi is built.\n\nAt its core, a blockchain is a hash-linked chain of blocks operating over a peer-to-peer (P2P) network~\\cite{bonneau2015sok}. A block is a timestamped data structure aggregating transactions, which record, e.g., asset transfers. To transfer assets, users need to broadcast digitally signed transactions through the P2P network. The so-called miners then collect transactions, pack transactions into blocks, and append blocks to the blockchain. The whole network follows a consensus protocol (e.g., Nakamoto consensus~\\cite{bitcoin}) allowing honest participants to agree on a consistent version of the blockchain. Transactions waiting to be confirmed on-chain are stored in the so-called mempool\\footnote{Note that there is no universal mempool across all network participants. Every node maintains its own mempool depending on the received transactions.}. We refer the reader to~\\cite{bonneau2015sok,bano2019sok} for a more thorough background on blockchains.\n\nSome blockchains, for example, Ethereum~\\cite{wood2014ethereum}, offer generic computation capabilities through smart contracts.\nIn essence, an Ethereum smart contract is an account controlled by an immutable program (i.e., bytecode). One can trigger the execution of the bytecode by sending a transaction, which contains the executing parameters specified by the transaction sender, to the smart contract account. The EVM, a quasi Turing-complete state machine~\\cite{atzei2017survey}, provides the runtime environment to the contract execution. Solidity~\\cite{dannen2017introducing}, which can be compiled into bytecode, is to date the most prevalent high-level language for implementing Ethereum smart contracts. Smart contracts are widely used to create cryptocurrencies (also known as tokens) on Ethereum in addition to the native coin ETH. Notably, WETH is a one-to-one equivalent token of ETH.\n\nTo submit a transaction on-chain, a user is required to pay a transaction fee. On Ethereum, the transaction fee equals the product of the gas (i.e., an integer measuring the computation complexity of a transaction) and the gas price (i.e., the amount of ETH that the transaction sender is willing to pay for a single unit of gas). Due to the limited space of an Ethereum block (i.e., the total amount of gas consumed in on block), a financially rational miner may include the transactions with the highest gas prices from the mempool into the next block. The blockchain network congests when the mempool grows faster than the transaction inclusion speed due to, for example, traders place substantial orders in a market collapse. Under such circumstances, users have to increase gas prices or wait longer than average to confirm their transactions.\n\n\\subsection{Decentralized Finance (DeFi)}\nSmart contracts allow, not only the creation of tokens, but further the construction of sophisticated on-chain financial systems, namely Decentralized Finance (DeFi). In DeFi, any entity can design a financial protocol, implement in smart contracts, and deploy on-chain. Compared to traditional finance, DeFi presents promising peculiarities, e.g., non-custody and public verifiability~\\cite{qin2021cefi}. Although most DeFi protocols are mirrored services from traditional finance (e.g., exchanges), a proper redesign appears to be necessary considering the special settlement mechanisms of the underlying blockchains. For instance, due to the limited computation capacity, a limit order book with a matching engine, which has been adopted in centralized exchanges for decades, is, however, inefficient on blockchains. This leads to the invention of the automated market maker, where traders, instead trading against other traders, only need to interact with a pool of assets reserved in a smart contract. Since the rise of DeFi, we have observed numerous such innovative DeFi design, most of which, however, have not been thoroughly studied. As a result, the risks and threats that DeFi users are exposed to are still unclear, necessitating empirical research to provide objective insights.\n\nAt the time of writing, Ethereum is the dominating permissionless blockchain hosting DeFi. The DeFi ecosystem on Ethereum reached a TVL of over $80$B~USD\\footnote{In comparison, the Binance Smart Chain (BSC), ranked the second in terms of TVL at the time of writing, reaches~$20$B USD (cf.\\ \\url{https:\/\/debank.com\/ranking\/locked_value}). We omit BSC in this work because BSC starts to grow from early 2021, which has not accumulated sufficient data.}, with more than $50\\%$ contributed by lending protocols. Lending and borrowing is a popular way to realize a leverage (amplifying the profit) in DeFi. A typical use case is outlined as follows. A trader collateralizes $\\numprint{5000}$~USDT (a USD-pegged stablecoin, cf.\\ Section~\\ref{sec:stablecoinbackground}) to borrow~$1$ ETH, when the ETH\/USDT price is $\\numprint[ETH]{1} = \\numprint[USDT]{3000}$. The borrower then sells the borrowed $\\numprint[ETH]{1}$ for $\\numprint[USDT]{3000}$. If the ETH price declines to, for example, $\\numprint[ETH]{1} = \\numprint[USDT]{2000}$, the trader can purchase~$1$ ETH with $\\numprint[USDT]{2000}$, repay the debt, redeem the collateral, and finally realize a profit of $\\numprint[USDT]{1000}$. The trader at the same time bears the liquidation risk if the ETH price increases and the USDT collateral is insufficient to back the $\\numprint[ETH]{1}$ debt. In a liquidation, a liquidator repays the ETH debt for the trader and acquires the USDT collateral. The acquired collateral exceeds the rapid debt in value incurring a loss to the trader. Such repayment-acquisition liquidation mechanisms are adopted by most DeFi lending platforms. However, the incentives, risks (e.g., to what extend borrowers have lost in liquidation events), and stabilities of these protocols have not been thoroughly studied, which motivates this work.\n\nWe outline the details of the liquidation mechanisms in Section~\\ref{sec:existing-protocols}. In the following, we introduce the essential components of DeFi that are relevant to lending protocols.\n\n\\subsubsection{Price Oracle}\nBecause lending protocols aim to liquidate collateralized assets upon collateral price declines, the lending smart contract is required to know the price of the collateral asset. Prices can either be provided through an on-chain oracle, such as smart contract based exchanges (e.g., Uniswap~\\cite{uniswap2018}), or via an off-chain oracle (such as Chainlink~\\cite{arijuel2017chainlink}). On-chain price oracles are known to be vulnerable to manipulation~\\cite{qin2020attacking}.\n\n\\subsubsection{Flash Loan} The atomicity of blockchain transactions (executions in a transaction collectively succeed or fail) enables flash loans. A flash loan represents a loan that is taken and repaid within a single transaction~\\cite{qin2020attacking,allen2020design}. A borrower is allowed to borrow up to all the available assets from a flash loan pool and execute arbitrary logic with the capital within a transaction. If the loan plus the required interests are not repaid, the whole transaction is reverted without incurring any state change on the underlying blockchain (i.e., the flash loan never happened). Flash loans are shown to be widely used in liquidations~\\cite{qin2020attacking}.\n\n\\subsubsection{Stablecoin}\\label{sec:stablecoinbackground}\nStablecoins are a class of cryptocurrencies designed to provide low price volatility~\\cite{clark2020demystifying}. The price of a stablecoin is generally pegged to some reference point (e.g., USD). The typical stablecoin mechanisms are reserve of the pegged asset (e.g., USDT and USDC), loans (e.g., DAI), dual coin, and algorithmic supply adjustments~\\cite{moin2020sok}.\n\n\\subsection{Terminology}\\label{sec:terminology}\nWe adhere to the following terminologies in this paper.\n\\begin{description}\n\\item[Loan\/Debt:] A borrower, secured by a collateral deposit, temporarily takes capital from a lender. The collateral is the insurance of the lender against defaults.\n\\item[Interest Rate:] A loan is repaid by repaying the lent amount, plus a periodic percentage of the loan amount. The interest rate can be governed by the scarcity\/surplus of the available asset supply within the lending smart contract.\n\\item[Over\/Under-collateralization:] Blockchain based loans are typically over-collateralized, i.e., the borrower has to provide collateral assets of higher total value than the granted loan. A loan is under-collateralized when the value of the collateral is inferior to the debt.\n\\item[Position:] In this work, the collateral and debts are collectively referred to as a position. A position may consist of multiple-cryptocurrency collaterals and debts.\n\\item[Liquidation:] In the event of a negative price fluctuation of the debt collateral (i.e., a move below the liquidation threshold), a position can be liquidated. In permissionless blockchains, anyone can repay the debt and claim the collateral.\n\\item[Liquidation Threshold ($\\mathbf{LT}$):] Is the percentage at which the collateral value is counted towards the borrowing capacity (cf. Equation~\\ref{eq:borrowing-capacity}).\n\\item[Liquidation Spread ($\\mathbf{LS}$):] Is the bonus, or discount, that a liquidator can collect when liquidating collateral (cf.\\ Equation~\\ref{eq:ls}). This spread incentivises liquidators to act promptly once a loan crosses the liquidation threshold.\n\n\\begin{equation}\\label{eq:ls}\n\\begin{aligned}\n &Value\\ of\\ Collateral\\ to\\ Claim \\\\&= Value\\ of\\ Debt\\ to\\ Repay\\times(1+\\mathbf{LS})\n\\end{aligned}\n\\end{equation}\n\\item[Close Factor ($\\mathbf{CF}$):] Is the maximum proportion of the debt that is allowed to be repaid in a single liquidation.\n\\item[Collateralization Ratio ($\\mathsf{CR}$):] Is the ratio between the total value of collateral and debt (cf.\\ Equation~\\ref{eq:collateralizationratio}) where $i$ represents the index of collateral or debt if the borrower owns collateral or owes debt in multiple cryptocurrencies.\n\n\\begin{equation}\\label{eq:collateralizationratio}\n \\mathsf{CR}=\\frac{\\sum{Value\\ of\\ Collateral_i}}{\\sum{Value\\ of\\ Debt_i}}\n\\end{equation}\nA debt is under-collateralized if $\\mathsf{CR}<1$, otherwise the debt is over-collateralized.\n\\item[Borrowing Capacity ($\\mathsf{BC}$):] Refers to the total value that a borrower is allowed to request from a lending pool, given its collateral amount.\nFor each collateral asset $i$ of a borrower, its borrowing capacity is defined in Equation~\\ref{eq:borrowing-capacity}.\n\n\\begin{equation}\\label{eq:borrowing-capacity}\n \\mathsf{BC} = \\sum{Value\\ of\\ Collateral_i\\times \\mathbf{LT}_i}\n\\end{equation}\n\\item[Health Factor ($\\mathsf{HF}$):] The health factor measures the collateralization status of a position, defined as the ratio of the borrowing capacity over the outstanding debts (cf.\\ Equation~\\ref{eq:health-factor}).\n\n\\begin{equation}\\label{eq:health-factor}\n\\mathsf{HF}=\\frac{\\mathsf{BC}}{\\sum{Value\\ of\\ Debt_i}}\n\\end{equation}\nIf $\\mathsf{HF}<1$, the collateral becomes eligible for liquidation.\n\\end{description}\n\n\n\n\n\n\n\\section{Systematization of Lending and Liquidation Protocols}\\label{sec:existing-protocols}\nIn this section, we systematize how the current borrowing mechanisms and their specific liquidation processes operate. \n\\subsection{Borrowing and Lending System Model}\\label{sec:borrowingandlendingsystemmodel}\n\n\\begin{figure}[tb!]\n \\centering\n \n \n \n \n \n \n \n \n \n \n \n \n \n \\includegraphics[width=0.9\\columnwidth]{figures\/high_level_system_diagrams.pdf}\n \\caption{High-level system diagram of a lending pool system.\n }\n \\label{fig:liquidation_system_diagrams}\n\\end{figure}\n\nThe following aims to summarize the different actors engaging in borrowing and lending on a blockchain (cf.\\ Figure~\\ref{fig:liquidation_system_diagrams}).\n\nA \\textbf{lender} is an actor with surplus capital who would like to earn interest payments on its capital by lending funds to a third party, e.g., a borrower.\n\nA \\textbf{borrower} provides collateral to borrow assets from a lender. The borrower is liable to pay regular interest fees to the lender (typically measured as percentage of the loan). As lending in blockchains is typically performed without KYC, the borrower has to collateralize a value that is greater than the borrowed loan. In pure lending\/borrowing platforms such as Aave, Compound and dYdX, the collateral from borrowers is also lent out as loans. Borrowers hence automatically act as lenders.\n\nWhile lending could be performed directly on a peer-to-peer basis, blockchain-based lending protocols often introduce a \\textbf{lending pool} governed by a smart contract. A pool can hold several cryptocurrencies, and users can interact with the pool to deposit or withdraw assets according to the rules defined by the smart contract code.\n\nA \\textbf{liquidator} observes the blockchain for unhealthy positions (i.e., the health factor is below~$1$) to be liquidated. Liquidators typically operate bots, i.e., automated tools which perform a blockchain lookup, price observation, and liquidation attempt, if deemed profitable. Liquidators are engaging in a competitive environment, where other liquidators may try to front-run each other~\\cite{daian2019flash}. Notably, an atomic liquidation (e.g., a fixed spread liquidation) is settled in one blockchain transaction, while non-atomic liquidation mechanisms (e.g., auctions) generally require liquidators to interact with the lending pool in multiple transactions.\n\n\n\\subsection{Systematization of Liquidation Mechanisms}\\label{sec:liquidationmechanisms}\nWe observe that existing lending platforms mainly adopt two distinct liquidation mechanisms. One mechanism is based on a non-atomic \\emph{English auction}~\\cite{krishna2009auction} process, and the other follows an atomic fixed spread strategy. We formalize the two existing dominating mechanisms as follows.\n\n\n\\subsubsection{Auction Liquidation}\nAn auction based liquidation mechanism follows the subsequent methodology:\n\\begin{enumerate}\n \\item A loan becomes eligible for liquidation (i.e., the health factor drops below $1$).\n \\item A liquidator starts the auction process (which can last several hours).\n \\item Interested liquidators provide their bids (e.g., the highest bid receives the loan collateral).\n \\item The auction ends according to the rules set forth in the auction contract.\n\\end{enumerate}\n\n\\begin{figure}[ht!]\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/makerdao.pdf}\n \\caption{Two-phase liquidation auction of MakerDAO. Any actor can invoke the public function \\emph{bite} to initiate the collateral auction and invoke the public function \\emph{deal} to finalize the liquidation after the auction terminates.}\n \\label{fig:maker-dao-auction}\n\\end{figure}\n\n\n\\paragraph{MakerDAO tend-dent auction} Specifically, MakerDAO employs a two-phase auction process, which we term \\emph{tend-dent auction} (cf.\\ Figure~\\ref{fig:maker-dao-auction}). \nWhen a position with $D$ value of debt and $C$ value of collateral is eligible for liquidation, i.e., $\\mathsf{HF}<1$, a liquidator is able to initiate the tend-dent auction. A liquidator is required to bid at a higher price than the last bid in the auction. The two-phase workflow is described as follows.\n\\begin{description}[leftmargin=10pt]\n \\item[Tend:] In the \\emph{tend} phase, liquidators compete by bidding to repay parts of the debt in exchange for the entire collateral. We denote the amount of debt committed to repay in each bid by $d_i$, s.t.~$d_i \\leq D$ and $d_i>d_{i-1}$. If the auction terminates in the tend phase, the winning bidder receives all the collateral (i.e., $C$). When $d_i$ reaches $D$, the auction moves into the \\emph{dent} phase.\n \\item[Dent:] In the dent phase, liquidators compete by bidding to accept decreasing amounts of collateral in exchange for the full debt (i.e., $D$) they will end up repaying. We denote the amount of collateral committed in each bid by $c_i$, s.t.~$c_i \\leq C$ and $c_i 1$}\\;\n}\n\\BlankLine\n\\Fn{\\Liquidate{$\\mathcal{POS}$, $repay$}}{\n $\\mathcal{POS}'\\leftarrow\\left \\langle C - repay\\times (1+\\mathbf{LS}) , D - repay \\right \\rangle$\\;\n \\KwRet{$\\mathcal{POS}'$}\\;\n}\n\\BlankLine\n$repay_1 \\leftarrow \\operatorname{argmax}_r$ \\IsLiquidatable{\\Liquidate{$\\mathcal{POS}, r$}}\\;\n$\\mathcal{POS}'\\leftarrow$ \\Liquidate{$\\mathcal{POS}, repay_1$}\\;\n$repay_2 \\leftarrow \\mathcal{POS}'.D \\times \\mathbf{CF}$\\;\n\n\\caption{Optimal fixed spread liquidation strategy.}\n\\label{alg:optimalfixedspreadliquidation}\n\\end{algorithm}\n\n\\subsubsection{Optimality Analysis} Given a liquidatable borrowing position with $C$ collateral value and $D$ debt (cf.\\ Equation~\\ref{eq:position}), we proceed to analyze the profit of our optimal strategy.\n\n\\begin{equation}\\label{eq:position}\n \\mathcal{POS} = \\left \\langle C, D \\right \\rangle\n\\end{equation}\n$\\mathbf{LT}$, $\\mathbf{LS}$, $\\mathbf{CR}$ denote the liquidation threshold, liquidation spread and close factor respectively (cf.\\ Section~\\ref{sec:terminology}).\n\nFollowing Algorithm~\\ref{alg:optimalfixedspreadliquidation}, the repaid debt amounts in the two successive liquidations are given in Equation~\\ref{eq:repay1} and~\\ref{eq:repay2}. Note that the $D-\\mathbf{LT}\\cdot C > 0$ because $\\mathcal{POS}$ is liquidatable (i.e., the debt is greater than the borrowing capacity). We show in Appendix~\\ref{app:reasonalconfiguration} that a reasonable fixed spread liquidation configuration satisfies $1-\\mathbf{LT}(1+\\mathbf{LS})>0$.\n\n\\begin{equation}\\label{eq:repay1}\n\\begin{aligned}\n repay_1 &= \\operatorname{argmax}_r \\frac{\\mathbf{LT}(C-r(1+\\mathbf{LS}))}{D-r}\\geq 1\\\\\n &=\\frac{D-\\mathbf{LT}\\cdot C}{1-\\mathbf{LT}(1+\\mathbf{LS})}\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\\label{eq:repay2}\n repay_2 = \\mathbf{CF}\\left(D-repay_1\\right)= \\mathbf{CF}\\left(D-\\frac{D-\\mathbf{LT}\\cdot C}{1-\\mathbf{LT}(1+\\mathbf{LS})}\\right)\n\\end{equation}\nThe overall profit of the two liquidations is shown in Equation~\\ref{eq:overallprofit}.\n\\begin{equation}\\label{eq:overallprofit}\n\\begin{aligned}\n profit_{o} &= (repay_1 + repay_2)\\times \\mathbf{LS}\\\\\n &=\\mathbf{LS}\\cdot\\mathbf{CF}\\cdot D+\\mathbf{LS}(1-\\mathbf{CF})\\left(\\frac{D-\\mathbf{LT}\\cdot C}{1-\\mathbf{LT}(1+\\mathbf{LS})}\\right)\n\\end{aligned}\n\\end{equation}\n\nIf the liquidator instead chooses to perform the up-to-close-factor strategy, the repay amount is $\\mathbf{CF}\\cdot D$ and the profit hence is $profit_{c}=\\mathbf{LS}\\cdot\\mathbf{CF}\\cdot D$. Therefore, the optimal strategy can yield more profit than the up-to-close-factor strategy. The increase rate of the liquidation profit is shown in Equation~\\ref{eq:increaserate}.\n\n\\begin{equation}\\label{eq:increaserate}\n \\Delta R_{profit} = \\frac{profit_o-profit_c}{profit_c}=\\frac{\\mathbf{CF}}{1-\\mathbf{CF}}\\cdot\\frac{1-\\mathbf{LT}\\cdot\\mathsf{CR}}{1-\\mathbf{LT}(1+\\mathbf{LS})}\n\\end{equation}\nwhere $\\mathsf{CR}=\\frac{C}{D}$ is the collateralization ratio (cf.\\ Section~\\ref{sec:terminology}). We notice that the optimal strategy is more effective when $\\mathsf{CR}$ is low. \n\n\\subsubsection{Case Study}\nIn the following, we study the most profitable fixed spread liquidation transaction (\\empirical{$4.04$M}~USD\\xspace)\\footnote{Transaction hash: \\etherscantx{0x53e09adb77d1e3ea593c933a85bd4472371e03da12e3fec853b5bc7fac50f3e4}} we detect and showcase how the optimal fixed spread liquidation strategy increases the profit of a liquidator. In this Compound liquidation, the liquidator first performs an oracle price update\\footnote{Compound allows any entity to update the price oracle with authenticated messages signed by, for example, off-chain price sources.}, which renders a borrowing position liquidatable. The liquidator then liquidates the position within the same transaction. In Table~\\ref{tab:casestudystatus}, we present the status change of the position following the price update. Before the price update (block~\\block{11333036}), the position owns a total collateral of $135.07$M~USD (with a borrowing capacity of $101.30$M~USD), and owes a debt of~$101.18$M~USD. After the price of DAI increases from $1.08$ to $1.095299$~USD\/DAI, the total debt reaches $102.61$M~USD, while the borrowing capacity is only $102.55$M~USD. The health factor drops below~$1$, and hence the position becomes liquidatable.\n\n\\begin{table}[tb!]\n\\centering\n\\caption{Status of the borrowing position (\\account{0x909b443761bbD7fbB876Ecde71a37E1433f6af6f}). Note we ignore the tiny amount of collateral and debt in USDT that the borrower owns and owes. The liquidation thresholds (i.e., $\\mathbf{LT}$) of DAI and USDC are both~$0.75$.}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}ccc|c|c@{}}\n\\toprule\n\\multirow{2}{*}{\\bf Token} & \\multirow{2}{*}{\\bf Collateral} & \\multirow{2}{*}{\\bf Debt} & \\multicolumn{2}{c}{\\bf Price (USD)} \\\\ \\cmidrule(l){4-5} \n& & & Block \\block{11333036} & After Price Update \\\\ \\midrule\nDAI & $108.51$M & $93.22$M & $1.08$ & $1.095299$ \\\\\nUSDC & $17.88$M & $506.64$k & $1$ & $1$ \\\\ \\midrule\\midrule\n\\multicolumn{3}{c|}{\\bf Total Collateral (USD)} & $135.07$M & $136.73$M \\\\\n\\multicolumn{3}{c|}{\\bf Borrowing Capacity (USD)} & $101.30$M & $102.55$M \\\\\n\\multicolumn{3}{c|}{\\bf Total Debt (USD)} & $101.18$M & $102.61$M \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\label{tab:casestudystatus}\n\\end{table}\n\n\\begin{table}[tb!]\n\\centering\n\\caption{The depiction of liquidation strategies. At the time of the original liquidation, the price of DAI is $1.095299$~USD\/DAI. The close factor is $50\\%$. The optimal strategy is the most profitable liquidation mechanism for the liquidator.}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}c|l|l@{}}\n\\toprule\n\\textbf{Original liquidation} & \\multicolumn{2}{l}{\\begin{tabular}[c]{@{}l@{}}Repay $46.14$M~USD\\\\ Receive $49.83$M~DAI\\\\ \\emph{Profit} $3.69$M~DAI\\end{tabular}} \\\\ \\midrule\\midrule\n\\textbf{Up-to-close-factor strategy} & \\multicolumn{2}{l}{\\begin{tabular}[c]{@{}l@{}}Repay $46.61$M~DAI\\\\ Receive $50.34$M~DAI\\\\ \\emph{Profit} $3.73$M~DAI\\end{tabular}} \\\\ \\midrule\\midrule\n\\multirow{4}{*}{\\textbf{Optimal strategy}} & Liquidation 1 & Liquidation 2 \\\\ \\cmidrule(l){2-3} \n & \\begin{tabular}[c]{@{}l@{}}Repay $296.61$K~DAI\\\\ Receive $320.34$K~DAI\\\\ \\emph{Profit} $23.73$K~DAI\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Repay $46.46$M~DAI\\\\ Receive $50.18$M~DAI\\\\ \\emph{Profit} $3.72$M~DAI\\end{tabular} \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\label{tab:liquidation-strategy-comparisons}\n\\end{table}\n\nTo evaluate the up-to-close-factor strategy and our optimal liquidation strategy, we implement the original liquidation and the two liquidation strategies\\footnote{We publish the smart contract code at \\url{https:\/\/anonymous.4open.science\/r\/An-Empirical-Study-of-DeFi-Liquidations-Anonymous\/CompoundLiquidationCaseStudy.sol}.} in Solidity v0.8.4\\footnote{\\url{https:\/\/docs.soliditylang.org\/en\/v0.8.4\/}}.\nWe execute them on the corresponding blockchain states\\footnote{We fork the Ethereum mainchain locally from block~\\block{11333036} and apply all the transactions executed prior to the original liquidation transaction in block~\\block{11333037}. We then execute the liquidation strategies to ensure that they are validated on the exact same state of the original liquidation.} and present the results in Table~\\ref{tab:liquidation-strategy-comparisons}. We find that the optimal strategy is superior to the up-to-close-factor strategy and can generate an additional profit of $49.26$K~DAI (\\empirical{$53.96$K}~USD\\xspace) compared to the original liquidation.\n\n\n\\subsubsection{Mitigation}\nThe aforementioned optimal strategy defeats the original intention of a close factor, which incurs undesirably additional losses to borrowers. A possible mitigation solution is that for every position only one liquidation is permitted within one block. Such a setting enforces a liquidator adopting the optimal strategy to settle the two liquidations in two blocks, which decreases the success probability.\n\nWe proceed to assume the existence of a mining liquidator with a mining power $\\alpha$. Given a liquidation opportunity, the up-to-close-factor strategy produces a profit of $profit_c$, while the optimal strategy yields $profit_{o_1}$ and $profit_{o_2}$ respectively in the two successive liquidations. We further assume that there is no ongoing consensus layer attack (e.g., double-spending), implying that a miner with an $\\alpha$ fraction mining power mines the next block is with a probability of $\\alpha$. We hence derive the the expected profit of the two strategies as shown in Equation~\\ref{eq:expected-profit-close} and~\\ref{eq:expected-profit-optimal}\\footnote{To ease understanding, we assume other competing liquidators adopt the up-to-close-factor strategy.}.\n\n\\begin{equation}\\label{eq:expected-profit-close}\n \\mathbb{E}[\\text{up-to-close-factor}] = \\alpha\\cdot profit_c\n\\end{equation}\n\\begin{equation}\\label{eq:expected-profit-optimal}\n \\mathbb{E}[\\text{optimal}] = \\alpha\\cdot profit_{o_1} + \\alpha^2\\cdot profit_{o_2}\n\\end{equation}\nThe liquidator is incentivized to perform the optimal strategy only when $\\mathbb{E}[\\text{optimal}] > \\mathbb{E}[\\text{up-to-close-factor}]$, leading to Equation~\\ref{eq:optimal-rational-condition}.\n\n\\begin{equation}\\label{eq:optimal-rational-condition}\n\\alpha > \\frac{profit_c-profit_{o_1}}{profit_{o_2}}\n\\end{equation}\nIntuitively, $profit_{o_1}$ is relatively small compared to $profit_{c}$ and $profit_{o_2}$ because the liquidator needs to keep the position unhealthy after the first liquidation. The expected profit in the second liquidation then should be sufficient to cover the opportunity cost in the first one, which is typically unattainable.\nInstantiating with our case study liquidation (cf.\\ Table~\\ref{tab:casestudystatus}), we show that a rational mining liquidator would attempt the optimal strategy in two consecutive blocks only if its mining power is over $99.68\\%$. Therefore, we conclude that the one liquidation in one block effectively reduces the expected profit of the optimal liquidation strategy, protecting borrowers from a further liquidation losses.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\\label{sec:related-work}\n\n\\point{Blockchains and DeFi}\nThere is a growing body of literature on blockchains and DeFi. Qin \\textit{et al.}~\\cite{qin2020attacking} study flash loan attacks and present an optimization approach to maximize the profit of DeFi attacks. Zhou \\textit{et al.}~\\cite{zhou2020high} analyze sandwich attacks in decentralized exchanges. Eskandari \\textit{et al.}~\\cite{eskandari2019sok} provide an overview of the blockchain front-running attacks. Daian \\textit{et al.}~\\cite{daian2019flash} investigate the front-running attacks in decentralized exchanges and propose the concept of \\emph{Miner Extractable Value} (MEV), a financial revenue miners can extract through transaction order manipulation. Qin \\textit{et al.}~\\cite{qin2021quantifying} quantify the extracted MEV on the Ethereum blockchain, including fixed spread liquidations, and present a generalized front-running algorithm, transaction replay. Zhou \\textit{et al.}~\\cite{zhou2021just} propose a framework called \\textsc{DeFiPoser} that allows to automatically create profit-generating transactions given the blockchain state.\n\n\\point{Blockchain Borrowing and Lending Markets}\nDarlin \\textit{et al.}~\\cite{darlin2020optimal} study the MakerDAO liquidation auctions. The authors optimize the costs for participating in the auctions and find that most auctions conclude at higher than optimal prices. The work appears real-world relevant, as it considers the transaction fees, conversion costs and cost of capital, yet it does not consider potential gas bidding contests by the end of MakerDAO auctions~\\cite{daian2019flash}. Kao \\textit{et al.}~\\cite{kao2020analysis} and ZenGo~\\cite{zengo-compound} are to our knowledge the first to have investigated Compound's liquidation mechanism (the third biggest lending protocol in terms of USD at the time of writing). Perez~\\textit{et al.}~\\cite{perez2020liquidations} follow up with a report that focuses on additional on-chain analytics of the Compound protocol. DragonFly Research provides a blog post~\\cite{medium-liquidation} about the liquidator profits on Compound, dYdX and MakerDAO. Minimizing financial deposit amounts in cryptoeconomic protocols, while maintaining the same level of security is studied in Balance~\\cite{harzbalance}.\n\n\\point{Liquidations in Traditional Finance} Liquidations are essential to traditional finance (TradFi) and are well studied in the related literature~\\cite{titman1984effect,shleifer1992liquidation,alderson1995liquidation,almgren1999value,reinhart2011liquidation}. We remark that liquidations in blockchain systems are fundamentally different from those in TradFi in terms of high-level designs and settlement mechanisms.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nDue to their significant volatility when compared to alternative financial vehicles cryptocurrencies are attracting speculators. Furthermore, because speculators seek to further their risk exposure, non-custodial lending and borrowing protocols on blockchains are thriving. The risks of borrowing, however, manifests themselves in the form of liquidation profits claimed by liquidators.\n\nIn this paper we study the lending platforms that capture $85$\\% of the blockchain lending market. We systematize the most prevalent liquidation mechanisms and find that many liquidations sell excessive amounts of borrower's collateral. In this work we provide extensive data analytics covering over $2$ years the prevalent $4$ lending protocols. We systematize their respective liquidation mechanisms and show that most liquidation systems are unfavorable to the borrowers. We finally show an optimal liquidation strategy which we have not yet observed in the wild.\n\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA \\emph{hole} in a graph is an induced cycle of length at least four. A \\emph{proper coloring} of a graph is a function that assigns to each vertex a color with the constraint that two adjacent vertices are not given the same color. The \\emph{chromatic number} of $G$, denoted by $\\chi(G)$, is the smallest number of colors needed to color the graph properly. All the colorings considered in the sequel are proper, so we just call them colorings. \nThe size of the largest clique of $G$ is denoted $\\omega(G)$.\nWe obviously have $\\omega(G)\\leq \\chi(G)$, and one may wonder whether the equality holds. In fact, it does not hold in the general case, and the simplest counter-example is any \\emph{odd hole}, i.e. any hole of odd length, for which $\\omega(G)=2$ but $\\chi(G)=3$. Graphs for which the equality $\\chi(G')=\\omega(G')$ holds for every induced subgraph $G'$ of $G$ are called \\emph{perfect}, and the Strong Perfect Graph Theorem \\cite{SPGT} proved that \na graph if perfect if and only if it is\n\\emph{Berge}, that is to say there is no odd hole in $G$ nor in its complement.\nIn order to get some upper bound on $\\chi(G)$, Gy\\'arf\\'as \\cite{G54} introduced the concept of $\\chi$-bounded class: a family $\\mathcal{G}$ of graphs is called $\\chi$\\emph{-bounded} if there exists a function $f$ such that $\\chi(G')\\leq f(\\omega(G'))$ whenever $G'$ is an induced subgraph of $G\\in \\mathcal{G}$.\n\nThis notion has widely been studied since then, in particular in hereditary classes (\\emph{hereditary} means closed under taking induced subgraph). \nA classical result of Erd\\H{o}s \\cite{E40} asserts that there exist graphs with arbitrarily large \\emph{girth} (that is, the length of the shortest induced cycle) and arbitrarily large chromatic number. Thus forbidding only one induced subgraph $H$ may lead to a $\\chi$-bounded class only if $H$ is acyclic. It is conjectured that this condition is also sufficient \\cite{G32, S52}, but it is proved only if $H$ is a path, a star \\cite{G54} or a tree of radius two \\cite{TreeRadius2} (or three, with additional conditions \\cite{TreeRadius3}). Scott \\cite{ScottSubdivTree} also proved it for any tree $H$, provided that we forbid every induced subdivision of $H$, instead of just $H$ itself.\n\nConsequently, forbidding holes in order to get a $\\chi$-bounded class is conceivable only if we forbid infinitely many hole lengths. \nTwo parameters should be taken into account: first, the length of the holes, and secondly, the parity of their lengths. \nIn this respect, Gy\\'arf\\'as \\cite{G54} made a famous series of three conjectures. The first one asserts the class of graphs with no odd hole is $\\chi$-bounded. The second one asserts that, for every $k$, the class of graphs with no hole of length at least $k$ is $\\chi$-bounded. The last one generalizes the first two conjectures and asserts that for every $k$, the class of graphs with no odd hole of length at least $k$ is $\\chi$-bounded. After several partial results \\cite{RS99, ScottCycles, CSSGyarfas}, the first and the second conjectures were recently solved by Chudnovsky, Scott and Seymour \\cite{SSOddHoles, CSSLongHoles}. Moreover, we learned while writing this article that Scott and Seymour claimed\\footnote{Their article is still in preparation.} to have proved a very general result implying the triangle-free case of the third conjecture (which would also imply the result of this paper): for every $k\\geq 0$, every triangle-free graph with large enough chromatic number admits a sequence of holes of $k$ consecutive lengths.\n\n\nThe class of even-hole-free graphs has extensively been studied from a structural point of view. A decomposition theorem together with a recognition algorithm have been found by Conforti, Cornu\\'ejols, Kapoor and Vu\\v{s}kovi\\'c \\cite{EvenHole1, EvenHole2,ChangLu}. Reed conjectured \\cite{R58} that every even-hole-free graphs has a vertex whose neighborhood is the union of two cliques (called a \\emph{bisimplicial} vertex), which he and his co-authors proved \\cite{EvenHoleBisimplicial} a few years later. As a consequence, they obtained that every even-hole-free graph $G$ satisfies $\\chi(G)\\leq 2\\omega(G)-1$.\n\nForbidding $C_4$ is in fact a strong restriction since $C_4$ can also be seen as the complete bipartite graph $K_{2,2}$: K\\\"uhn and Osthus \\cite{KuhnOsthus} proved that for every graph $H$ and for every integer $s$, every graph of large average degree (with respect to $H$ and $s$) with no $K_{s,s}$ as a (non-necessarily induced) subgraph contains an induced subdivision of $H$, where each edge is subdivided at least once. This strong result implies that $\\chi$ is bounded in any class $\\mathcal{C}$ defined as graphs with no triangles, no induced $C_4$ and no cycles of length divisible by $k$, for any fixed integer $k$. Indeed, let $G\\in \\mathcal{C}$ be a minimal counter-example to $\\chi(G)\\leq t$ (with $t$ chosen large enough with respect to $k$), then it has large minimum degree. Moreover it has neither induced $C_4$ nor triangles, consequently it has no $C_4$ subgraphs. By K\\\"uhn and Osthus' theorem, there exists an induced subdivision $H$ of $K_\\ell$ for some well-chosen integer $\\ell$ depending on $k$. Consider $K_\\ell$ as an auxiliary graph where we color each edge with $c\\in\\{1,\\ldots, k \\}$ if this edge is subdivided $c$ times modulo $k$ in $H$. By Ramsey's theorem \\cite{Ramsey}, if $\\ell$ is large enough, then we can find a monochromatic clique $K$ of size $k$. Let $C_0$ be a Hamiltonian cycle through $K$ and call $C$ the corresponding cycle in the subdivided edges in $H$. Since $K$ was monochromatic in $K_\\ell$, the edges used in $C_0$ are subdivided the same number of times modulo $k$, consequently $C$ has length divisible by $k$. Moreover, it is an induced cycle since each edge is subdivided at least once in $H$.\n\n\nThis is why we are interested in finding a $\\chi$-boundedness result when every even hole except $C_4$ is forbidden, which was conjectured by Bruce Reed \\cite{ReedPrivate}. In this paper, we achieve a partial result by forbidding also triangles. This is a classical step towards $\\chi$-boundedness, and Thomass\\'e \\emph{et al.} \\cite{FPTBullFree} even asked whether this could always be sufficient, namely: does there exist a function $f$ such that for every class $\\mathcal{C}$ of graphs and any $G\\in \\mathcal{C}$, $\\chi(G)\\leq f(\\chi_T(G), \\omega(G))$, where $\\chi_T(G)$ denotes the maximum chromatic number of a triangle-free induced subgraph of $G$?\n\n\n\nThe result of this paper is closely related to the following recent one, by Bonamy, Charbit and Thomass\\'e, answering to a question by Kalai and Meshulam on the sum of Betti numbers of the stable set complex (see \\cite{Thomasse0mod3} for more details):\n\n\\begin{theorem}[\\cite{Thomasse0mod3}]\nThere exists a constant $c$ such that every graph $G$ with no induced cycle of length divisible by 3 satisfies $\\chi(G) < c$.\n\\end{theorem}\n\nIndeed, the so-called Parity Changing Path (to be defined below) is directly inspired by their Trinity Changing Path. The structure of the proofs also have several similarities.\n\n\n\n\n\\bigskip\n\n\\paragraph{Contribution} We prove the following theorem:\n\n\\begin{theorem} \\label{th: C chi borne}\nThere exists a constant $c$ such that every graph $G$ with no triangle and no induced cycle of even length at least 6 satisfies $\\chi(G) < c$.\n\\end{theorem}\n\nThe outline is to prove the result when the 5-hole is also forbidden (see Lemma \\ref{lem: sans C5} below), which should intuitively be easier, and then deduce the theorem for the general case.\n\n\n\nTo begin with, let us introduce and recall some notations:\nthe class under study, namely graphs with no triangle and no induced $C_{2k}$ with $k\\geq 3$ (meaning that every even hole is forbidden except $C_4$) will be called \\noindent {\\bf Definitions. }{$\\mathcal{C}_{3, 2k\\geq 6}$} for short. Moreover, we will consider in Subsection \\ref{sec: sans C5} the subclass \\noindent {\\bf Definitions. }{$\\mathcal{C}_{3, 5, 2k\\geq 6}$} of $\\mathcal{C}_{3, 2k\\geq 6}$ in which the 5-hole is also forbidden. For two subsets of vertices $A, B \\subseteq V$, $A$ \\noindent {\\bf Definitions. }{dominates} $B$ if $B\\subseteq N(A)$.\nA \\noindent {\\bf Definitions. }{major connected component} of $G$ is a connected component $C$ of $G$ for which $\\chi(C)=\\chi(G)$. Note that such a component always exists. \nFor any induced path $P=x_1x_2\\cdots x_\\ell$ we say that $P$ is a path from its \\noindent {\\bf Definitions. }{origin} $x_1$ to its \\noindent {\\bf Definitions. }{end} $x_\\ell$ or an \\noindent {\\bf Definitions. }{$x_1x_\\ell$-path}. Its \\noindent {\\bf Definitions. }{interior} is $\\{x_2, \\ldots, x_{\\ell-1}\\}$ and its \\noindent {\\bf Definitions. }{length} is $\\ell-1$. \n\nMoreover, we use a rather common technique called a \\emph{levelling} \\cite{SSOddHoles, CSSGyarfas} :\ngiven a vertex $v$, the \\noindent {\\bf Definitions. }{$v$-levelling} is the partition $(N_0, N_1, \\ldots, N_k, \\ldots)$ of the vertices according to their distance to $v$: $N_k$ is the set of vertices at distance exactly $k$ from $v$ and is called the \\noindent {\\bf Definitions. }{$k$-th level}. In particular, $N_0=\\set{v}$ and $N_1=N(v)$. We need two more facts about levellings: \nif $x$ and $y$ are in the same part $N_k$ of a $v$-levelling, we call an \\noindent {\\bf Definitions. }{upper $x y$-path} any shortest path from $x$ to $y$ among those with interior in $N_0\\cup \\cdots \\cup N_{k-1}$. Observe that it always exists since there is an $xv$-path and a $vy$-path (but it may take shortcuts; in particular, it may be just one edge). Moreover, in any $v$-levelling, there exists $k$ such that $\\chi(N_k)\\geq \\chi(G)\/2$: \nindeed, if $t$ is the maximum of $\\chi(N_i)$ over all levels $N_i$, one can color $G$ using $2t$ colors by coloring $G[N_{i}]$ with the set of colors $\\{1, \\ldots, t\\}$ if $i$ is odd, and with the set of colors $\\{t+1, \\ldots , 2t\\}$ if $i$ is even. Such a level with chromatic number at least $\\chi(G)\/2$ is called a \\noindent {\\bf Definitions. }{colorful} level.\n\n\n\nLet us now introduce the main tool of the proof, called \\noindent {\\bf Definitions. }{Parity Changing Path} (\\noindent {\\bf Definitions. }{PCP} for short) which, as already mentioned, is inspired by the \\emph{Trinity Changing Path (TCP)} appearing in \\cite{Thomasse0mod3}: intuitively (see Figure \\ref{fig: PCP} for an unformal diagram), a PCP is an induced sequence of induced subgraphs and paths $(G_1, P_1, \\ldots, G_\\ell, P_\\ell, H)$ such that each block $G_i$ can be crossed by two possible paths of different parities, and the last block $H$ typically is a stock of big chromatic number.\nFormally, a \\noindent {\\bf Definitions. }{PCP of order $\\ell$} in $G$ is a sequence of induced subgraphs $G_1, \\ldots, G_{\\ell}, H$ (called \\noindent {\\bf Definitions. }{blocks}; the $G_i$ are the \\noindent {\\bf Definitions. }{regular} blocks) and induced paths $P_1, \\ldots , P_{\\ell}$ such that the origin of $P_i$ is some vertex $y_i$ in $G_i$, and the end of $P_i$ is some vertex $x_{i+1}$ of $G_{i+1}$ (or of $H$ if $i=\\ell$). Apart from these special vertices which belong to exactly two subgraphs of the PCP, the blocks and paths $G_1, \\ldots, G_\\ell, H, P_1, \\ldots, P_\\ell$ composing the PCP are pairwise disjoint. The only possible edges have both endpoints belonging to the same block or path.\nWe also have one extra vertex $x_1\\in G_1$ called the \\noindent {\\bf Definitions. }{origin} of the PCP. \nMoreover in each block $G_i$, there exists one induced $x_i y_i$-path of odd length, and one induced $x_i y_i$-path of even length (these paths are not required to be disjoint one from each other). In particular $x_i \\neq y_i$ and $x_i y_i$ is not an edge. For technical reasons that will appear later, we also require that $H$ is connected, every $G_i$ has chromatic number at most 4 and every $P_i$ has length at least 2. Finally the chromatic number of $H$ is called the \\noindent {\\bf Definitions. }{leftovers}. \n\nIn fact in Subsection \\ref{sec: Yes C4 avec C5}, we need a slightly stronger definition of PCP: a \\noindent {\\bf Definitions. }{strong PCP} is a PCP for which every $G_i$ contains an induced $C_5$.\n\n\\begin{figure}\n\\center\n\\includegraphics[scale=1]{fig\/PCP}\n\\caption{An informal diagram for a PCP of order 3. Grey curved lines stand for the even and odd length $x_iy_i$-paths.}\n\\label{fig: PCP}\n\\end{figure}\n\n\\bigskip\n\nWe first bound the chromatic number in $\\mathcal{C}_{3, 5, 2k\\geq 6}$ (see Lemma \\ref{lem: sans C5} below), which is easier because we forbid one more cycle length, and then deduce the theorem for $\\mathcal{C}_{3, 2k\\geq 6}$. The proofs for $\\mathcal{C}_{3, 2k\\geq 6}$ and $\\mathcal{C}_{3, 5, 2k\\geq 6}$ follow the same outline, which we informally describe here:\n\n\\begin{enumerate}\n\\item If $\\chi(G)$ is large enough, then for every vertex $v$ we can grow a PCP whose origin is $v$ and whose leftovers are large (Lemmas \\ref{lem: debut PCP sans C5}, \\ref{lem: existence PCP sans C5} and then Lemma \\ref{lem: debut PCP}).\n\\item Using (i), if $\\chi(G)$ is large enough and $(N_0, N_1, \\ldots)$ is the $v$-levelling,\nwe can grow a \\noindent {\\bf Definitions. }{rooted} PCP:\nit is a PCP in a level $N_k$, which has a \\noindent {\\bf Definitions. }{root}, \\emph{i.e.} a vertex in the previous level $N_{k-1}$ whose unique neighbor in the PCP is the origin (Lemma \\ref{lem: rooted PCP sans C5} and then Lemma \\ref{lem: rooted PCP}).\n\\item Given a rooted PCP in a level $N_k$, if a vertex $x\\in N_{k-1}$ has a neighbor in the last block $H$, then it has a neighbor in every regular block $G_i$ (Lemma \\ref{obs: sommets riches sans C5}).\n\\item Given a rooted PCP of order $\\ell$ in a level $N_k$ and a stable set $S$ in $N_{k-1}$, the set of neighbors of $S$ inside $N_k$ can not have a big chromatic number. Consequently, the \\noindent {\\bf Definitions. }{active lift} of the PCP, defined as $N(G_\\ell)\\cap N_{k-1}$, has high chromatic number (Lemmas \\ref{lem: shadow stable bornee sans C5}, \\ref{lem: active lift gros si neighb stable gros} and \\ref{lem: shadow borne bornee sans C5} and then Lemmas \\ref{lem: active lift gros si neighb stable gros}, \\ref{lem: shadow stable bornee}, \\ref{lem: shadow borne bornee}).\n\\item The final proofs put everything together: consider a graph of $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ (resp. $\\mathcal{C}_{3, 2k\\geq 6}$) with chromatic number large enough. Then pick a vertex $v$, let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level. By (ii), grow inside $N_k$ a rooted PCP $P$. Then by (iv), get an active lift $A$ of $P$ inside $N_{k-1}$ with big chromatic number. Grow a rooted PCP $P'$ inside $A$, and get an active lift $A'$ of $P'$ inside $N_{k-2}$ with chromatic number big enough to find an edge $xy$ (resp. a 5-hole $C$) in $A'$ . Then ``clean\" $P'$ in order to get a stable set $S$ inside the last regular block of $P'$, dominating this edge (resp. hole). Now find an even hole of length $\\geq 6$ in $\\set{x,y}\\cup S\\cup P$ (resp. $C\\cup S\\cup P$), a contradiction.\n\\end{enumerate}\n\n\n\\section{Forbidding 5-holes}\n\\label{sec: sans C5}\n\nThis section is devoted to the proof of the following lemma :\n\n\\begin{lemma} \\label{lem: sans C5}\nThere exists a constant $c'$ such that every graph $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ satisfies $\\chi(G)< c'$.\n\\end{lemma}\n\n\nWe follow the outline described above. Let us start with step (i):\n\n\\begin{lemma} \\label{lem: debut PCP sans C5}\nLet $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ be a connected graph and $v$ be any vertex of $G$. For every $\\delta$ such that $\\chi(G)\\geq \\delta \\geq 18$, there exists a PCP of order 1 with origin $v$ and leftovers at least $h(\\delta)=\\delta\/2 -8$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is illustrated on Figure \\ref{fig: build PCP overview}. Let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level.\nLet $N'_k$ be a major connected component of $G[N_k]$, so $\\chi(N'_k)\\geq \\delta\/2$.\n Let $xy$ be an edge of $N'_k$, and $x'$ (resp. $y'$) be a neighbor of $x$ (resp. $y$) in $N_{k-1}$.\nLet $Z' = N(\\{x',y',x,y\\})\\cap N'_k$ and $Z=Z' \\setminus \\{x,y\\}$. \n Let $z\\in Z$ be a vertex having a neighbor $z_1$ in a major connected component $M_1$ of $N'_k \\setminus Z'$. Observe that $N'_k \\setminus Z'$ is not empty since $\\chi(Z')\\leq 6$ (the neighborhood of any vertex is a stable set).\n The goal is now to find two $vz$-paths $P$ and $P'$ of different parities with interior in $G[N_0 \\cup \\ldots \\cup \\{ x', y' \\} \\cup \\{x,y\\}]$. Then we can set $G_1=G[P\\cup P']$, $P_1=G[\\{z,z_1\\}]$ and $H=G[M_1]$ as parts of the wanted PCP. In practice, we need to be a little more careful to ensure the condition on the length of $P_1$ and the non-adjacency between $z$ and $H$, which is described after finding such a $P$ and a $P'$.\n\nLet $P_0$ (resp. $P'_0$) be a $vx'$-path (resp. $vy'$-path) of length $k-1$ (with exactly one vertex in each level).\nBy definition of $Z$, $z$ is connected to $\\{x',y',x,y\\}$.\n\\begin{enumerate}\n\\item (see Figure \\ref{fig: build PCP case 1}) If $z$ is connected to $x$ or $y$, say $x$, then $z$ is connected neither to $x'$ nor to $y$, otherwise it creates a triangle. We add the path $x'xz$ to $P_0$ to form $P$. Similarly, we add either the edge $y'z$ if it exists, or else the path $y' y x z$ to $P'_0$ to form $P'$. Observe that $P'$ is indeed an induced path since there is no triangle.\n\\item (see Figure \\ref{fig: build PCP case 2}) Otherwise, $z$ is connected neither to $x$ nor to $y$, thus $z$ is connected to exactly one of $x'$ and $y'$, since otherwise it would either create a triangle $x', y', z$ or a 5-hole $z x' x y y'$, so say $zx'\\in E$ and $zy'\\notin E$. We add the edge $x'z$ to $P_0$ to form $P$. We add the path $y' x' z$ if $y'x'\\in E$, otherwise add the path $y' y x x' z$ to $P'_0$ to form $P'$. Observe that this is an induced path since $G$ has no triangle and no 5-hole.\n\\end{enumerate}\nNow come the fine tuning.\nChoose in fact $z_1\\in M_1\\cap N(z)$ so that $z_1$ is connected to a major connected component $M_2$ of $M_1 \\setminus N(z)$. Choose $z_2$ a neighbor of $z_1$ in $M_2$ such that $z_2$ is connected to a major connected component $M_3$ of $M_2\\setminus N(z_1)$. We redefine $H=G[\\{z_2 \\cup M_3\\}]$ and $P_1=G[\\{z,z_1,z_2\\}]$. Then $P_1$ is a path of length 2, $G_1$ is colorable with 4 colors as the union of two induced paths, and $H$ is connected. Moreover $H$ has chromatic number at least $\\chi(N'_k)-\\chi(Z')-\\chi(N(z))-\\chi(N(z_1))$. Since the neighborhood of any vertex is a stable set, $\\chi(Z')\\leq 6$ and $\\chi(N(z)), \\chi(N(z_1))\\leq 1$. Thus $\\chi(H)\\geq \\delta\/2- 8$.\n\n\\end{proof}\n\n\\begin{figure}\n\\center\n\n\\subfigure[Overview of the situation\\label{fig: build PCP overview}]{\\includegraphics[scale=1, page=1]{fig\/buildPCP}}\n\\subfigure[Case (i)\\label{fig: build PCP case 1}]{\\includegraphics[scale=1, page=2]{fig\/buildPCP}}\n\n\\subfigure[Case (ii)\\label{fig: build PCP case 2}]{\\includegraphics[scale=1, page=3]{fig\/buildPCP}}\n\\caption{Illustrations for the proof of Lemma \\ref{lem: debut PCP sans C5}. Dashed edges stand for non-edges, and grey edges stand for edges that may or may not exist.}\n\\label{fig: build PCP}\n\\end{figure}\n\nWe can iterate the previous process to grow some longer PCP. In the following, for a function $f$ and an integer $k$, $f^{(k)}$ denotes the $k$-th iterate of $f$, that is to say that $f^{(k)}(x)=\\underbrace{(f \\circ \\ldots \\circ f)}_{k \\text{ times}} (x)$.\n\n\\begin{lemma}\n\\label{lem: existence PCP sans C5}\nLet $h(x)=x\/2-8$ be the function defined in Lemma \\ref{lem: debut PCP sans C5}. For every positive integers $\\ell, \\delta\\in \\mathbb{Z}_+$, if $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ is connected and satisfies $\\chi(G)\\geq \\delta$ and $h^{(\\ell-1)}(\\delta)\\geq 18$, then from any vertex $x_1$ of $G$, one can grow a PCP of order $\\ell$ with leftovers at least $h^{(\\ell)}(\\delta)$.\n\\end{lemma}\n\n\\begin{proof}\nWe prove the result by induction on $\\ell$. For $\\ell=1$, the result follows directly from Lemma \\ref{lem: debut PCP sans C5}. Now suppose it is true for $\\ell-1$, and let $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ be such that $\\chi(G)\\geq \\delta$ and $h^{(\\ell-1)}(\\delta)\\geq 18$. Then $\\delta\\geq h^{(\\ell-1)}(\\delta)\\geq 18$, so we can apply Lemma \\ref{lem: debut PCP sans C5} to get a PCP of order 1 and leftovers at least $h(\\delta)$ from any vertex $x_1$. Let $x_2$ be the common vertex between the last block $H$ and the first path $P_1$ of the PCP (as in the definition). Now apply the induction hypothesis to $H$, knowing that $H$ is connected, $\\chi(H)\\geq h(\\delta)=\\delta'$ and $h^{(\\ell-2)}(\\delta')\\geq 18$. Then we obtain a PCP of order $\\ell-1$ with origin $x_2$ and leftovers at least $h^{(\\ell-2)}(\\delta')$, which finishes the proof by gluing the two PCP together.\n\\end{proof}\n\n\n\nNow we grow the PCP in a level $N_k$ of high chromatic number, and we want the PCP to be rooted (\\emph{i.e.} there \nexists a root $u'\\in N_{k-1}$ that is adjacent to the origin $u$ of the PCP, but to no other vertex of the PCP). This is step (ii).\n\n\n\\begin{lemma}\n\\label{lem: rooted PCP sans C5}\nLet $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ be a connected graph, $v\\in V(G)$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $h$ be the function defined in Lemma \\ref{lem: debut PCP sans C5}. For every $k, \\delta$ such that $\\chi(N_k)\\geq \\delta+1$ and $h^{(\\ell-1)}(\\delta)\\geq 18$, there exists a rooted PCP of order $\\ell$ in $N_k$ with leftovers at least $h^{(\\ell)}(\\delta)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $N'_k$ be a major connected component of $N_k$ and $u\\in N'_k$. Consider a neighbor $u'$ of $u$ in $N_{k-1}$. Since there is no triangle, $N'_k\\setminus N(u')$ still has big chromatic number (at least $\\delta$), and let $N''_k$ be a major connected component of $N'_k\\setminus N(u')$. Let $z$ be a vertex of $N(u)\\cap N'_k$ having a neighbor in $N''_k$. Then we apply Lemma \\ref{lem: existence PCP sans C5} in $\\{z\\}\\cup N''_k$ to grow a PCP of order $\\ell$ from $z$ with leftovers at least $h^{(\\ell)}(\\delta)$. Now $u'$ has an only neighbor $z$ on the PCP, which is the origin.\n\\end{proof} \n \nLet us observe the properties of such a rooted PCP. We start with step (iii):\n\n\\begin{lemma} \\label{obs: sommets riches sans C5}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 2k\\geq 6}$, $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a rooted PCP $(G_1, P_1, \\ldots, G_\\ell, P_\\ell, H)$ of order $\\ell$ in a level $N_k$ for some $k$. If $x'\\in N_{k-1}$ has a neighbor $x$ in $H$, then $x$ has a neighbor in every $G_i$ for $1\\leq i \\leq \\ell$.\n\\end{lemma}\n\n\n\n\\begin{proof}\nLet $u$ be the origin of the PCP and $u'$ its root. Since $x'\\neq u'$ by definition of the root, there exists an upper $x'u'$-path $P_{up}$ of length at least one. \nConsider a $ux$-path $P$ inside the PCP. Let $v_1, \\ldots, v_r$ be the neighbors of $x'$ on this path, different from $x$ (if any), in this order (from $u$ to $x$).\nNow we can show that any regular block $G_i$ contains at least one $v_j$: suppose not for some index $i$, let $j$ be the greatest index such that $v_j$ is \\emph{before} $x_i$, \n\\emph{i.e.} $v_j\\in G_1\\cup P_1 \\cup \\cdots \\cup G_{i-1} \\cup P_{i-1}$.\n\nIf such an index does not exist (\\emph{i.e.} all the $v_j$ are after $G_i$), then there is an odd and an even path from $u$ to $v_1$ of length at least 3 by definition of a regular block,\nand this path does not contain any neighbor of $x'$. Close them to build two induced cycles by going through \n$x'$, $P_{up}$ and $u'$: one of them is an even cycle, and its length is at least 6.\n\nIf $j=r$ (\\emph{i.e.} all the $v_j$ are before $G_i$), then we can use the same argument with a path of well-chosen parity from $v_r$ to $x$, crossing $G_i$. \n\n\nOtherwise, there is an odd and an even path in the PCP between $v_j$ and $v_{j+1}$, crossing $G_i$, and its length is at least 4 because $x_i$ and $y_i$ are at distance at least 2 one from each other. We can close the even path path by going back and forth to $x$: this gives an even hole of length at least 6.\n\\end{proof}\n\nNote that, in the lemma above, $G$ is taken in $\\mathcal{C}_{3, 2k\\geq 6}$ and not in $\\mathcal{C}_{3, 5, 2k\\geq 6}$. In particular, we will use Lemma \\ref{obs: sommets riches sans C5} in the next section as well. Let us now continue with step (iv):\n \n\\begin{lemma} \\label{lem: shadow stable bornee sans C5}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. \nLet $S\\subseteq N_{k-1}$ be a stable set. Then $\\chi(N(S)\\cap N_k)\\leq 52$.\n\n\\end{lemma}\n\n\\begin{proof}\nLet $\\delta=\\chi(N(S)\\cap N_{k-1})-1$. Suppose by contradiction that $\\delta\\geq 52 $, then $h(\\delta)\\geq 18$ hence by Lemma \\ref{lem: rooted PCP sans C5}, we can grow a rooted PCP of order 2 inside $N(S)\\cap N_k$. Let $u$ be the origin of the PCP and $u'$ its root. Observe in particular that $S$ dominates $G_2$. Let $xy$ be an edge of $G_2$, and let $x'$ (resp. $y'$) be a neighbor of $x$ (resp. $y$) in $S$. \nBy Lemma \\ref{obs: sommets riches sans C5}, both $x'$ and $y'$ have a neighbor in $G_1$.\nThis gives an $x'y'$-path $P_{down}$ with interior in $G_1$. In order not to create an even hole nor a 5-hole by closing it with $x' x y y'$, we can ensure that $P_{down}$ is an even path of length at least 4. Moreover, there exists an upper $x'y'$-path $P_{up}$. Then either the hole formed by the concatenation of $P_{up}$ and $x' x y y'$, or the one formed by the concatenation of $P_{up}$ and $P_{down}$ is an even hole of length $\\geq 6$, a contradiction.\n\\end{proof}\n\nThe previous lemma allows us to prove that one can \\emph{lift} the PCP up into $N_{k-1}$ to get a subset of vertices with high chromatic number. We state a lemma that will be reused in the next section:\n\n\n\n\\begin{lemma}\n\\label{lem: active lift gros si neighb stable gros}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a rooted PCP of order $\\ell$ in a level $N_k$ (for some $k\\geq 2$) with leftovers at least $\\delta$. Let $A=N(G_\\ell)\\cap N_{k-1}$ (called the active lift of the PCP).\nSuppose that for every stable set $S\\subseteq A$, we have $\\chi(N(S)\\cap N_{k})\\leq \\gamma$,\n then $\\chi(A)\\geq {\\delta}\/{\\gamma}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $r=\\chi(A)$, suppose by contradiction that $r<\\delta\/\\gamma$ and decompose $A$ into $r$ stable sets $S_1, \\ldots, S_r$. Then $N(A)\\cap N_k$ is the (non-necessarily disjoint) union of $r$ sets $N(S_1)\\cap N_k, \\ldots, N(S_r)\\cap N_k$, and each of them has chromatic number at most $\\gamma$ by assumption. Consequently $\\chi(N(A)\\cap N_k)\\leq r \\gamma < \\delta$ and hence $\\chi(H\\setminus N(A)) \\geq \\chi(H) - \\chi(N(A)\\cap N_k) \\geq 1$.\nLet $x$ be any vertex of $H\\setminus N(A)$ and $x'$ be a neighbor of $x$ in $N_{k-1}$. By construction, $x'\\notin A$ so $x'$ has no neighbor in $G_\\ell$. This is a contradiction with Lemma \\ref{obs: sommets riches sans C5}.\n\\end{proof}\n\nBy Lemmas \\ref{lem: shadow stable bornee sans C5} and \\ref{lem: active lift gros si neighb stable gros} with $\\gamma=52$, we can directly deduce the following:\n\n\\begin{lemma} \\label{lem: shadow borne bornee sans C5}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a rooted PCP of order $\\ell$ in a level $N_k$ (for some $k\\geq 2$) with leftovers at least $\\delta$. Let $A=N(G_\\ell)\\cap N_{k-1}$ be the active lift of the PCP, then we have $\\chi(A)\\geq g(\\delta)={\\delta}\/{52}$.\n\\end{lemma}\n\n\n\n\nWe can now finish the proof, this is step (v). Recall that a sketch was provided, and it may help to understand the following proof.\n\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem: sans C5}] \nLet $c'$ be a constant big enough so that \n$$g\\left(h^{(2)}\\left(g\\left(h^{(2)}\\left(\\frac{c'}{2}-1\\right)\\right)-1\\right)\\right)\\geq 5 \\ .$$\nSuppose that $\\chi(G)\\geq c'$. Pick a vertex $v$, let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level, so $\\chi(N_k)\\geq \\chi(G)\/2\\geq c_1+1$ where $c_1=c'\/2-1$. By Lemma \\ref{lem: rooted PCP sans C5}, grow a rooted PCP $P=(G_1, P_1, G_2, P_2, H)$ inside $N_k$ of order 2 with leftovers at least $c_2=h^{(2)}(c_1)$. Then apply Lemma \\ref{lem: shadow borne bornee sans C5} and get an active lift $A$ of $P$ inside $N_{k-1}$ with chromatic number at least $c_3=g(c_2)$. Since $h(c_3 -1)\\geq 18$, apply again Lemma \\ref{lem: rooted PCP sans C5} to get a rooted PCP $P'=(G'_1, P'1, G'_2, P'_2, H')$ of order 2 inside $N_{k-1}$ with leftovers at least $c_4=h^{(2)}(c_3 -1)$. Now apply Lemma \\ref{lem: shadow borne bornee sans C5} to get an active lift $A'$ of $P'$ inside $N_{k-2}$ with chromatic number at least $c_5=g(c_4)$.\n\nBecause of the chromatic restriction in the definition of the PCP, one can color $G'_2$ with 4 colors. Moreover, $G'_2$ dominates $A'$ by definition. Thus there exists a stable set $S\\subseteq G'_2$ such that $\\chi(N(S)\\cap A')\\geq c_6=c_5\/4$ (since $A'$ is the union of the $N(S')\\cap A'$ for the four stable sets $S'$ that partition $G_2'$).\n\nNow $c_6>1$ so there is an edge $xy$ inside $N(S) \\cap A'$. Call $x'$ (resp. $y'$) a vertex of $S$ dominating $x$ (resp. $y$). Both $x'$ and $y'$ have a neighbor in $G_2$ by definition of $A$ and, by Lemma \\ref{obs: sommets riches sans C5}, both $x'$ and $y'$ also have a neighbor in $G_1$. This gives an $x'y'$-path $P_1$ (resp. $P_2$) with interior in $G_1$ (resp. $G_2$). Due to the path $x'x y y'$ of length 3, $P_1$ and $P_2$ must be even paths of length at least 4. Thus the concatenation of $P_1$ and $P_2$ is an even hole of length at least 6, a contradiction.\n\\end{proof}\n\n\\section{General case}\n\\label{sec: Yes C4 avec C5}\n\nThis section aims at proving Theorem \\ref{th: C chi borne}, using the result of the previous section. As already mentioned, we follow the same outline, except that we now need the existence of a $C_5$ several times. Let us start by a technical lemma to find both an even and an odd path out of a 5-hole and its dominating set:\n\n\n\n\\begin{lemma}\n\\label{lem: C5 chemin pair chemin impair}\nLet $G$ be a triangle-free graph inducing a 5-hole $C$. Let $S\\subseteq V(G)$ be a minimal dominating set of $C$, assumed to be disjoint from $C$. If we delete the edges with both endpoints in $S$, then for every vertex $t\\in S$, there exists a vertex $t'\\in S$ such that \none can find an induced $tt'$-path of length 4 and an induced $tt'$-path of length 3 or 5, both with interior in $C$.\n\\end{lemma}\n\n\\begin{proof}\nLet $t\\in S$, call $v_1$ a neighbor of $t$ on the cycle and number the others vertices of $C$ with $v_2, \\ldots , v_5$ (following the adjacency on the cycle). Since $G$ is triangle-free, $t$ can not be adjacent to both $v_3$ and $v_4$, so up to relabeling the cycle in the other direction we assume that $t$ is not adjacent to $v_3$. Let $t'\\in S$ be a vertex dominating $v_3$. Then $tv_1v_2v_3t'$ is an induced path of length 4 between $t$ and $t'$. Moreover, $tv_1v_5v_4t'$ is a (non-necessarily induced) path of length 5 between $t$ and $t'$. If this path is not induced, the only possible chords are $tv_4$ and $t'v_5$ since $G$ is triangle-free, which in any case gives an induced $tt'$-path of length 3.\n\\end{proof}\n\n\n\n\nRecall that in this section, we are interesting in strong PCP, \\emph{i.e.} PCP, all regular blocks $G_i$ of which contain an induced $C_5$. We start with step (i):\n\n\n\\begin{lemma} \\label{lem: debut PCP}\nLet $c'$ be the constant of Lemma \\ref{lem: sans C5}, let $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ and $v$ be any vertex of $G$. For every $\\delta \\in \\mathbb{N}$ such that $\\chi(G)\\geq \\delta \\geq 2c'$, there exists a strong PCP of order 1 with origin $v$ and leftovers at least $f(\\delta)=\\delta\/2-15$.\n\n\\end{lemma}\n\n\\begin{proof}\n\n\nLet $(N_0, N_1, \\ldots)$ be the $v$-levelling, $N_k$ be a colorful level and let $N'_k$ be a major connected component of $G[N_k]$, so $\\chi(N'_k)\\geq c'$. Using Lemma \\ref{lem: sans C5}, there exists a 5-hole $C$ in $G[N'_k]$. Consider a minimum dominating set $D$ of $C$ inside $N_{k-1}$. \n\n\nFrom now on, the proof is very similar to the one of Lemma \\ref{lem: debut PCP sans C5}. Similarly, we define $Z'=N(D\\cup C)\\cap N'_k$ and $Z=Z' \\setminus C$. Let $z\\in Z$ be a vertex having a neighbor $z_1$ in a major connected component $M_1$ in $N'_k \\setminus Z'$. The goal is now to find two $vz$-paths $P$ and $P'$ of different parity with interior in $N_0 \\cup \\ldots \\cup D \\cup C$, then we can set $G_1=G[P\\cup P'\\cup C]$, $P_1=G[\\{z,z_1\\}]$ and $H=G[M_1]$ as parts of the wanted PCP. In practice, we need to be a little more careful to ensure the condition on the length of $P_1$ and the non-adjacency between $z$ and $H$.\n\n\nLet us now find those two paths $P$ and $P'$. By definition of $Z$, $z$ also has a neighbor in $D$ or in $C$. \n\\begin{enumerate}\n\\item If $z$ has a neighbor $x\\in C$, let $y\\in C$ be a vertex adjacent to $x$ on the hole. Let $x'$ and $y'$ be respectively a neighbor of $x$ and a neighbor of $y$ in $D$. Observe that $z$ is connected neither to $x'$ nor to $y$, otherwise it creates a triangle. We grow $P$ by starting from an induced path of length $k-1$ from $v$ to $x'$ and then add the path $x'x z$. Similarly, we grow $P'$ by starting from an induced path of length $k-1$ from $v$ to $y'$, and then add the edge $y'z$ if it exists, or else the path $y' y x z$. Observe that $P'$ is indeed an induced path since there is no triangle.\n\\item If $z$ has no neighbor in $C$, then it has at least one neighbor $x'$ in $D$. Apply Lemma \\ref{lem: C5 chemin pair chemin impair} to get a vertex $y'\\in D$ such that there exists an $x'y'$-path of length 3 or 5, and another one of length 4, both with interior in $C$. Observe that \n$x'$ and $y'$ cannot have a common neighbor $u$ in $N_{k-2}\\cup \\{z\\}$, otherwise there would be either a triangle $x', u, y'$ (if $x'y'\\in E$), or a $C_6$ using the \\mbox{$x'y'$-path} of length 4 with interior in $C$. Now we grow $P$ by starting from an induced path of length $k-1$ from $v$ to $x'$, and add the edge $x'z$. We grow $P'$ by starting from an induced path of length $k-1$ from $v$ to $y'$, and then add the edge $x'y'$ if it exists, otherwise add the $x'y'$-path of length 3 or 5 with interior in $C$, and then finish with the edge $x'z$.\n\\end{enumerate}\n\n\nNow come the fine tuning.\nChoose in fact $z_1\\in M_1\\cap N(z)$ so that $z_1$ is connected to a major connected component $M_2$ of $M_1 \\setminus N(z)$. Choose $z_2$ a neighbor of $z_1$ in $M_2$ such that $z_2$ is connected to a major connected component $M_3$ of $M_2\\setminus N(z_1)$. We redefine $H=G[\\{z_2 \\cup M_3\\}]$ and $P_1=G[\\{z,z_1,z_2\\}]$. Then $P_1$ is a path of length 2, $H$ is connected and $G_1$ is colorable with 4 colors (it is easily 7-colorable as the union of a 5-hole and two paths; a careful case analysis shows that it is 4-colorable). Moreover $H$ has chromatic number at least $\\chi(N'_k)-\\chi(Z')-\\chi(N(z))-\\chi(N(z_1))$. Since the neighborhood of any vertex is a stable set, $\\chi(Z')\\leq |D|+|C| +\\chi(C)\\leq 13$ and $\\chi(N(z)), \\chi(N(z_1))\\leq 1$. Thus $\\chi(H)\\geq \\delta\/2- 15$.\n\n\\end{proof}\n\nWe go on with step (ii): find a strong rooted PCP.\nThe following lemma is proved in the same way as Lemma \\ref{lem: rooted PCP sans C5} by replacing the use of Lemma \\ref{lem: existence PCP sans C5} by Lemma \\ref{lem: debut PCP}, so we omit the proof here.\n\n\\begin{lemma}\n\\label{lem: rooted PCP}\nLet $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ be a connected graph, $f$ be the function defined in Lemma \\ref{lem: debut PCP}, $v$ be a vertex of $G$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. For every $k, \\delta$ such that $\\chi(N_k)\\geq \\delta+1\\geq 2c'+1$, there exists a strong rooted PCP of order 1 in $N_k$ with leftovers at least $f(\\delta)$.\n\\end{lemma}\n\nStep (iii) is proved by Lemma \\ref{obs: sommets riches sans C5} from the previous section, and was valid not only for $G\\in \\mathcal{C}_{3, 5, 2k\\geq 6}$ but also for $G\\in \\mathcal{C}_{3, 2k\\geq 6}$. So we continue with step (iv):\n\n\\begin{lemma} \\label{lem: shadow stable bornee}\nLet $v$ be a vertex of a graph $G\\in \\mathcal{C}_{3, 2k\\geq 6}$, $(N_0, N_1, \\ldots)$ be the $v$-levelling. \nLet $S$ be a stable set inside $N_{k-1}$. Then $\\chi(N(S)\\cap N_k)\\leq 2c'$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose by contradiction that $\\chi(N(S)\\cap N_k)\\geq 2c'+1$. By Lemma \\ref{lem: rooted PCP}, we can grow in $N(S)\\cap N_k$ a rooted PCP of order 1, and in particular $S$ dominates $G_1$.\nBy definition of a strong PCP, there is a 5-hole $C$ in $G_1$. Since $S$ is a dominating set of $C$, we can apply Lemma \\ref{lem: C5 chemin pair chemin impair} to get two vertices $t, t'\\in S$ such that one can find both an even and an odd $tt'$-path with interior in $C$ and length at least 3. Then any upper $tt'$-path close a hole of even length $\\geq 6$.\n\\end{proof}\n\nIn fact, as in previous section, we can directly deduce from Lemmas \\ref{lem: active lift gros si neighb stable gros} and \\ref{lem: shadow stable bornee} that one can lift the PCP up into $N_{k-1}$ to get a subset of vertices with high chromatic number:\n\n\n\n\\begin{lemma} \\label{lem: shadow borne bornee}\nLet $G\\in \\mathcal{C}_{3, 2k\\geq 6}$, $v\\in V(G)$ and $(N_0, N_1, \\ldots)$ be the $v$-levelling. Let $P$ be a strong rooted PCP of order $1$ in a level $N_k$ (for some $k\\geq 2$) with leftovers $\\delta$. Let $A=N(G_1)\\cap N_{k-1}$ be the active lift of the PCP. If $\\delta \\geq 2c'$, then $\\chi(A)\\geq \\varphi(\\delta)=\\frac{\\delta}{2c'}$.\n\\end{lemma}\n\n\n\n\nWe are now ready to finish the proof, this is step (v). Recall that a sketch was given and may be useful to have a less technical overview of the proof.\n\n\\begin{proof}[Proof of Theorem \\ref{th: C chi borne}] \nLet $c$ be a constant such that \n$$\\varphi\\left(f\\left(\\varphi\\left(f\\left(\\frac{c}{2} -1\\right)\\right)-1\\right)\\right)\\geq 4c' \\ .$$ \nSuppose that $G\\in \\mathcal{C}_{3, 2k\\geq 6}$ has chromatic number $\\chi(G)\\geq c$. Then pick a vertex $v$, let $(N_0, N_1, \\ldots)$ be the $v$-levelling and $N_k$ be a colorful level, consequently $\\chi(N_k)\\geq c_1+1=c\/2$.\nApply Lemma \\ref{lem: rooted PCP} and grow inside $N_k$ a strong rooted PCP $P=(G_1, P_1, H)$ of order 1 with leftovers at least $c_2=f(c_1)$. Then apply Lemma~\\ref{lem: shadow borne bornee} and get an active lift $A=N(G_1)$ of $P$ inside $N_{k-1}$ with chromatic number at least $c_3=\\varphi(c_2)$. By Lemma \\ref{lem: rooted PCP}, we can obtain a strong rooted PCP $P'=(G'_1, P'_1, H')$ inside $A$ with leftovers at least $c_4=f(c_3-1)$, and by Lemma~\\ref{lem: shadow borne bornee} we obtain an active lift $A'$ of $P'$ inside $N_{k-2}$ with chromatic number at least $c_5=\\varphi(c_4)$. Because of the chromatic restriction in the definition of the PCP, one can color $G'_1$ with 4 colors. Moreover, $G'_1$ dominates $A'$ by definition. Thus there exists a stable set $S\\subseteq P'$ such that $\\chi(N(S)\\cap A')\\geq c_6=c_5\/4$.\nNow $c_6\\geq c'$ thus Lemma \\ref{lem: sans C5} proves the existence of a 5-hole $C$ inside $N(S) \\cap A'$. Let us give an overview of the situation: we have a 5-hole $C$ inside $N_{k-2}$, dominated by a stable set $S$ inside $N_{k-1}$, and every pair of vertices $t,t'$ of $S$ can be linked by a $tt'$-path $P_{down}$ with interior in $G_1\\subseteq N_k$. Lemma \\ref{lem: C5 chemin pair chemin impair} gives the existence of two vertices $t, t'\\in S$ linked by both an odd path and an even path of length $\\geq 3$ with interior in $C$. Closing one of these paths with $P_{down}$ gives an induced even hole of length $\\geq 6$, a contradiction.\n\\end{proof}\n\n\n\n\\section*{Acknowledgment}\n\nThe author would like to sincerely thank St\\'ephan \\textsc{Thomass\\'e} for bringing this problem to her knowledge and for useful discussions about Trinity\/Parity Changing Paths.\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nGiven a spin manifold $M$ of dimension $n$, we consider the universal spinor bundle $\\Sigma M$. This is the bundle whose sections consist of pairs of metrics $g \\in \\Gamma(\\odot^2_+ T^*M)$ and spinor fields $\\varphi \\in \\Gamma(\\Sigma_g M)$.\nWe denote by $\\mathcal{N}$ the set\n$$\\{ (g, \\varphi) \\in \\Gamma( \\Sigma M) : |\\varphi| = 1\\}$$\nand define the spinorial energy functional\n$$\\mathcal{E}: \\mathcal{N} \\to \\mathbb R$$\n$$\\mathcal{E}(g, \\varphi) = \\frac{1}{2}\\int_M |\\nabla^g \\varphi|^2 \\operatorname{vol}_g.$$\nIf the dimension of $M$ is at least three, the only critical points of $\\mathcal{E}$ are absolute minimizers. This implies $\\nabla^g \\varphi = 0$ for a critical point $(g, \\varphi)$, i.e. $\\varphi$ is a parallel spinor with respect to the metric $g$. Existence of parallel spinors is a strong constraint on the metric $g$. Indeed, such a metric is necessarily Ricci flat and of special holonomy. Conversely, Ricci flat manifolds with special holonomy admit a parallel spinor. Given that manifolds with such metrics are difficult to construct, it is natural to consider the negative gradient flow\n$$\\partial_t \\Phi_t = Q(\\Phi_t)$$\nof $\\mathcal{E}$ to find such a metric. Here $Q: \\mathcal{N} \\to T \\mathcal{N}$ is the negative gradient of $\\mathcal{E}$ with respect to the natural $L^2$ metric on $\\mathcal{N}$. It turns out that $Q$ is weakly elliptic and has negative symbol.\nThe spinorial energy and the associated negative gradient flow, called spinor flow, were first examined in \\cite{Ammann2015}. There, short time existence of this flow on closed manifolds was established.\nFrom here on we assume $M$ to be a closed manifold and $\\dim M = n \\geq 3$.\nWe will prove that critical points of $\\mathcal{E}$, i.e. pairs of Ricci flat metrics and parallel spinor fields, are stable with respect to the spinor flow, that is:\n\\begin{thm}\nSuppose $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ is a critical point of $\\mathcal{E}$ and suppose $\\bar{g}$ has no Killing fields. Then there exists a $C^{\\infty}$ neighborhood $U$ of $\\bar{\\Phi}$, such that a solution of the negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi$ smoothly converges to a critical point. In any $C^k$ norm the speed of convergence is exponential.\n\\end{thm}\nA Ricci flat manifold is up to a finite covering a product of irreducible Ricci-flat manifolds and a flat torus. Hence the condition on the Killing fields can also be read as saying that $\\bar{g}$ has no torus factor. This can for instance be ruled out by the topological condition that the fundamental group of $M$ be finite.\nThe strategy of the proof will be roughly as follows: first, we establish a \u0141ojasiewicz-Simon type inequality for the spinorial energy. This inequality implies exponential decay of the energy along the flow. We will then show that this implies convergence to a critical point. The inequality depends in its optimal form on the fact that the critical set of $\\mathcal{E}$ is smooth. This was shown in \\cite{Ammann2015b}.\n\nWe will also consider the stability of volume constrained critical points. A section $\\Phi \\in \\mathcal{N}$ is a {\\em volume constrained critical point}, if\n$$\\frac{d}{dt}\\Bigr|_{t=0} \\mathcal{E}(\\Phi_t) = 0$$\nfor all volume preserving variations $\\Phi_t$ of $\\Phi$. Such a critical point evolves under the spinor flow by rescaling. A {\\em volume constrained minimizer} $\\Phi = (g, \\varphi)$ is a volume constrained critical point, such that for any $\\Psi \\in \\mathcal{N}$ close to $\\Phi$ we have $\\mathcal{E}(\\Phi) \\leq \\mathcal{E}(\\Psi)$, provided the metrics induced by $\\Phi$ and $\\Psi$ have equal volume.\n\\begin{thm}\nLet $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ be a volume constrained minimizer of $\\mathcal{E}$. Suppose that the critical set near $\\bar{\\Phi}$ is a manifold and suppose $\\bar{g}$ has no Killing fields. Then there exists a $C^{\\infty}$ neighborhood $U$ of $\\bar{\\Phi}$, such that the volume normalized spinor flow converges smoothly to a volume constrained minimizer, if the initial condition is in $U$. The convergence speed in $C^k$ is exponential. \n\\end{thm}\nThe strategy for the proof is essentially the same as in the case of critical points. However, here both the condition on Killing fields and the assumption that the critical set near $\\bar{\\Phi}$ is a manifold are strong restrictions. Indeed, suppose $(g, \\varphi)$ is such that\n$$\\nabla^g_X \\varphi = \\lambda X \\cdot \\varphi \\text{ for all } X \\in \\Gamma(TM)$$\nwith $\\lambda \\in \\mathbb R$.\nThen $(g,\\varphi)$ is a volume constrained critical point. The spinor $\\varphi$ is called a Killing spinor.\nIf $g$ carries a Killing spinor, then the cone $((0,\\infty)\\times M, dr^2 + r^2g)$ carries a parallel spinor.\nA large class of metrics with Killing spinors is supplied by Sasaki--Einstein manifolds.\nSince the Reeb vector field is a Killing vector field, all Sasaki--Einstein manifolds carry Killing fields.\nFurthermore, the moduli space of Sasaki--Einstein manifolds is not known to be smooth in general.\nWhat's more, in contrast to the space of parallel spinors, whose dimension is locally constant under Ricci flat deformations of the metric, the dimension of the space of killing spinors can jump under Einstein deformations of the metric. Indeed, a 3-Sasakian manifold admits three linearly independent Killing spinors. Van Coevering found that a toric 3-Sasakian manifold has Einstein deformations $g_t$, such that the space of Killing spinors is two-dimensional for any $t\\neq 0$.\n\nSince the spinor flow is a generalization of the heat flow for $G_2$-structures introduced in \\cite{Weiss2012}, our result is a generalization of the stability result proven there. However, the arguments of our proof are closer in spirit to the proofs in \\cite{Haslhofer2011}, \\cite{Haslhofer2014}, \\cite{Krncke2014}, where stability of Ricci-flat and Einstein metrics with respect to the Ricci flow is shown.\n\n\\section*{Acknowledgements}\nThe author thanks Hartmut Wei\u00df for posing the problem and numerous discussions related to it.\n\n\n\\section{The universal spinor bundle and the spinor flow}\nFor convenience and completeness, we recall the precise definitions of the spinor flow as well as results on short time existence of the flow. Details may be found in \\cite{Ammann2015}. \nWe defined the spinor energy to be a functional on sections of the universal spinor bundle. We will now construct this universal spinor bundle.\nBefore we do this, let us first recall the ordinary spinor bundle on a spin manifold. \nThe orientation preserving component of the general linear group $\\operatorname{GL}_+(n)$ has fundamental group $\\mathbb Z_2$ and hence there exists a universal double covering group $\\widetilde{\\operatorname{GL}}_+(n)$ together with a covering map $\\xi: \\widetilde{\\operatorname{GL}}_+(n) \\to \\operatorname{GL}_+(n)$, which is also a homomorphism.\nLet $M$ be a spin manifold of dimension $n$. By this we mean a manifold $M$ and a $\\widetilde{\\operatorname{GL}}_+(n)$ principal bundle $\\tilde{P}$ which covers the $\\operatorname{GL}_+(n)$ frame bundle $P$, $\\pi: \\tilde{P} \\to P$, so that for any $g \\in \\widetilde{\\operatorname{GL}}_+(n), p \\in \\tilde{P}$ we have\n$$\\pi(p \\cdot g) = \\pi(p) \\cdot \\xi(g).$$\nNow let $g$ be a metric on $M$. The metric induces a reduction of the structure group of $P$ to the oriented orthonormal frame bundle $P_{\\operatorname{SO}(n)}$. The group $\\operatorname{Spin}(n) = \\xi^{-1}(\\operatorname{SO}(n))$ is called the spin group. Thus the structure group of $\\tilde{P}$ reduces to $\\operatorname{Spin}(n)$ and we call this bundle $P_{\\operatorname{Spin}(n)}$, which double covers $P_{\\operatorname{SO}(n)}$.\nNow we define the (complex) spinor bundle as the associated vector bundle\n$$\\Sigma_g M = P_{\\operatorname{Spin}(n)} \\times_{\\Delta_n} \\Sigma_n$$\nwhere $\\Delta_n : \\operatorname{Spin}(n) \\to \\operatorname{End}(\\Sigma_n)$, $\\Sigma_n = \\mathbb C^{2^{[n\/2]}}$, is the standard complex spin representation. Up to scaling, there exists one $\\operatorname{Spin}(n)$ invariant Hermitian product on $\\Sigma_n$. This turns $\\Sigma_g M$ into a Hermitian bundle.\nThe universal spinor bundle gives us a way to compare spinors over different metrics.\nRecalling that\n$$\\faktor{\\operatorname{GL}_+(n)}{\\operatorname{SO}(n)} \\cong \\odot^2_+ \\mathbb R^n,$$\nwe conclude\n$$\\odot^2_+ T^*M = P \\times_{\\operatorname{GL}_+(n)} \\faktor{\\operatorname{GL}_+(n)}{\\operatorname{SO}(n)} = \\tilde{P} \\times_{\\widetilde{\\operatorname{GL}}_+(n)} \\faktor{\\widetilde{\\operatorname{GL}}_+(n)}{\\operatorname{Spin}(n)} = \\faktor{\\tilde{P}}{\\operatorname{Spin}(n)}.$$\nWe define\n$$\\Sigma M = \\tilde{P} \\times_{\\Delta_n} \\Sigma_n.$$\nThis is a vector bundle over $\\odot^2_+ T^*M$, i.e. we have the structure of two nested fibrations:\n$$\\Sigma M \\xrightarrow{\\pi_{\\Sigma}} \\odot^2_+ T^*M \\xrightarrow{\\pi_{\\mathcal{M}}} M.$$\nGiven a metric $g$ we can identify $\\pi_{\\Sigma}^{-1}(g)$ and $\\Sigma_g M$. Using this identification any element $\\Phi \\in\\Sigma M$ can be considered as a pair of a metric $g_{\\Phi} = \\pi_{\\Sigma}(\\Phi)$ and a spinor $\\varphi_{\\Phi} \\in \\Sigma_{g_{\\Phi}} M$.\nAs above we also get a Hermitian inner product $h$ on $\\Sigma M$.\nWe denote by $\\langle \\cdot, \\cdot \\rangle = \\operatorname{Re} h$ the real part of $h$ and $| \\cdot |$ the associated norm.\nNow the definition\n$$\\mathcal{N} = \\{\\Phi \\in \\Gamma(\\Sigma M) : |\\Phi| = 1\\}$$\nfrom the introduction is fully explained. To make sense of the gradient of $\\mathcal{E}$ we need to compute the tangent spaces of $\\mathcal{N}$. For this we need to compare spinors in different fibers $\\Sigma_{g_1} M$ and $\\Sigma_{g_2} M$. This can be done using the Bourgignon--Gauduchon connection.\n\nSuppose we have a vector space $V$ and two inner products $\\langle \\cdot, \\cdot \\rangle_1$ and $\\langle \\cdot, \\cdot \\rangle_2$. Then there exists a unique endomorphism $A^2_1 : V \\to V$, such that\n$$\\langle v, w \\rangle_2 = \\langle A_1^2 v, w \\rangle_1 \\text{ for all } v,w \\in V.$$\nDenote by $B_1^2$ the square root of $A_1^2$. The operator $B_1^2$ maps orthonormal bases of $(V, \\langle \\cdot, \\cdot \\rangle_1)$ to orthonormal bases of $(V, \\langle \\cdot, \\cdot \\rangle_2)$. Since no choices are involved and $B_1^2$ depends smoothly on the inner products, we can transfer this construction to Riemannian manifolds $(M,g_i)$, $i=1,2$, and consequently obtain a smooth principal bundle isomorphism\n$$P^{g_2}_{\\operatorname{SO}(n)} \\to P^{g_1}_{\\operatorname{SO}(n)}.$$\nThis map lifts to the spinor bundle and hence induces a isomorphism $\\hat{B}^{g_2}_{g_1}:\\Sigma_{g_2} M \\to \\Sigma_{g_1} M$.\nSince the metric on $\\Sigma_n$ is $\\operatorname{Spin}(n)$-invariant, this is an isometry.\nNotice that the restriction of $\\hat{B}_{g_1}^{g_2}$ to a fibre over a point $x \\in M$ only depends on the scalar products $g_1(x), g_2(x)$ on $T_xM$.\nWe now have a canonical isometry between two spinor bundles over the same manifold with two distinct metrics. From this we can derive a horizontal distribution\n$$\\mathcal{H}_{\\Phi} = \\left\\{ \\frac{d}{dt}\\Bigr|_{t=0} \\hat{B}^{g}_{g_t} \\varphi \\; \\Big| \\; g: (-\\epsilon, \\epsilon) \\to \\odot^2_+ T_x^*M, g(0) = g \\right\\} \\subset T_{\\Phi} \\Sigma M $$\nwhere $\\Phi = (g, \\varphi) \\in \\Sigma M_x$. By construction, $\\mathcal{H}_{\\Phi} \\cong \\odot^2 T_x^*M$.\nThis distribution yields a splitting of the tangent bundle\n$$T_{\\Phi} \\Sigma M_x = \\mathcal{H}_{\\Phi} \\oplus \\Sigma_g M \\cong \\odot^2 T_x^*M \\oplus \\Sigma_g M.$$\nTurning to sections of the universal spinor bundle, this implies that for $\\Phi = (g, \\varphi) \\in \\Gamma (\\Sigma M)$ we have the splitting\n$$T_{\\Phi} \\Gamma(\\Sigma M) = \\Gamma(\\odot^2 T^* M) \\oplus \\Gamma( \\Sigma_g M)$$\nand if $\\Phi \\in \\mathcal{N}$\n$$T_{\\Phi} \\mathcal{N} = \\Gamma(\\odot^2 T^* M) \\oplus \\Phi^{\\perp},$$\nwhere\n$$\\Phi^{\\perp} = \\{ \\psi \\in \\Gamma(\\Sigma_g M) : \\langle \\varphi, \\psi\\rangle \\equiv 0 \\}.$$\nNow we define for $\\Phi \\in \\Gamma (\\Sigma M)$ and $\\Psi_1, \\Psi_2 \\in T_{\\Phi} \\Gamma( \\Sigma M)$\n$$\\left( \\Psi_1, \\Psi_2 \\right)_{L^2} = \\int_M g(h_1, h_2) \\operatorname{vol}_g + \\int_M \\langle \\psi_1, \\psi_2 \\rangle_{\\Sigma_g M} \\operatorname{vol}_g$$\nwhere $(h_i, \\psi_i) \\in \\Gamma(\\odot^2_+ T^*M) \\oplus \\Gamma (\\Sigma_g M)$ are the sections corresponding to $\\Psi_i$ according to the isomorphisms above. From now on we will use these isomorphisms implicitly.\nNow the negative gradient\n$$Q: \\mathcal{N} \\to T\\mathcal{N}$$\nis defined by the property\n$$\\left(Q(\\Phi), \\Psi \\right) = - \\frac{d}{dt}\\Bigr|_{t=0} \\mathcal{E}(B^g_{g+th} (\\varphi + t \\psi)),$$\nwhere $\\Phi = (g, \\varphi) \\in \\mathcal{N}$ and $\\Psi = (h, \\psi) \\in T_{\\Phi} \\mathcal{N}$.\n\n\\section{Diffeomorphism invariance, the gauged spinor flow and a slice theorem}\nWe denote by $\\operatorname{Diff}_s(M)$ the group of spin diffeomorphisms, i.e. the orientation preserving diffeomorphisms of $M$, which lift to $\\tilde{P}$. To be more precise, by a lift of a orientation preserving diffeomorphism $f: M \\to M$ to $\\tilde{P}$, we mean a lift of the map\n$$ P_x \\ni [e_1, ..., e_n] \\mapsto [Df e_1,..., Df e_n] \\in P_{f(x)}$$\ninduced by $f$ on the oriented frame bundle $P$ to the topological spin bundle $\\tilde{P}$.\nSince $\\tilde{P}$ is a $\\mathbb Z_2$ bundle over $P$, there is a choice of lift and the group of lifts of spin diffeomorphisms $\\widehat{\\operatorname{Diff}}_s(M)$ fits into an exact sequence\n$$0 \\to \\mathbb Z_2 \\to \\widehat{\\operatorname{Diff}}_s(M) \\to \\operatorname{Diff}_s(M) \\to 0.$$\nThe group $\\widehat{\\operatorname{Diff}}_s(M)$ acts on $\\Gamma( \\Sigma M)$ in the following way. Let $F \\in \\widehat{\\operatorname{Diff}}_s(M)$, $\\Phi = (g, \\varphi) \\in \\Gamma(\\Sigma M)$. The map $F : \\tilde{P} \\to \\tilde{P}$ is a lift of a diffeomorphism $f: M \\to M$. Restricting $\\tilde{P}$ to $P_{\\operatorname{Spin}(n)}^g$ we obtain an isomorphism\n$$F : P_{\\operatorname{Spin}(n)}^g \\to P^{f_* g}_{\\operatorname{Spin}(n)}.$$\nThen we define locally\n$$F_*\\varphi = [F \\circ b \\circ f^{-1}, \\varphi \\circ f^{-1}] \\in \\Gamma(\\Sigma_{f_* g} M),$$\nif $\\varphi = [b, \\varphi]$, $b$ a local section of $P_{\\operatorname{Spin}(n)}^g$, $\\varphi$ a $\\Sigma_n$ field. The push forward preserves the metric in the following sense:\n$$|F_* \\varphi|_{f_* g} (x) = |\\varphi|_g (f^{-1}(x)).$$\nIn particular $F_*$ preserves $\\mathcal{N}$. Moreover, we have\n$$\\mathcal{E}(F_* \\Phi) = \\mathcal{E}(\\Phi),$$\n$$Q(F_* \\Phi) = F_* Q(\\Phi).$$\nIn particular, the spinor flow is not strongly parabolic, since $Q$ is invariant under an infinite dimensional group. This invariance is reflected on the infinitesimal level by the following Bianchi-type identity\n$$\\lambda_{g,\\varphi} Q(g,\\varphi) = 0$$\nwhere\n$$\\lambda_{g, \\varphi} : \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M) \\to \\Gamma(TM)$$\nis defined as the formal adjoint of\n$$\\lambda_{g,\\varphi}^* : \\Gamma(TM) \\to \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M)$$\n$$X \\mapsto \\left(2 \\delta^*_g X^{\\flat}, \\nabla^g_X \\varphi - \\frac{1}{4} dX^{\\flat} \\cdot \\varphi\\right) = \\left( \\mathcal{L}_X g, \\tilde{\\mathcal{L}}_X \\varphi\\right) =: \\tilde{\\mathcal{L}}_X \\Phi.$$\nIndeed, the tangent space of the orbit $\\widehat{\\operatorname{Diff}}_s(M).(g,\\varphi)$ is the image of $\\lambda_{g,\\varphi}^*$.\nAt a critical point $(g,\\varphi) \\in \\Gamma( \\Sigma M)$, we get the following exact sequence\n$$0 \\to \\Gamma(TM) \\xrightarrow{\\lambda_{g, \\varphi}^*} \\Gamma(\\odot^2 T^* M) \\oplus \\Gamma(\\varphi^{\\perp}) \\xrightarrow{L_{g,\\varphi}} \\Gamma(\\odot^2 T^* M) \\oplus \\Gamma(\\varphi^{\\perp}) \\xrightarrow{\\lambda_{g, \\varphi}} \\Gamma(TM) \\to 0,$$\nwhere $L_{g,\\varphi} = DQ (g,\\varphi)$. It turns out that with\n$$X_{\\bar{g}} : \\Gamma(\\odot^2 T^*M) \\to \\Gamma(TM)$$\n$$g \\mapsto -2 (\\delta_{\\bar{g}} g)^{\\sharp}$$\nand $\\bar{g}$ any given metric the operator\n$$\\tilde{Q}_{\\bar{g}}(\\Phi) = Q(\\Phi) + \\lambda^*_{g,\\varphi}(X_{\\bar{g}}(\\Phi))$$\nis strongly parabolic for any $\\Phi = (\\bar{g}, \\varphi) \\in \\mathcal{N}_{\\bar{g}}$ and hence the flow\n$$\\partial_t \\Phi_t = \\tilde{Q}(\\Phi_t)$$\nexists for short time. We call this flow {\\em gauged spinor flow} or {\\em spinor-DeTurck flow}.\nMoreover, the spinor flow and the gauged spinor flow differ only by a family of diffeomorphisms, i.e.\nif $\\Phi_t = (g_t, \\varphi_t)$ is a solution of the spinor flow and $\\tilde{\\Phi}_t = (\\tilde{g}_t, \\tilde{\\varphi}_t)$ is a solution of the gauged spinor flow with $\\Phi_0 = \\tilde{\\Phi}_0$, then there exists a family $F_t \\in \\widehat{\\operatorname{Diff}}_s(M)$, induced by $f_t \\in \\operatorname{Diff}(M)$, such that\n$$\\tilde{\\Phi}_t = F_{t*} \\Phi_t.$$\nThis family obeys the partial differential equation\n$$\\partial_t f_t = P_{g_t, \\bar{g}}(f_t)$$\nwith initial condition $f_0 = \\operatorname{id}_M$, where\n$$P_{g,\\bar{g}} : \\mathcal{C}^{\\infty}(M,M) \\to T \\mathcal{C}^{\\infty}(M,M)$$\n$$f \\mapsto -df (X_{f^* \\bar{g}} (g)).$$\nFor future reference we note that the linearization of $P_{\\bar{g},\\bar{g}}$ at $\\operatorname{id}_M$ is given by\n$$\\Gamma(TM) \\ni X \\mapsto -4 (\\delta_{\\bar{g}} \\delta^*_{\\bar{g}} X^{\\flat})^{\\sharp} \\in \\Gamma(TM).$$\nBecause\n$$T_{\\Phi} \\Gamma(\\Sigma M) = \\ker \\lambda_{\\Phi} \\oplus \\operatorname{im} \\lambda_{\\Phi}^* = \\ker \\lambda_{\\Phi} \\oplus T_{\\Phi} \\widehat{\\operatorname{Diff}}_s(M).\\Phi,$$\nwe can consider $\\ker \\lambda_{\\Phi}$ to be an infinitesimal slice to the diffeomorphism action.\n Indeed, we will prove that, in a weak sense, $\\ker \\lambda_{g,\\varphi}$ parametrizes a slice in a simple way. To see this we first need a parametrization of $\\Gamma(\\Sigma M)$ by the set $T_{(g,\\varphi)} \\Gamma(\\Sigma M) = \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M)$ near $(g,\\varphi)$. This will be frequently useful and throughout the rest of the article $\\Xi = \\Xi_{g,\\varphi}$ denotes this parametrization.\nWe define\n$$\\Xi_{g,\\varphi}: (U_g \\subset \\Gamma(\\odot^2 T^*M)) \\times \\Gamma(\\Sigma_g M) \\to \\Gamma(\\Sigma M)$$\n$$(h, \\psi) \\mapsto (g+h, \\hat{B}^g_{g+h}(\\varphi + \\psi))$$\nand its inverse\n$$\\Xi^{-1}: \\Gamma(\\Sigma M) \\to U_g \\times \\Gamma(\\Sigma_g M)$$\n$$(g', \\varphi') \\mapsto (g'-g, \\hat{B}^{g'}_g (\\varphi') - \\varphi).$$\nHere $U_g = \\{h \\in \\Gamma(\\odot^2 T^*M) : g+h \\text{ is a metric}\\}$.\nIn terms of this parametrization we can formulate the following slice theorem:\n\\begin{prop}\nLet $\\Phi = (g,\\varphi) \\in \\Gamma(\\Sigma M)$ and assume $g$ has no Killing fields. Then there exists a $C^{k+1,\\alpha}$ neighborhood $U$ of $\\Phi$, such that for any $\\tilde{\\Phi} \\in U$, there exists a $C^{k+2,\\alpha}$ diffeomorphism $f: M\\to M$, such that\n$$\\lambda_{\\Phi}(\\Xi^{-1}(F^* \\tilde{\\Phi})) = 0.$$\n\\end{prop}\n\\begin{proof}\nWe base the proof on \\cite{Viaclovsky13}, theorem 3.6.\nConsider the map\n$$G: \\Gamma^{k+1, \\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M) \\times \\Gamma^{k+2, \\alpha}(TM) \\to \\Gamma^{k,\\alpha}(TM)$$\n$$((h, \\psi), X) \\mapsto \\lambda_{\\Phi}(\\phi^{X*}_1(g+h, B^g_{g+h}(\\varphi + \\psi)))$$\nThen the derivative of $G$ at $((0,0), 0)$ in $X$ direction is given by\n$$\\frac{d}{dt}\\Bigr|_{t=0} \\lambda_{\\Phi}(\\phi^{tV*}_1 \\Phi) = \\lambda_{\\Phi}(\\tilde{\\mathcal{L}}_V \\Phi) = \\lambda_{\\Phi} \\lambda_{\\Phi}^* V$$\nfor $V \\in \\Gamma^{k+2, \\alpha}(TM)$. Since $g$ posesses no Killing fields, $\\lambda_{\\Phi} \\lambda_{\\Phi}^*$ is injective, because in the first component $\\lambda_{\\Phi} \\lambda_{\\Phi}^* X$ is just $\\delta_g \\delta_g^* X^{\\flat}$. Additionally, $\\lambda_{\\Phi} \\lambda_{\\Phi}^*$ is an elliptic operator. It is selfadjoint and hence it must also be surjective. Thus we may apply the implicit function theorem and we find that there exists a neighborhood $U \\subset \\Gamma^{k+1, \\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M)$ of $(0,0)$ and a map $H: U \\to \\Gamma^{k+2, \\alpha}(TM)$, such that $G((h,\\psi), H(h,\\psi)) = 0$.\nNow let $\\tilde{\\Phi} \\in \\Xi(U)$.\nThen denote by $f$ the time-$1$ map of the vector field $H(\\Xi^{-1}(\\tilde{\\Phi}))$. Then\n$$\\lambda_{\\Phi}(F^*(\\Xi^{-1}(\\tilde{\\Phi}))) = 0$$\nby construction.\nThe statement then follows, because\n$$F^*(\\Xi^{-1}(\\tilde{\\Phi})) = \\Xi^{-1}(F^*(\\tilde{\\Phi})).$$\n\\end{proof}\n\n\n\\section{Volume normalized spinor flow}\nVolume constrained critical points evolve by rescaling under the spinor flow. We expect similar behavior near such a point. To address convergence questions in this situation, it is thus useful to rescale the solutions to a fixed volume. In this section, we introduce the volume normalized spinor flow and describe its evolution equation.\nLet $\\Phi_t = (g_t, \\phi_t)$ be a solution to the spinor flow. We denote by $\\mu(t)$ the normalizing factor $\\left(\\int_M \\operatorname{vol}_{g_t}\\right)^{-2\/n}$. Then $\\int_M \\operatorname{vol}_{\\mu(t) g_t} = 1$.\nNow let $\\tilde{\\Phi}(t) = (\\tilde{g}(t), \\tilde{\\varphi}(t))$, where\n$$\\tilde{g}(t) = \\mu (\\tau(t)) g_{\\tau(t)},$$\n$$\\tilde{\\varphi}(t) = \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (\\varphi_{\\tau(t)}),$$\nwhere $\\tau: I \\subset \\mathbb R \\to J \\subset \\mathbb R$ is some time reparametrization.\nThen we have\n$$\\partial_t \\tilde{g}_t = \\dot{\\mu}(\\tau(t)) \\tau'(t) g_{\\tau(t)} + \\mu(\\tau(t)) \\dot{g}_{\\tau(t)} \\tau'(t),$$\n$$\\partial_t \\tilde{\\varphi}_t = \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (\\dot{\\varphi}_{\\tau(t)}) \\tau'(t).$$\nSolving a separable ordinary differential equation, we can arrange $\\tau'(t) \\mu(\\tau(t)) = 1$.\nWe call $\\tilde{\\Phi}_t$ with this choice of time rescaling the {\\em volume normalized spinor flow}.\nFor any $h\\in \\Gamma(\\odot^2 T^*M)$, we denote by $\\mathring{h}$ the tensor \n$$h - \\frac{\\int_M \\operatorname{tr}_g h \\operatorname{vol}_g}{n \\int_M \\operatorname{vol}_g} g.$$\nSince $\\tilde{g}_t$ has constant volume $1$, it follows that $\\int_M \\partial_t g_t \\operatorname{vol}_{g_t} = 0$. \nThus we have\n$$\\partial_t \\tilde{g}_t = \\mathring{Q}_1(g_{\\tau(t)}, \\varphi_{\\tau(t)}).$$\nBy corollary 4.5 in \\cite{Ammann2015}, we moreover have $Q_1(c^2 g, \\hat{B}^g_{c^2 g}(\\varphi)) = Q_1(g, \\varphi)$, which implies\n$$\\partial_t \\tilde{g}_t = \\mathring{Q}_1(g_{\\tau(t)}, \\varphi_{\\tau(t)}) = \\mathring{Q}_1\\left(\\mu (\\tau(t)) g_{\\tau(t)}, \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (\\varphi_{\\tau(t)})\\right) = \\mathring{Q}_1(\\tilde{\\Phi}_t).$$\nAgain by corollary 4.5 in op. cit., we have $Q_2(c^2 g, \\hat{B}^g_{c^2 g} (\\varphi)) = c^{-2} \\hat{B}^g_{c^2 g} Q_2(g, \\varphi)$. Thus\n$$\\partial_t \\tilde{\\varphi}_t = \\mu(t)^{-1} \\hat{B}^{g_{\\tau(t)}}_{\\mu(\\tau(t)) g_{\\tau(t)}} (Q_2(g_{\\tau(t)}, \\varphi_{\\tau(t)})) = Q_2(\\tilde{\\Phi}_t).$$\nWe define\n$$\\mathring{Q}(\\Phi) = (\\mathring{Q}_1(\\Phi), Q_2(\\Phi))$$\nand can rewrite the evolution of $\\tilde{\\Phi}_t$ as\n$$\\partial_t \\tilde{\\Phi}_t = \\mathring{Q}(\\tilde{\\Phi}_t).$$\nSince $\\mathring{Q}$ is the negative gradient of $\\mathcal{E}$ restricted to the set\n$$\\mathcal{N}^1 = \\left\\{ \\Phi = (g, \\varphi) \\in \\mathcal{N} : \\int_M \\operatorname{vol}_g = 1\\right\\},$$\nwe conclude that the volume normalized spinor flow coincides with the negative gradient flow of $\\mathcal{E}$ restricted to $\\mathcal{N}^1$.\n\n\\section{Analytical setup}\nIn the following proof of stability we will analyze three flows: the spinor flow, the gauged spinor flow and the mapping flow. Each of these flows is defined on an infinite dimensional manifold rather than a vector space and we feel it is appropiate to clarify our analytic setup, so that we can proceed in a somewhat more formal manner later on without bypassing rigor altogeher. \n\nThe set of unit spinors $\\mathcal{N}$ forms a Fr\u00e9chet manifold with the $\\mathcal{C}^{\\infty}$ topology. We will however never use this topology directly. Instead, we will typically restrict to a chart and work with the Sobolev or $C^{k,\\alpha}$ topologies. We do this as follows.\nFix $\\Phi_0 = (g_0, \\varphi_0) \\in \\Gamma(\\Sigma M)$. We already constructed the chart \n$$\\Xi^{-1}_{\\Phi_0} : U \\subset \\Gamma(\\Sigma M) \\to V \\subset \\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_{g_0} M).$$\nThe metric $g_0$ then induces the usual $H^k$ and $C^{k,\\alpha}$ norms on $\\Gamma(\\odot^2 T^*M) \\oplus \\Gamma(\\Sigma_g M)$ and we simply pull them back via the chart.\nLocally we can now consider the spinor energy $\\mathcal{E}$ as a map $V \\to \\mathbb R$ and $Q$ as a map $V \\to V$.\nWhenever we use a $C^k$ or $H^s$ norm we implicitly use this construction. In particular, when we write $\\|\\Phi - \\Phi_0\\|_{X}$ for a fixed $\\Phi_0$ and a nearby $\\Phi$, we mean $\\|\\Xi^{-1}_{\\Phi_0} (\\Phi)\\|_{X}$, where $X$ is one of the discussed Banach spaces.\n\nFor the mapping flow we proceed in a similar manner. Note first that for $f_0 \\in \\mathcal{C}^{\\infty}(M,M)$, there is a local chart around $f_0$ given by\n$$U \\subset \\mathcal{C}^{\\infty}(M,M) \\to V \\subset \\Gamma(f_0^* TM)$$\n$$f \\mapsto (x \\mapsto (\\exp_{f_0(x)})^{-1}(f(x))),$$\nwhere $\\exp$ is the exponential map of some Riemannian metric on $g$ and $V$ is a neighborhood of the $0$ section in $TM$, such that $exp_x$ is a diffeomorphism from $V_x = T_x M \\cap V$ to $\\exp (V_x)$ for every $x \\in M$. \nThen we define\n$$U = \\{f: M\\to M \\Big| (f_0,f)(M) \\subset \\exp(V)\\}.$$\nWe can define appropiate norms in the standard manner using some Riemannian metric on $M$, for example\n$$\\left(X,Y\\right)_{L^2} = \\int_M g_{f_0(p)}(X(p), Y(p)) \\operatorname{vol}_g$$\nfor $X,Y \\in \\Gamma(f_0^* TM)$.\n\nFor future reference we also quote a standard parabolic estimate and prove an interior estimate following from this.\n\\begin{thm}\nSuppose $A_t$ is an elliptic differential operator of order $m$, uniformly elliptic in $t$, with $\\mathcal{C}^{\\infty}$ coefficients in $x$ and $t$.\nThen for any $s \\in \\mathbb R$ and $T > 0$, there exists $C > 0$ such that\n$$\\|u_t\\|_{H^s}^2 + \\int_0^T \\|u_{t'}\\|_{H^{s+m'}}^2 dt' \\leq C \\left( \\|u_0\\|_{H^s}^2 + \\int_0^T \\|\\partial_t u_{t'} - A_{t'} u_{t'}\\|^2_{H^{s-m'}} dt' \\right)$$\nfor any $t \\in [0, T]$ and $u \\in C^1([0,T], H^s) \\cap C^0([0,T], H^{s+m'})$, where $m' = m\/2$.\n\\end{thm}\nFor a proof, see 6.5.2 in \\cite{Chazarain1982}.\nWe will need the following estimate for solutions, derived from this inequality:\n\\begin{cor}\n\\label{PE}\nFor any $\\delta > 0$ and any \n $A_t$ as above, there exists $C, \\tilde{C}>0$, such that for any $u_t$ a solution of\n$$\\partial_t u_t = A_t u_t,$$\nwe have\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|_{H^r}^2d \\tau \\leq C \\int_0^T \\|u_{\\tau}\\|_{H^{s}}^2 d\\tau,$$\nas well as\n$$\\|u_{t}\\|_{H^r}^2 \\leq \\tilde{C} \\int_0^T \\|u_{\\tau}\\|_{H^{s}}^2 d\\tau$$\nfor any $r,s \\in \\mathbb R$ and any $t \\in [\\delta, T]$.\n\\end{cor}\n\\begin{proof}\nFor $r < s$ the inequality is trivial.\nFor $r > s$ the claim follows inductively from\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|^2_{H^{s+m'}} d\\tau \\leq C \\int_0^T \\|u_{\\tau}\\|_{H^{s-m'}}^2 d\\tau.$$\nFor this consider $f:[0,T] \\to [0,1]$ smooth such that $f(0) = 0, f(\\delta) = 1$.\nThen\n$$\\partial_t (f(t) u_t) - A_t u_t = (\\partial_t f(t)) u_t.$$\nHence the above estimate yields\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|_{s+m'}^2 d\\tau \\leq C \\int_0^T \\|u_{\\tau}\\|_{s-m'}^2 d\\tau,$$\nwhere $C = \\max |\\partial_t f|$.\n\nWe have shown that\n$$\\int_{\\delta}^T \\|u_{\\tau}\\|_{H^r}^2 d\\tau \\leq \\tilde{C} \\int_0^T \\|u_{\\tau}\\|_{H^s}^2 d\\tau.$$\nSince $\\partial_t u_t = A_t u_t$ and $u_t$ is a differential operator of order $m$ this implies\n$$\\int_{\\delta}^T \\|\\partial_{\\tau} u_{\\tau}\\|^2_{H^{r-m}} d\\tau \\leq \\tilde{C} \\int_0^T \\|u_{\\tau}\\|^2_{H^s} d\\tau$$\nand hence by the Sobolev embedding $W^{1,2}([a,b]; H^{l+1}, H^l) \\hookrightarrow C^0([a,b]; H^l)$ (cf. \\cite{Cherrier12}, Theorem 1.7.4 and (1.7.62)) we conclude\n$$\\|u_t\\|_{H^{r-m}} \\leq \\hat{C} \\int_0^T \\|u_{\\tau}\\|^2_{H^s} d\\tau.$$\n(Here\n$$W^{1,2}([a,b]; H^{l+1}, H^l) = L^2([a,b]; H^{l+1}) \\cap \\{u: [a,b] \\to H^l : \\partial_t u \\in L^2([a,b]; H^l) \\}$$\nwith the obvious norm.)\n\\end{proof}\n\n\n\\section{The \u0141ojasiewicz inequality and gradient estimates}\nThe \u0141ojasiewicz inequality relates the norm of the gradient of a differentiable function to its value near a critical point in a way that allows us to show convergence of the gradient flow. There are two situations when \u0141ojasiewicz inequalities are known to hold. The optimal situation is when the function is a Morse function or less restrictively a Morse--Bott function. Then we have\n$$|f(x) - f(x_0)| \\leq C \\|\\operatorname{grad} f(x)\\|^2$$\nfor $x_0$ a critical point of $f$ and some constant $C > 0$. This can be easily seen by applying the Morse--Bott lemma: near a critical manifold we may write a Morse--Bott function as\n$$f(x_1, ..., x_n) = c + x_1^2 + ... + x_r^2 - x_{r+1}^2 - ... - x_s^2,$$\nwhere $(x_1, ..., x_n)$ are coordinates with $x_0$ at the origin and critical manifold $\\{x_{s+1} = ... = x_{n} = 0\\}$.\nBecause in a small neighborhood the Riemannian metric is very close to being Euclidean, we get the inequality\n$$|f(x) - c| \\leq C |\\operatorname{grad} f(x)|^2$$\nfor some $C > 0$.\nThe other case is that $f$ is analytic. Then there exists $\\theta \\in (1,2)$, such that\n$$|f(x) - f(x_0)| \\leq \\|\\operatorname{grad} f(x)\\|^{\\theta}.$$\nWe will make use of both versions. The inequality for analytic functions is a difficult theorem in the theory of semianalytic sets, due to \u0141ojasiewicz. The first version will be employed to demonstrate stability of parallel spinors, since there we know $\\mathcal{E}$ to be Morse--Bott. For volume constrained critical points we do not know this and instead use the weaker inequality for analytic functions.\nBoth inequalities are known in this general form only for functions on finite dimensional domains. We will spend most of the rest of the section justifying these inequalities for the spinor energy functional.\n\n\\begin{prop}[Optimal \u0141ojasiewicz inequality for parallel spinors]\n\\label{LIP}\nLet $\\bar{\\Phi}$ be a critical point of $\\mathcal{E}$. (Hence $\\bar{\\Phi}$ is an absolute minimiser with $\\mathcal{E}(\\bar{\\Phi}) = 0$.) Then there exists a $C^{2,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$ and some constant $C>0$, such that for any $\\Phi \\in U$ we have\n$$\\mathcal{E}(\\Phi) \\leq C \\|Q(\\Phi)\\|_{L^2}^2.$$\n\\end{prop}\n\n\\begin{prop}[\u0141ojasiewicz inequality for volume constrained critical points]\n\\label{LIVC}\nLet $\\bar{\\Phi} = (\\bar{g},\\bar{\\varphi})$ be a volume constrained critical point of $\\mathcal{E}$. Then there exists a $C^{2,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$ and some constant $\\theta \\in (1,2)$, such that for any $\\Phi=(g,\\varphi)$ with $\\int_M \\operatorname{vol}_g = \\int_M \\operatorname{vol}_{\\bar{g}}$ we have\n$$|\\mathcal{E}(\\Phi) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\|\\mathring{Q}(\\Phi)\\|_{L^2}^{\\theta}.$$\nIf the set of volume constrained critical points near $\\bar{\\Phi}$ is a manifold, this can be improved to\n$$|\\mathcal{E}(\\Phi) - \\mathcal{E}(\\bar{\\Phi})| \\leq C \\|\\mathring{Q}(\\Phi)\\|_{L^2}^2.$$\n\\end{prop}\nThe proofs of both propositions rely on the following infinite-dimensional form of the \u0141ojasiewicz inequality, due to Colding and Minicozzi II, see \\cite{Colding2013}.\n\\begin{thm}\n\\label{CML}\n\\begin{enumerate}\n\\item Suppose $E \\subset L^2$ is a closed subspace, $U$ is an open neighborhood of $0$ in $C^{2, \\beta} \\cap E$.\n\\item Suppose $G: U \\to \\mathbb R$ is an analytic function or that there is a neighborhood $V$ of $0$, such that $\\{x \\in V : \\operatorname{grad} G (x) = 0\\}$ is a finite dimensional submanifold.\n\\item Suppose the gradient $\\operatorname{grad} G : U \\to C^{\\beta} \\cap E$ is $C^1$, $\\operatorname{grad} G (0) = 0$ and\n$$\\|\\operatorname{grad} G(x) - \\operatorname{grad} G(y) \\|_{L^2} \\leq C \\|x - y\\|_{H^2}$$\n\\item $L = D \\operatorname{grad} G (0)$ is symmetric, bounded from $C^{2, \\beta} \\cap E$ to $C^{\\beta} \\cap E$ and from $H^2 \\cap E$ to $L^2 \\cap E$ and Fredholm from $C^{2, \\beta} \\cap E$ to $C^{\\beta} \\cap E$.\n\\end{enumerate}\nThen there exists $\\theta \\in (1,2)$ so that for all $x\\in E$ sufficiently small\n$$|G(x) - G(0)| \\leq \\|\\operatorname{grad} G(x)\\|_{L^2}^\\theta$$\nIf there is a neighborhood $V$ of $0$, such that $\\{x \\in V : \\operatorname{grad} G (x) = 0\\}$ is a finite dimensional submanifold, we get the stronger inequality\n$$|G(x) - G(0)| \\leq C \\|\\operatorname{grad} G(x)\\|_{L^2}^2$$\nfor some $C>0$.\n\\end{thm}\n{\\em Remark.} Colding and Minicozzi II prove this for $G$ analytic. The alternative condition we give is essentially that $G$ is Morse--Bott at $0$. The proof in that case is the same except that when the finite dimensional \u0141ojasiewicz inequality is used, we instead invoke the stronger inequality for Morse--Bott functions.\n\nSince this theorem requires the linearisation of the gradient to be Fredholm we will be working on a slice of the spin diffeomorphism group. \n\\begin{lemma}\nLet $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ be a critical point. Let $\\iota: \\ker \\lambda_{\\bar{\\Phi}} \\to \\Gamma(\\Sigma M)$ be the inclusion. $f = \\mathcal{E} \\circ \\Xi_{\\bar{\\Phi}} \\circ \\iota$ fulfills the conditions of theorem \\ref{CML}.\nIn particular we have\n$$|f(x)| \\leq C \\|\\operatorname{grad} f(x)\\|_{L^2}^2$$\n\\end{lemma}\n\\begin{proof}\nWe equip $\\Gamma(\\odot^2 T^* M) \\oplus \\Gamma(\\Sigma_g M)$ with the $L^2$ metric induced by $\\bar{g}$, and similarly we define the $C^{2,\\alpha}$ norm in terms of $\\bar{g}$.\nThen clearly $\\mathcal{E} \\circ \\Xi_{\\bar{\\Phi}} \\circ \\iota$ is a smooth function and by \\cite{Ammann2015b} its critical set is smooth, thus the second condition in theorem \\ref{CML} is fulfilled.\nMoreover $0$ corresponds to $\\bar{\\Phi}$ and hence is a critical point, i.e. $\\operatorname{grad} f(0) = 0$.\nThe gradient of $f$ can be considered as a nonlinear second order differential operator. In fact, it is a smooth map \n$$\\operatorname{grad} f: \\Gamma^{2,\\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M) \\to \\Gamma^{\\alpha}(\\odot^2 T^*M \\oplus \\Sigma_g M).$$\nOn any bounded $C^{2,\\alpha}$ neighborhood $U$ of $0$ we have\n$$\\|\\operatorname{grad} f(x) - \\operatorname{grad} f(y)\\|_{L^2} \\leq C \\|x - y\\|_{H^2}.$$\nThis is a simple consequence of the fact that $Q(g,\\varphi)$ can be locally represented as a polynomial expression\nin the coordinate expressions of $g$ and $\\varphi$ and their first and second derivatives. In a bounded $C^{2,\\alpha}$ neighborhood we then estimate terms as needed to get an expression which is bounded by $\\|(g,\\varphi)\\|_{H^2}$.\nThis concludes the argument for conditions 1,2 and 3.\n\nSince $DQ(\\bar{\\Phi})$ is symmetric (by \\cite{Ammann2015}), so is $L$. Since $L$ is a linear second order differential operator, it induces continuous maps $C^{2,\\alpha} \\to C^{\\alpha}$ and $H^2 \\to L^2$.\nIt remains to be shown that $L$ is Fredholm.\nTo see this, remember that we have a splitting\n$$T_{\\bar{\\Phi}} \\mathcal{N} = \\ker \\lambda_{\\bar{\\Phi}} \\oplus \\operatorname{im} \\lambda^*_{\\bar{\\Phi}}.$$\nWith respect to these operators, we know the two identities\n$$DQ(\\bar{\\Phi}) \\circ \\lambda^*_{\\bar{\\Phi}} = 0 \\text{ and } \\lambda_{\\bar{\\Phi}} \\circ DQ(\\bar{\\Phi}) = 0,$$\nboth of which reflect diffeomorphism invariance of $Q$.\nMoreover, we introduced the perturbed gradient $\\tilde{Q}_{\\bar{\\Phi}}$, which we know is strongly elliptic and thus its linearization is Fredholm. Its linearization is also symmetric.\nThus we conclude that $DQ(\\bar{\\Phi})$ has the form\n$$\\bordermatrix{\n & \\ker \\lambda_{\\bar{\\Phi}} & \\operatorname{im} \\lambda_{\\bar{\\Phi}}^* \\cr\n\\ker \\lambda_{\\bar{\\Phi}} & P & 0 \\cr\n\\operatorname{im} \\lambda_{\\bar{\\Phi}}^* & 0 & 0 \\cr\n},$$\nwhereas $D\\tilde{Q}_{\\bar{\\Phi}}(\\bar{\\Phi})$ has the form \n$$\\bordermatrix{\n & \\ker \\lambda_{\\bar{\\Phi}} & \\operatorname{im} \\lambda_{\\bar{\\Phi}}^* \\cr\n\\ker \\lambda_{\\bar{\\Phi}} & P & 0 \\cr\n\\operatorname{im} \\lambda_{\\bar{\\Phi}}^* & 0 & R \\cr\n}.$$\nSince $D\\tilde{Q}_{\\bar{\\Phi}}(\\bar{\\Phi})$ is Fredholm, so is $P = \\pi \\circ DQ(\\Phi) \\circ \\iota$, where $\\pi: T_{\\bar{\\Phi}} \\mathcal{N} \\to \\ker \\lambda_{\\bar{\\Phi}}$ denotes the orthogonal projection.\nWe compute\n$$D\\operatorname{grad} f (0) = D(\\Xi \\circ \\iota)(0)^* DQ(\\Xi \\circ \\iota(x)) = \\pi \\circ D\\Xi(0)^* DQ(\\bar{\\Phi}).$$\nSince the domain is restricted to $\\ker \\lambda_{\\bar{\\Phi}}$ and $D\\Xi(0) = \\operatorname{id}$, we conclude that\n$$D \\operatorname{grad} f(0) = P,$$\nand hence $L = D\\operatorname{grad} f(0)$ is Fredholm as required. Thus we have checked all conditions in theorem \\ref{CML}, and the inequality holds.\n\\end{proof}\n\n\\begin{proof}[Proof of proposition \\ref{LIP}]\nWhat remains to be shown is that the inequality\n$$|f(x)| \\leq \\|\\operatorname{grad} f(x)\\|_{L^2}^2$$\nimplies the inequality\n$$|\\mathcal{E}(\\Phi)| \\leq C \\|Q(\\Phi)\\|^2_{L^2}.$$\nFirst, by the slice theorem there exists a $C^{k+1,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ there exists a diffeomorphism $f: M\\to M$, such that\n$$\\lambda_{\\bar{\\Phi}}(\\Xi^{-1}(F_* \\Phi)) = 0.$$\nSince\n$$\\mathcal{E}(F_* \\Phi) = \\mathcal{E}(\\Phi), \\quad F_* Q(\\Phi) = Q(F_* \\Phi)$$\nand since the $L^2$ metric is diffeomorphism invariant, we can assume that $\\Phi$ lies in the slice, i.e. $\\lambda_{\\bar{\\Phi}}(\\Xi^{-1}(\\Phi)) = 0$. Then we have\n$$|\\mathcal{E}(\\Phi)| = f(\\Xi^{-1}(\\Phi)) \\leq \\|\\operatorname{grad} f(\\Xi^{-1}(\\Phi))\\|_{L^2}^2.$$\nHence we must show\n$$\\|\\operatorname{grad} f (\\Xi^{-1}(\\Phi))\\|_{L^2}^2 \\leq \\|Q(\\Phi)\\|_{L^2}.$$\nFirst note that the metric on $\\ker \\lambda_{\\bar{\\Phi}}$ is the metric induced by $\\bar{\\Phi}$. By making the neighborhood smaller if necessary, we can assume that all $L^2$ metrics in that neighborhood are uniformly equivalent. We have\n$$\\operatorname{grad} f(\\Xi^{-1}(\\Phi)) = D(\\Xi \\circ \\iota) (\\Xi^{-1}(\\Phi))^* Q(\\Phi).$$\nSince $D(\\Xi \\circ \\iota)$ is clearly Lipschitz, we obtain our estimate. This concludes the proof of the \u0141ojasiewicz inequality in this case.\n\\end{proof}\n\\begin{proof}[Proof of proposition \\ref{LIVC}]\nFor the purposes of the following discussion, read the spaces of smooth mappings as the spaces of $C^{2,\\alpha}$ mappings, so that they are Banach spaces or Banach manifolds.\nBy the analytic regular value theorem, we can find an analytic parametrization of\n$$\\left\\{ g \\in \\Gamma(\\odot^2_+ T^*M) : \\int_M \\operatorname{vol}_g = 1\\right\\}$$\nby\n$$\\left\\{ h \\in \\Gamma(\\odot^2_+ T^* M) : \\int_M \\operatorname{tr}_g h \\operatorname{vol}_g = 0\\right\\}.$$\n(For a treatment of the implicit function theorem in the analytic category on Banach spaces, take for example \\cite{Hajek14}, theorem 174.)\nWe combine this parametrization with $\\Xi_{g,\\varphi}$ to obtain an analytic parametrization\n$$\\Psi: U \\subset V_0 \\to \\mathcal{N}^1$$\nwhere\n$$V_0 = \\left\\{(h, \\psi) \\in \\ker \\lambda_{g,\\varphi} : \\int_M \\operatorname{tr}_g h \\operatorname{vol}_g = 0\\right\\}$$\nand\n$$\\mathcal{N}^1 = \\left\\{\\Phi = (g,\\varphi) \\in \\mathcal{N} : \\int_M \\operatorname{vol}_g = 1\\right\\}.$$\nDefine $f = \\mathcal{E} \\circ \\iota \\circ \\Psi$, with $\\iota: \\mathcal{N}^1 \\to \\mathcal{N}$ the inclusion.\nThen $f$ fulfills the conditions of theorem \\ref{CML}, which can be shown as in the previous lemma. \nApplying the theorem, we thus obtain\n$$|f(x) - f(0)| \\leq \\|\\operatorname{grad} f (x)\\|_{L^2}^{\\theta},$$\nwhere $\\theta \\in (1,2)$. If the critical set is a manifold near $\\bar{\\Phi}$, we use the optimal version theorem of theorem \\ref{CML} and obtain\n$$|f(x) - f(0)| \\leq C \\|\\operatorname{grad} f(x)\\|_{L^2}^2$$\nfor some $C > 0$.\n What remains to be shown is\n$$\\|\\operatorname{grad} f(\\Psi^{-1}(\\Phi))\\|_{L^2} \\leq C \\|\\mathring{Q}(\\Phi)\\|_{L^2}.$$\nAs in the previous proposition, we compute\n$$\\operatorname{grad} f(\\Psi^{-1}(\\Phi)) = (D\\Psi )^*(\\Psi^{-1}(\\Phi)) \\operatorname{grad} (\\mathcal{E} \\circ \\iota)(\\Phi).$$\nThen the claim follows, since, on the one hand, $D\\Psi$ is Lipschitz by the regular value theorem, and on the other hand\n$$\\operatorname{grad}(\\mathcal{E} \\circ \\iota)(\\Phi) = D\\iota(\\Phi)^* \\operatorname{grad} \\mathcal{E}(\\Phi) = D\\iota(\\Phi)^* Q(\\Phi).$$\nSince $D\\iota(\\Phi)^* : T_{\\Phi} \\mathcal{N} \\to T_{\\Phi} \\mathcal{N}^1$ is the orthogonal projection,\nthis implies\n$$\\operatorname{grad}(\\mathcal{E} \\circ \\iota)(\\Phi) = \\mathring{Q}(\\Phi).$$\n\\end{proof}\n\n\n\\begin{thm}[Energy decay]\n\\label{L2cv}\nSuppose $M$ is a compact manifold.\n\\begin{enumerate}\n\\item Suppose $\\bar{\\Phi}$ is a critical point of $\\mathcal{E}$. Then there exists a $C^{2,\\alpha}$ neigborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ the following inequalities hold\n$$\\mathcal{E}(\\Phi_t) \\leq C e^{-\\alpha t},$$\n$$\\int_T^{\\infty} \\|Q(\\Phi_t)\\|^2_{L^2} dt \\leq C e^{-\\alpha T},$$\nand\n$$\\int_T^{\\infty} \\|Q(\\Phi_t)\\|_{L^2} dt \\leq C e^{-\\alpha T},$$\nwhere $C, \\alpha>0$ and $\\Phi_t$ is the solution of\n$$\\partial_t \\Phi_t = Q(\\Phi_t), \\Phi_0 = \\Phi.$$\n\\item Suppose $\\bar{\\Phi}$ is a volume constrained minimizer of $\\mathcal{E}$. Then there exists a $C^{2,\\alpha}$ neighborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ it holds\n$$|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\frac{C}{1+T^{\\beta}},$$\n$$\\int_T^{\\infty} \\|\\mathring{Q}(\\Phi_t)\\|^2_{L^2} dt \\leq \\frac{C}{1 + T^{\\beta}}$$\nand\n$$\\int_T^{\\infty} \\|\\mathring{Q}(\\Phi_t)\\|_{L^2} dt \\leq \\frac{C}{1+T^{\\gamma}},$$\nfor some $C, \\beta > 1$. If the set of volume constrained critical sets is a manifold near $\\bar{\\Phi}$, we can instead choose exponential bounds as in the first case. Here we assume $\\Phi_t$ is the volume normalized spinor flow with initial condition $\\Phi_0 = \\Phi \\in U$.\n\\end{enumerate}\nThe integrals are to be read as the integral from $T$ to the maximal time of existence in the neighborhood $U$. The constants $C, \\alpha, \\beta$ only depend on the constants $C$ and $\\theta$ in the \u0141ojasiewicz inequalities.\n\\end{thm}\n{\\em Remark. } The constants $\\beta$ and $\\gamma$ can be computed from the constant $\\theta$ in the \u0141ojasiewicz inequality as $\\beta = \\frac{\\theta }{2 - \\theta}$ and $\\gamma = \\frac{\\theta - 1}{2 - \\theta}$. As $\\theta$ tends to $2$, $\\beta$ tends to infinity, i.e. the convergence rate improves. As $\\theta$ tends to $1$, $\\beta$ tends to $1$, i.e. the convergence rate gets worse. Likewise, $\\gamma$ tends to $\\infty$ if $\\theta$ tends to $2$, but $\\gamma$ tends to $0$ as $\\theta$ tends to $1$. \n\\begin{proof}\nFirst we note that\n$$\\frac{d}{dt} \\mathcal{E}(\\Phi_t) = -\\|Q(\\Phi_t)\\|_{L^2}^2$$\nimplies that the integral of the gradient over all future time is controlled by the energy at a fixed time.\nNow applying the optimal \u0141ojasiewicz inequality, we obtain \n$$\\frac{d}{dt} \\mathcal{E}(\\Phi_t) \\leq -\\frac{1}{C} \\mathcal{E}(\\Phi_t).$$\nIntegrating this differential inequality, we obtain\n$$\\mathcal{E}(\\Phi_t) \\leq \\mathcal{E}(\\Phi_0) e^{-(1\/C) t}.$$\nChoosing the neighborhood so that $\\mathcal{E}$ is bounded, we obtain the desired inequality.\n\nFor the second case consider\n$$\\frac{d}{dt} |\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| = -\\|Q(\\Phi_t)\\|_{L^2}^2 \\leq -|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})|^{2\/\\theta}.$$\nIntegrating this differential inequality, we obtain\n$$|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\left(\\frac{2}{\\theta} - 1\\right) \\frac{1}{(C + t)^{\\beta}}$$\nwhere $\\beta = \\frac{1}{2\/\\theta - 1}$ and $C = |\\mathcal{E}(\\Phi_0) - \\mathcal{E}(\\bar{\\Phi})|^{1-2\/\\theta}$.\nBy continuity of $\\mathcal{E}$ we can find a lower bound for $C$ on a small neighborhood, and using this lower bound we obtain the desired inequality. The bound for the integral of $\\|\\mathring{Q}(\\Phi_t)\\|_{L^2}$ follows as above.\n\nFor the estimates of $\\int_T^{\\infty} \\|Q(\\Phi_t)\\| dt$, notice that the \u0141ojasiewicz inequality implies $\\mathcal{E}(\\Phi)^{-1\/\\theta} \\geq C \\|Q(\\Phi)\\|^{-1}$. (Here we actually have $\\theta = 2$. The case of volume constrained minimizers is analogous with $\\theta \\neq 2$ in general.)\nThis implies\n\\begin{eqnarray*}\n-\\frac{d}{dt}\\mathcal{E}(\\Phi_t)^{1-1\/\\theta} & = & (1-1\/\\theta) \\mathcal{E}(\\Phi_t)^{-1\/\\theta} \\|Q(\\Phi_t)\\|^2\\\\\n& \\geq & C \\|Q(\\Phi_t)\\| \n\\end{eqnarray*}\nHence\n$$\\int_T^{\\infty} \\|Q(\\Phi_t)\\|_{L^2} dt \\leq C \\mathcal{E}(\\Phi_T)^{1-1\/\\theta}.$$\nPlugging in the estimate for $\\mathcal{E}(\\Phi_T)$ then gives the desired result.\n\\end{proof}\n\n\\section{Mapping flow estimates}\nSuppose $\\Phi_t$ solves\n$$\\partial_t \\Phi_t = Q(\\Phi_t).$$\nIn the previous section we proved a strong estimate of the gradient along the flow in the $L^2$ norm, provided $\\Phi_t$ is near a critical point. We would now like to improve this to an estimate in some higher regularity norm. Since the gradient $Q_t = Q(\\Phi_t)$ satisfies the linear parabolic equation\n$$\\partial_t Q_t = DQ(\\Phi_t) Q_t,$$\nthis is reasonable by parabolic regularity. Unfortunately, this equation is only weakly parabolic and hence we can not directly apply parabolic regularity. However, we recall that $\\tilde{\\Phi}_t = F_{t*} \\Phi_t$ obeys the strongly parabolic equation\n$$\\partial_t \\tilde{\\Phi}_t = \\tilde{Q}(\\tilde{\\Phi}_t)$$\nif $f_t$ satisfies the mapping flow equation\n$$\\partial_t f_t = P_{g_t, g_0}(f_t), f_0 = \\operatorname{id}_M.$$\nThe gauged gradient $\\tilde{Q}_t = \\tilde{Q}(\\tilde{\\Phi}_t)$ satisfies the linear strongly parabolic equation\n$$\\partial_t \\tilde{Q}_t = D\\tilde{Q}(\\tilde{\\Phi}_t) \\tilde{Q}_t.$$\nParabolic regularity applies to $\\tilde{Q}_t$, but we have no estimate of $\\tilde{Q}_t$! To obtain such an estimate, we will now show how to control $\\partial_t f_t$ along the mapping flow. In the next section, we will combine this estimate with the gradient estimate of the previous section to obtain an estimate of $\\tilde{Q}_t$.\n\n\\begin{lemma}\n\\label{MFE}\nLet $\\tilde{g} \\in \\Gamma(\\odot^2_+ T^* M)$ and $k > \\frac{n}{2} + 2$. Suppose $\\tilde{g}$ has no Killing fields.\nThen there exists a $H^k$ neighborhood $U \\times V$ of $(\\operatorname{id}_M, \\tilde{g})$ and constants $C, \\lambda > 0$, such that for a solution $f_t$ and a metric $g_t \\in V$, $g_t$ once differentiable in time, of an initial value problem\n$$f_0 = \\operatorname{id}_M$$\n$$\\dot{f_t} = P_{g_t, \\tilde{g}}(f_t)$$\nwe have\n$$\\int_{t_1}^{t_2} \\|P_{g_t, \\tilde{g}}\\|_{H^{-2}} dt \\leq C \\left( \\int_0^{t_1} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda (t - t_1)} dt + \\int_{t_1}^{t_2} \\|\\dot{g}_t\\|_{L^2} dt + e^{-\\lambda t_1}\\right)$$\nfor some $C, \\lambda > 0$, provided the flow exists until time $t_2$ in the neighborhood $U \\times V$.\n\\end{lemma}\n\\begin{proof}\nAs computed in \\cite{Ammann2015},\n$$DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M) X = -4 (\\delta_{\\tilde{g}} \\delta_{\\tilde{g}}^* X^{\\flat})^{\\sharp}.$$\nA computation of the symbol then shows that this operator is strongly elliptic. \nFurthermore, this formula implies\n$$\\left(DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M) X, X\\right)_{L^2} = -4 \\left(\\delta^*_{\\tilde{g}} X^{\\flat}, \\delta^*_{\\tilde{g}} X^{\\flat} \\right) = -4 \\left(\\mathcal{L}_X \\tilde{g}, \\mathcal{L}_X \\tilde{g} \\right)_{L^2}.$$\nSince we assume $\\tilde{g}$ has no Killing fields, this implies $DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M)$ is strictly negative definite, i.e. there exists $\\mu > 0$, such that\n$$\\left(DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M) X, X\\right)_{L^2} \\leq -\\mu \\left(X,X\\right)_{L^2}.$$\nSince the coefficients of the operator $P_{g_1, g_2}(f)$ are continuous in $f$ and the first derivatives of $g_1$ and $g_2$ and recalling that by the Sobolev embedding theorem $H^k$ continuously embeds in $C^2$, we conclude that there is a $H^k$ neighborhood $U$ of $\\tilde{g}$, a neighborhood $V$ of $\\operatorname{id}_M$ and a constant $0 < \\lambda < \\mu$, such that $DP_{g, \\tilde{g}}(f)$ is strongly elliptic and strictly negative definite with a constant $\\lambda$.\n\nSince $L = DP_{\\tilde{g}, \\tilde{g}}(\\operatorname{id}_M)$ is strictly negative definite, it induces an invertible operator from $H^{s+2} \\to H^s$. We have, up to equivalence,\n$$\\|f\\|_{H^{-2}} = \\|L^{-1}f\\|_{L^2}.$$\nThis implies, in particular, that $DP_{g, \\tilde{g}}(f)$ is also strictly negative definite with respect to the Sobolev inner product $\\langle \\cdot, \\cdot \\rangle_{H^{-2}}$.\n\nWe will now derive a differential inequality for $\\|\\dot{f}_t\\|_{H^{-2}}^2$, where\n$$\\dot{f}_t = P_{g_t, \\tilde{g}}(f_t).$$\nFor brevity, we let $P_{g_t, \\tilde{g}}(f_t) = P_{g_t}(f_t)$.\nIn what follows, we tacitly assume $g_t \\in U$, $f_t \\in V$ for all $t$, as per the statement of the lemma. \nWe calculate\n\\begin{eqnarray*}\n\\frac{1}{2} \\frac{d}{dt} \\langle P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}} & = & \\langle \\frac{d}{dt} P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}\\\\\n& = & \\langle P_{\\dot{g_t}}(f_t) + DP_{g_t}(f_t) \\dot{f}_t, P_{g_t}(f_t) \\rangle_{H^{-2}} \\\\\n& = & \\langle P_{\\dot{g_t}}(f_t), P_{g_t}(f_t) \\rangle + \\langle DP_{g_t}(f_t) P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}.\n\\end{eqnarray*}\nThe map\n$$g \\mapsto P_g(f) = 2 df(\\delta_{f^*\\tilde{g}} g),$$\nis a linear first order differential operator with bounds dependent on $\\|f\\|_{C^1}$ and $\\|\\tilde{g}\\|_{C^1}$. As such we can estimate, using that bound and the Cauchy-Schwarz inequality\n$$|\\langle P_{\\dot{g_t}}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}| \\leq \\|P_{\\dot{g_t}} (f_t)\\|_{H^{-2}} \\|P_{g_t}(f_t)\\|_{H^{-2}} \\leq C \\|\\dot{g}_t\\|_{L^2} \\|P_{g_t}(f_t)\\|_{H^{-2}}.$$\nThen we obtain for\n$$a(t) = \\langle P_{g_t}(f_t), P_{g_t}(f_t) \\rangle_{H^{-2}}$$\nthe inequality\n$$\\frac{1}{2} \\dot{a}(t) \\leq C \\|\\dot{g}_t\\|_{L^2} \\sqrt{a(t)} - \\lambda a(t).$$\nLet $b(t) = \\sqrt{a(t)}$. The function $b$ then satisfies the following differential inequality\n$$\\dot{b}(t) \\leq -\\lambda b(t) + \\|g_t\\|_{L^2}.$$\nDefine\n$$\\beta(t) = e^{-\\lambda t} \\left( b(0) + \\int_0^t e^{\\lambda s} \\|\\dot{g}_s\\|_{L^2} ds \\right).$$\nThen we have\n$$\\dot{\\beta}(t) = -\\lambda \\beta(t) + \\|\\dot{g}_t\\|_{L^2}.$$\nWe deduce\n$$\\frac{d}{dt} (b-\\beta) \\leq -\\lambda (b-\\beta),$$\nand since $b(0) = \\beta(0)$, $b(t) \\leq \\beta(t)$ follows.\nTo obtain the claim of the lemma, we will now estimate the integral of $\\beta(t)$. For brevity, we denote $\\gamma(t) = \\|\\dot{g}_t\\|_{L^2}$. Define $\\chi(s,t) = 1$ if $0 \\leq s \\leq t$ and $\\chi(s,t) = 0$ otherwise. Then we calculate\n\\begin{eqnarray*}\n\\int_{t_1}^{t_2} e^{-\\lambda t} \\int_0^t e^{\\lambda s} \\gamma(s) ds dt\n& = & \\int_{t_1}^{t_2} \\int_0^t e^{\\lambda(s-t)} \\gamma(s) ds dt \\\\\n& = & \\int_{t_1}^{t_2} \\int_0^{t_2} \\chi(s,t) e^{\\lambda(s-t)} \\gamma(s) ds dt\\\\\n& = & \\int_0^{t_2} \\gamma(s) \\int_{t_1}^{t_2} \\chi(s,t) e^{\\lambda(s-t)} dt ds\\\\\n& = & \\int_0^{t_2} \\gamma(s) \\int_{\\max\\{s, t_1\\}}^{t_2} e^{\\lambda(s-t)} dt ds\\\\\n& = & \\int_0^{t_1} \\gamma(s) \\int_{t_1}^{t_2} e^{\\lambda(s-t)} dt ds + \\int_{s}^{t_2} \\gamma(s) \\int_{t_1}^{t_2} e^{\\lambda (s-t)} dt ds\\\\\n& \\leq & \\lambda^{-1} \\left( \\int_0^{t_1} e^{\\lambda (s-t_1)}\\gamma(s) ds + \\int_{t_1}^{t_2} \\gamma(s) ds \\right) \n\\end{eqnarray*}\nThe integral of the term $b(0) e^{-\\lambda t}$ is\n$$\\int_{t_1}^{t_2} b(0) e^{-\\lambda t} dt = \\lambda^{-1} b(0) \\left( e^{-\\lambda t_1} - e^{-\\lambda t_2} \\right).$$\nThus\n\\begin{eqnarray*}\n \\int_{t_1}^{t_2} \\beta(t) dt & \\leq & \\lambda^{-1} \\left( b(0) e^{-\\lambda t_1} + \\int_0^{t_1} e^{\\lambda (s-t_1)}\\gamma(s) ds + \\int_{t_1}^{t_2} \\gamma(s) ds \\right) \n\\end{eqnarray*}\nand the claim of the lemma follows.\n\\end{proof}\n\n\\section{Smooth convergence of the flow}\nNow everything is in place to prove stability of the spinor flow. We obtain slightly sharper theorems than in the introduction:\n\\begin{thm}\n\\label{stabP}\nSuppose $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ is a critical point of $\\mathcal{E}$, such that $\\bar{g}$ has no Killing fields. Then for any $k > \\frac{n}{2} + 5$ there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that any solution of the negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi \\in U$ converges in $H^k$ to a critical point. The speed of convergence is exponential.\n\\end{thm}\n\\begin{thm}\n\\label{stabVC}\nSuppose $\\bar{\\Phi} = (\\bar{g}, \\bar{\\varphi})$ is a volume constrained minimizer of $\\mathcal{E}$ and suppose the set of critical points is a manifold near $\\bar{\\Phi}$. Suppose furthermore, that $\\bar{g}$ has no Killing fields and $k > \\frac{n}{2} + 5$. Then there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that a solution of the volume constrained negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi \\in U$ converges in $H^k$ to a critical point. The speed of convergence is exponential.\n\nIf the critical set is not a manifold, but $\\theta$ in proposition \\ref{LIVC} can be chosen to be larger than $3\/2$, then there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that a solution of the volume constrained negative gradient flow $\\Phi_t$ with initial condition $\\Phi_0 = \\Phi \\in U$ converges in $H^k$ to a critical point. The speed of convergence is $O(T^{-\\kappa})$, $\\kappa = \\frac{2\\theta -3}{2-\\theta} >0$.\n\\end{thm}\n\nWe will reduce the proof of these theorems to the following two lemmas:\n\\begin{lemma}[Existence near critical points]\n\\label{UniformExistence}\nLet $\\bar{\\Phi}$ be a critical point of $\\mathcal{E}$ and let $T, \\epsilon > 0, k > \\frac{n}{2} + 2$.\nThen there exists $\\delta > 0$, such that for any $\\Phi$ with $\\|\\Phi - \\bar{\\Phi}\\|_{H^k} < \\delta$, the flow\n$$\\partial_t \\Phi_t = \\tilde{Q}(\\Phi_t), \\Phi_0 = \\Phi$$\nexists until time $T$ and $\\|\\Phi_T - \\bar{\\Phi}\\|_{H^k} < \\epsilon$. The same result holds for volume constrained critical points and the volume constrained flow. \n\\end{lemma}\nThe proof is analogous to the proof of corollary 8.6 in \\cite{Weiss2012}\n\\begin{lemma}[Decay of the gradient in a Sobolev norm]\n\\label{GradientDecay}\nSuppose $\\bar{\\Phi}$ is a critical point of $\\mathcal{E}$. Then for any $k > \\frac{n}{2} + 5$ there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, a neighborhood $V$ of $\\operatorname{id}_M$ in $\\operatorname{Diff}(M)$, constants $C, \\alpha > 0$, such that for $\\Phi \\in U$ the gauged spinor flow $\\tilde{\\Phi}_t$ with initial condition $\\Phi$ fulfills the following estimate \n\\begin{equation}\n\\label{GEa}\n\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq C e^{-\\alpha T}\n\\end{equation}\nas long as $\\Phi_t$ and $f_t$ remain in the neighborhoods $U$ and $V$ respectively.\n\nAnalogously, if $\\bar{\\Phi}$ is a volume constrained critical point of $\\mathcal{E}$ and the critical set near $\\bar{\\Phi}$ is a manifold, then for any $k > \\frac{n}{2} + 5$ there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, a neighborhood $V$ of $\\operatorname{id}_M$ in $\\operatorname{Diff}(M)$, constants $C, \\alpha > 0$, such that for $\\Phi \\in U$ the volume normalized gauged spinor flow $\\tilde{\\Phi}_t$ with initial condition $\\Phi$ fulfills the following estimate \n\\begin{equation}\n\\label{GEb}\n\\|\\mathring{\\tilde{Q}}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq C e^{-\\alpha T}\n\\end{equation}\nas long as $\\Phi_t$ and $f_t$ remain in the neighborhoods $U$ and $V$ respectively. If the critical set is not a manifold we instead find $C, \\beta > 0$, such that\n\\begin{equation}\n\\label{GEc}\n\\|\\mathring{\\tilde{Q}}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq \\frac{C}{1+ T^\\beta}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}[Proof of the lemma]\nWe start with the first case.\nWe will show this estimate by combining the gradient estimate from the \u0141ojasiewicz inequality and the estimate of the mapping flow. This will give us an estimate of the time integral of $\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^s}$ for $s=-3$, which we will then improve via parabolic regularity.\nWe consider the spinor flow\n$$\\partial_t \\Phi_t = Q(\\Phi_t), \\Phi_0 = \\Phi,$$\nthe gauged spinor flow\n$$\\partial_t \\tilde{\\Phi}_t = \\tilde{Q}(\\tilde{\\Phi}_t), \\tilde{\\Phi}_0 = \\Phi$$\nand the mapping flow\n$$\\partial_t f_t = P_{g_t, \\bar{g}}(f), f_0 = \\operatorname{id}_M.$$\nThen we have that\n$$\\tilde{\\Phi}_t = F_t^* \\Phi_t$$\nand hence\n\\begin{eqnarray*}\n\\tilde{Q}(\\tilde{\\Phi}_t) & = & \\partial_t (F_t^* \\Phi_t) \\\\\n& = & F_t^* \\tilde{\\mathcal{L}}_{X_t} \\Phi_t + F_t^* \\dot{\\Phi}_t\n\\end{eqnarray*}\nwhere $X_t = \\frac{d}{dt} f_t$ and $\\tilde{\\mathcal{L}}$ is the spinorial Lie derivative.\n\nMultiplication of Sobolev functions $H^k \\times H^s \\to H^s$ for negative $s$ and positive $k$ is continous, if $k > -s$ and $k > n\/2$, where $n$ is the dimension of the manifold, see theorem 2 (i), sect. 4.4.3 in \\cite{Runst96}. In particular, our choice of $k$ allows any $s \\geq -3$.\n\nWe will use this to estimate $\\tilde{\\mathcal{L}}_{X_t} \\Phi_t$ in the $H^s$ norm.\nRecall that \n$$\\tilde{\\mathcal{L}}_X \\Phi = (\\mathcal{L}_X g, \\tilde{\\mathcal{L}}_X \\varphi) = (2 \\delta_g^* X^{\\flat}, \\nabla_X^g \\varphi - \\frac{1}{4} dX^{\\flat} \\cdot \\varphi).$$\nIn local coordinates we have\n$$\\mathcal{L}_X g = p_1(g_{jk}, \\partial_l g_{mn}, X^i) + p_2(g_{ij}, \\partial_k X^l)$$\nfor some polynomials $p_1, p_2$, which are linear in the partial derivative terms and the $X^i$ terms. Likewise we have\n$$\\tilde{\\mathcal{L}}_X \\varphi = q_1(X^i, \\partial_j\\varphi^{\\alpha}) + q_2(g_ij, \\partial_l g_{mn}, X^k, \\varphi^{\\alpha})$$\nfor polynomials $q_1, q_2$, linear in the partial derivative terms and the $X^i$ terms.\nFrom this follows, using the multiplication theorem above and the fact that $H^{k-1}$ is a Banach algebra (since it embeds into $C^2$),\n\\begin{eqnarray*}\n\\|\\tilde{\\mathcal{L}}_X \\Phi\\|_{H^s} & \\leq & C \\left( \\|DX\\|_{H^s} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k-1}}^d + \\|X\\|_{H^s} \\sum_{d=1}^r \\|D\\Phi\\|_{H^{k-1}}^d \\right)\\\\\n& \\leq & C \\left( \\|X\\|_{H^{s+1}} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k-1}}^d + \\|X\\|_{H^s} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k}}^d \\right)\\\\\n& \\leq & C \\left(\\|X\\|_{H^{s+1}} \\sum_{d=1}^r \\|\\Phi\\|_{H^{k}}^d\\right)\n\\end{eqnarray*}\nfor $k > -s + n\/2 + 2$, where $r$ is the maximal degree of the polynomials $p_1, p_2, q_1, q_2$.\nSince we will choose $s=-3$ and $k > n\/2 + 5$, this will be the case.\n\nFurthermore, given a diffeomorphism $f: M \\to M$ and a lift to the topological spin structure $F: \\tilde{P} \\to \\tilde{P}$, we have\n$$F^* \\Phi = \\Phi \\circ F,$$\nwhere we view $\\Phi$ as an equivariant map $\\Phi: \\tilde{P} \\to \\left(\\widetilde{\\operatorname{GL}_n^+} \\times \\Sigma_n\\right)\/\\operatorname{Spin}(n)$.\nUsing the transformation rule, we can derive an estimate\n$$\\|u \\circ f\\|_{W^{k,p}(M)} \\leq \\nu(\\|f\\|_{C^{\\max \\{k, 1\\}}}) \\|u\\|_{W^{k,p}(M)}$$\nfor the integral Sobolev spaces.\nFor real $s$, we conclude the following inequality by interpolation and duality\n$$\\|F^* \\Phi\\|_{H^s} \\leq \\tilde{\\nu} (\\|F\\|_{C^{\\lceil|s|\\rceil}}) \\|\\Phi\\|_{H^s},$$\nwhere $\\nu, \\tilde{\\nu}: [0, \\infty) \\to [0, \\infty)$ are continuous functions.\n \nIn conclusion we obtain\n\\begin{eqnarray*}\n\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^s} & = & \\|F_t^* \\tilde{\\mathcal{L}}_{X_t} \\Phi_t + F_t^* \\dot{\\Phi}_t\\|_{H^s} \\\\\n& \\leq & C \\nu(\\|F_t\\|_{C^{\\lceil|s|\\rceil}}) ( \\|X_t\\|_{H^{s+1}} \\|\\Phi_t\\|_{H^k} + \\|\\dot{\\Phi}_t\\|_{H^s} )\n\\end{eqnarray*}\nWe will assume both $f_t$ and $\\Phi_t$ to remain in a bounded $H^k$ neighborhood, thus we can estimate their norms by a constant, hence we obtain\n$$\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^s} \\leq C (\\|\\dot{f}_t\\|_{H^{s+1}} + \\|\\dot{\\Phi}_t\\|_{H^s}).$$\n\nIt remains to choose a neighborhood of $\\bar{\\Phi}$ so that we can also estimate the terms $\\|\\dot{f}_t\\|_{H^{s+1}}$ and $\\|\\dot{\\Phi}_t\\|_{H^s}$.\n\nBy theorem \\ref{L2cv} there exists a $H^k$ neighborhood $U$ of $\\bar{\\Phi}$, such that for any $\\Phi \\in U$ it holds\n$$\\int_T^{T_{\\max}} \\|Q(\\Phi_t)\\|_{L^2} dt \\leq C e^{-\\alpha T}.$$\n\nChoose a neighborhood $U\\times V_m$ of $(\\operatorname{id}_M, \\bar{g})$ such that we have the mapping flow estimate \\ref{MFE}.\nChoose a neighborhood $V_s$ of $\\bar{\\Phi}$, such that we have the $L^2$ estimate of the gradient along the spinor flow as in theorem \\ref{L2cv}. We may assume that $\\pi_{\\Sigma}(V_s) = V_m$.\nFurthermore, we choose the neighborhoods to be bounded in $H^k$.\n\nNow choose $\\Phi \\in V_s$ as initial condition for the spinor and the spinor-DeTurck flow. As above we denote these flows by $\\Phi_t$ and $\\tilde{\\Phi}_t$ respectively and by $f_t$ we mean the associated mapping flow.\nWe will now estimate the integral of the $H^{-3}$ norm of $\\tilde{Q}(\\tilde{\\Phi}_t)$.\nRecall that we have \n$$\\int_{T_1}^{T_2} \\|\\dot{\\Phi}_t\\|_{L^2} dt \\leq C e^{-\\alpha T_1}$$\nfrom theorem \\ref{L2cv}. For $\\dot{f_t}$ we get the estimate\n$$\\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} dt \\leq C \\left( \\int_0^{T_1} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda(t-T_1)} dt + \\int_{T_1}^{T_2} \\|\\dot{g}_t\\|_{L^2} dt + e^{-\\lambda T_1} \\right).$$\nThe second term can be bounded by $C e^{-\\alpha T_1}$ by the previous estimate, since $\\|\\dot{g}_t\\|_{L^2} \\leq \\|\\dot{\\Phi}_t\\|_{L^2}$. The first term we decompose into\n$$\\int_0^{T_1\/2} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda(t-T_1)} dt < C e^{-\\lambda T_1\/2}$$\nand\n$$\\int_{T_1\/2}^{T_1} \\|\\dot{g}_t\\|_{L^2} e^{\\lambda(t-T_1)} dt < C e^{-\\alpha T_1\/2}$$\nagain using the estimate for $\\|\\dot{g}_t\\|$.\nThus\n$$\\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} dt < C e^{-\\mu T}$$\nfor some $C>0, \\mu >0$. We will use the same constants in the estimate of $\\dot{g}_t$.\nPutting these estimates together we obtain\n\\begin{eqnarray*}\n\\int_{T_1}^{T_2} \\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}} dt & \\leq & C \\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} + \\|\\dot{\\Phi}_t\\|_{H^{-3}} dt\\\\\n& \\leq & C e^{-\\mu T_1}\n\\end{eqnarray*}\nSince $\\tilde{Q}$ is a continuous map from $H^k$ to $H^{k-2}$, because $H^k$ embeds into $C^3$, and $\\tilde{\\Phi}_t$ is in a bounded $H^k$ neighborhood, we obtain that $\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}} \\leq \\tilde{C}$.\nHence we may estimate\n$$\\int_{T_1}^{T_2} \\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}}^2 dt \\leq \\tilde{C} \\int_{T_1}^{T_2} \\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^{-3}} dt \\leq C \\tilde{C} e^{-\\mu T_1}.$$\n\nSince $\\tilde{Q}_t = \\tilde{Q}(\\tilde{\\Phi}_t)$ fulfills the linear strongly parabolic equation\n$$\\partial_t \\tilde{Q}_t = D\\tilde{Q}(\\tilde{\\Phi}_t) \\tilde{Q}_t,$$\nwe may now apply the parabolic estimate \\ref{PE} to obtain\n$$\\|\\tilde{Q}(\\tilde{\\Phi}_{T + \\delta})\\|_{H^k} \\leq C e^{-\\mu T} = \\tilde{C} e^{-\\mu (T+\\delta)}.$$\n(Since $\\tilde{\\Phi}_t$ remains in a bounded neighborhood of $\\bar{\\Phi}$, the parabolic inequality for $D\\tilde{Q}(\\tilde{\\Phi}_t)$ can be chosen independent of $\\tilde{\\Phi}_t$.\nIn particular $\\delta$ can be chosen independently of $T$ and $\\Phi$, hence the estimate gets worse by a constant factor $e^{\\mu \\delta}$.)\n\nThe argument for the estimate (\\ref{GEb}) is identical and for the estimate (\\ref{GEc}) the argument runs in parallel until we apply the gradient estimate. Then we get the following estimate:\n$$\\int_{T_1}^{T_2} \\|\\dot{\\Phi}_t\\|_{L^2} dt \\leq \\frac{C}{1 + T^{\\beta}}.$$\nSimilarly as above, we can estimate\n$$\\int_{T_1}^{T_2} \\|\\dot{f}_t\\|_{H^{-2}} dt \\leq \\frac{C}{1 + T^{\\beta}}.$$\nThus\n$$\\int_{T_1}^{T_2} \\|\\tilde{\\mathring{Q}}(\\tilde{\\Phi}_t)\\|_{H^{-3}} dt \\leq \\frac{C}{1 + T^{\\beta}}$$\nand hence\n$$\\|\\tilde{Q}(\\tilde{\\Phi}_t)\\|_{H^k} \\leq \\frac{C}{1 + T^{\\beta}}$$\nas claimed.\n\\end{proof}\n\n\n\\begin{proof}[Proof of theorem \\ref{stabP}]\nIn the following $B_\\rho$ denotes the ball of radius $\\rho$ around $\\bar{\\Phi}$ with respect to the $H^k$ norm, and in this proof ``flow'' always refers to the gauged spinor flow. \nUsing lemmas \\ref{UniformExistence} and \\ref{GradientDecay}, choose $0 < \\gamma < \\delta < \\epsilon$ and $T$, such that\n\\begin{enumerate}[label=(\\roman*)]\n\\item The estimate from lemma \\ref{GradientDecay} holds on $B_{\\epsilon}$.\n\\item For any $\\Phi \\in B_{\\delta}$ the flow exists until time $1$ and stays in $B_{\\epsilon}$\n\\item $\\int_T^{\\infty} C e^{-\\alpha t} dt < \\frac{\\delta}{3}$, where $C$ and $\\alpha$ as in lemma \\ref{GradientDecay}\n\\item For any $\\Phi \\in B_{\\gamma}$ the flow exists until time $T$ and remains in $B_{\\delta\/3}$.\n\\end{enumerate}\nNow let $\\Phi \\in B_{\\gamma}$.\nThen denote by $\\Phi_t$ the flow\n$$\\partial_t \\Phi_t = \\tilde{Q}(\\Phi_t), \\Phi_0 = \\Phi.$$\nDenote by $\\hat{T} \\in (0, \\infty]$ the maximal time, such that the flow with initial condition $\\Phi$ exists in $B_{\\delta}$. The condition on $B_{\\delta}$ ensures that $\\Phi_{\\hat{T}}$ exists and $\\|\\Phi_{\\hat{T}} - \\bar{\\Phi}\\|_{H^k} = \\delta$.\nOn the other hand,\n\\begin{eqnarray*}\n\\|\\bar{\\Phi} - \\Phi_{\\hat{T}}\\|_{H^k} & \\leq & \\|\\bar{\\Phi} - \\Phi_T\\|_{H^k} + \\|\\Phi_T - \\Phi_{\\hat{T}}\\|_{H^k} \\\\\n& \\leq & \\frac{\\delta}{3} + \\int_T^{\\hat{T}} \\|\\tilde{Q}(\\Phi_t)\\|_{H^k} dt \\\\\n& \\leq & \\frac{\\delta}{3} + \\int_T^{\\hat{T}} C e^{-\\alpha t} dt \\\\\n& \\leq & \\frac{2}{3} \\delta\n\\end{eqnarray*}\nThis is a contradiction and we conclude $\\hat{T} = \\infty$.\nAdditionally, \n$$\\int_T^{\\infty} \\|\\tilde{Q}(\\Phi_t)\\|_{H^k} dt \\leq \\frac{\\delta}{3},$$\nand we conclude that the limit\n$$\\Phi_{\\infty} = \\Phi_T + \\int_T^{\\infty} \\tilde{Q}(\\Phi_t) dt$$\nexists in $H^k$ and\n$$\\|\\Phi_{\\infty} - \\Phi_t\\|_{H^k} \\leq \\int_t^{\\infty} \\|\\tilde{Q}(\\Phi_t)\\|_{H^k} dt \\leq C e^{-\\alpha t}.$$\nSince \n$$\\lim_{t\\to \\infty} \\mathcal{E}(\\Phi_t) = 0,$$\n$\\Phi_{\\infty}$ is a critical point.\nWe have shown that the gauged spinor flow converges for $\\Phi \\in B_{\\gamma}$ to a critical point in $B_{\\delta}$.\nGiven that the mapping flow is a strongly parabolic equation, the velocity along the flow solves a linear strongly parabolic equation and we can apply the parabolic regularity estimate and the mapping flow estimate to obtain that the mapping flow converges exponentially in any $H^k$ norm. Since the spinor flow is given by $(F_t^{-1})^* \\Phi_t$, the spinor flow also converges exponentially.\n\\end{proof}\n\\begin{proof}[Proof of theorem \\ref{stabVC}]\nWhen the critical set is a manifold, the proof is entirely analogous to the previous proof. If the critical set is not a manifold, we have the weaker estimate\n$$\\|\\tilde{Q}(\\Phi_t)\\|_{H^k} \\leq \\frac{C}{1 + T^{\\gamma}}.$$\nThe exponent $\\gamma$ can be computed from $\\theta$ in the \u0141ojasiewicz inequality as $\\gamma = \\frac{\\theta - 1}{2 - \\theta}$. Hence if $\\theta > 3\/2$, $\\gamma > 1$. In that case we find\n$$\\int_T^{\\infty} \\frac{C}{1 + t^{\\gamma}} dt \\leq C\\frac{1}{T^{\\gamma-1}} \\xrightarrow{T \\to \\infty} 0$$\nand we can show existence and convergence of the flow as in the previous proof.\nWe define\n$$\\Phi_{\\infty} = \\Phi_T + \\int_T^{\\infty} \\tilde{Q}(\\Phi_t) dt$$\nand using that\n$$|\\mathcal{E}(\\Phi_t) - \\mathcal{E}(\\bar{\\Phi})| \\leq \\frac{C}{1+ T^{\\beta}}$$\nwe obtain\n$$\\mathcal{E}(\\Phi_{\\infty}) = \\lim_{t\\to \\infty} \\mathcal{E}(\\Phi_t) = \\mathcal{E}(\\bar{\\Phi})$$\nand hence $\\Phi_{\\infty}$ is also a local minimum, and in particular a critical point of $\\mathcal{E}|_{\\mathcal{N}^1}$.\nThe speed of convergence is then given by $\\frac{1}{T^{\\gamma-1}}$.\n\\end{proof}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}