# Bah, Faraggi, Jottar, Leigh ## Fermions and Type IIB Supergravity On Squashed Sasaki-Einstein Manifolds \[Links: [arXiv](https://arxiv.org/abs/1009.1615), [PDF](https://arxiv.org/pdf/1009.1615.pdf)\] \[Abstract: We discuss the dimensional reduction of fermionic modes in a recently found class of consistent truncations of type IIB supergravity compactified on squashed five-dimensional Sasaki-Einstein manifolds. We derive the lower dimensional equations of motion and effective action, and comment on the [[0359 Supersymmetry|supersymmetry]] of the resulting theory, which is consistent with $N=4$ gauged supergravity in $d=5$, coupled to two vector multiplets. We compute fermion masses by linearizing around two AdS$_{5}$ vacua of the theory: one that breaks $N=4$ down to $N=2$ spontaneously, and a second one which preserves no supersymmetries. The truncations under consideration are noteworthy in that they retain massive modes which are charged under a $U(1)$ subgroup of the R-symmetry, a feature that makes them interesting for applications to condensed matter phenomena via [[0001 AdS-CFT|gauge/gravity duality]]. In this light, as an application of our general results we exhibit the coupling of the fermions to the type IIB [[0431 Holographic superconductor|holographic superconductor]], and find a consistent further truncation of the fermion sector that retains a single spin-1/2 mode.\] # Bah, Faraggi, Jottar, Leigh, Zayas ## Fermions and $D=11$ Supergravity On Squashed Sasaki-Einstein Manifolds \[Links: [arXiv](https://arxiv.org/abs/1008.1423), [PDF](https://arxiv.org/pdf/1008.1423.pdf)\] \[Abstract: We discuss the dimensional reduction of fermionic modes in a recently found class of consistent truncations of $D=11$ supergravity compactified on squashed seven-dimensional Sasaki-Einstein manifolds. Such reductions are of interest, for example, in that they have (2+1)-dimensional [[0001 AdS-CFT|holographic duals]], and the [[0461 Fermions in AdS-CFT|fermionic]] content and their interactions with charged scalars are an important aspect of their applications. We derive the lower-dimensional equations of motion for the fermions and exhibit their couplings to the various bosonic modes present in the truncations under consideration, which most notably include charged scalar and form fields. We demonstrate that our results are consistent with the expected [[0359 Supersymmetry|supersymmetric]] structure of the lower dimensional theory, and apply them to a specific example which is relevant to the study of (2+1)-dimensional [[0431 Holographic superconductor|holographic superconductors]].\] ## Refs - immediate follow-up: [[2010#Bah, Faraggi, Jottar, Leigh]] # Banks, Seiberg ## Symmetries and Strings in Field Theory and Gravity \[Links: [arXiv](https://arxiv.org/abs/1011.5120), [PDF](https://arxiv.org/pdf/1011.5120.pdf)\] \[Abstract: We discuss aspects of global and gauged symmetries in quantum field theory and quantum gravity, focusing on discrete gauge symmetries. An effective Lagrangian description of $\mathbb{Z}_p$ gauge theories shows that they are associated with an emergent $\mathbb{Z}_p$ one-form (Kalb-Ramond) gauge symmetry. This understanding leads us to uncover new observables and new phenomena in nonlinear $\sigma$-models. It also allows us to expand on Polchinski's classification of cosmic strings. We argue that in models of quantum gravity, there are [[0187 Global symmetries in QG|no global symmetries]], all continuous gauge symmetries are compact, and all charges allowed by Dirac quantization are present in the spectrum. These conjectures are not new, but we present them from a streamlined and unified perspective. Finally, our discussion about string charges and symmetries leads to a more physical and more complete understanding of recently found consistency conditions of [[0332 Supergravity|supergravity]].\] # Bekaert, Boulanger, Sundell (Review) ## How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples \[Links: [arXiv](https://arxiv.org/abs/1007.0435), [PDF](https://arxiv.org/pdf/1007.0435)\] \[Abstract: Aiming at non-experts, we explain the key mechanisms of [[0421 Higher-spin gravity|higher-spin]] extensions of ordinary gravity. We first overview various no-go theorems for low-energy scattering of massless particles in flat spacetime. In doing so we dress a dictionary between the S-matrix and the Lagrangian approaches, exhibiting their relative advantages and weaknesses, after which we high-light potential loop-holes for non-trivial massless dynamics. We then review positive yes-go results for non-abelian cubic higher-derivative vertices in constantly curved backgrounds. Finally we outline how higher-spin symmetry can be reconciled with the equivalence principle in the presence of a cosmological constant leading to the Fradkin--Vasiliev vertices and Vasiliev's higher-spin gravity with its double perturbative expansion (in terms of numbers of fields and derivatives).\] # Burnell, Simon ## Space-Time Geometry of Topological phases \[Links: [arXiv](https://arxiv.org/abs/1004.5586), [PDF](https://arxiv.org/pdf/1004.5586.pdf)\] \[Abstract: The 2+1 dimensional lattice models of Levin and Wen [PRB 71, 045110 (2005)] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the special properties of these models are not obvious. In the current paper, we present a geometrical space-time picture of the partition function of the Levin-Wen models which can be described as doubles (two copies with opposite chiralities) of underlying Anyon theories. Our space-time picture describes the partition function as a knot invariant of a complicated link, where both the lattice variables of the microscopic Levin-Wen model and the terms of the hamiltonian are represented as labeled strings of this link. This complicated link, previously studied in the mathematical literature, and known as Chain-Mail, can be related directly to known topological invariants of 3-manifolds such as the so called Turaev-Viro invariant and the Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle excitations of the Levin-Wen models and we see how they can be understood by adding additional strings to the Chain-Mail link representing quasi-particle world-lines. Our construction gives particularly important new insight into how a doubled theory arises from these microscopic models.\] # Castro, Maloney, Strominger ## Hidden Conformal Symmetry of the Kerr Black Hole \[Links: [arXiv](https://arxiv.org/abs/1004.0996), [PDF](https://arxiv.org/pdf/1004.0996.pdf)\] \[Abstract: Extreme and very-near-extreme spin $J$ Kerr black holes have been conjectured to be holographically dual to two-dimensional (2D) conformal field theories (CFTs) with left and right [[0033 Central charge|central charges]] $c_L=c_R=12J$. In this paper it is observed that the 2D conformal symmetry of the scalar wave equation at low frequencies persists for generic non-extreme values of the mass $M$. Interestingly, this conformal symmetry is not derived from a conformal symmetry of the spacetime geometry except in the extreme limit. The periodic identification of the azimuthal angle is shown to correspond to a spontaneous breaking of the [[0028 Conformal symmetry|conformal symmetry]] by left and right temperatures $(T_L,T_R)$. The well-known low-frequency scalar-Kerr scattering amplitudes coincide with correlators of a 2D CFT at these temperatures. Moreover the CFT microstate degeneracy inferred from the [[0406 Cardy formula|Cardy formula]] agrees exactly with the Bekenstein-Hawking area law for all $M$ and $J$. These observations provide evidence for the conjecture that the Kerr black hole is dual to a $c_L=c_R=12J$ 2D CFT at temperatures $(T_L,T_R)$ for every value of $M$ and $J$.\] ## Refs - this is the original paper that generalises [[2008#Guica, Hartman, Song, Strominger]]'s proposal of the [[0520 Kerr-CFT correspondence|Kerr/CFT correspondence]] to the non-extremal case # Charmousis, Gouteraux, Kim, Kiritsis, Meyer ## Effective Holographic Theories for low-temperature condensed matter systems \[Links: [arXiv](https://arxiv.org/abs/1005.4690), [PDF](https://arxiv.org/pdf/1005.4690.pdf)\] \[Abstract: The IR dynamics of effective holographic theories capturing the interplay between charge density and the leading relevant scalar operator at strong coupling are analyzed. Such theories are parameterized by two real exponents $(\gamma,\delta)$ that control the IR dynamics. By studying the thermodynamics, spectra and conductivities of several classes of charged dilatonic black hole solutions that include the charge density back reaction fully, the landscape of such theories in view of condensed matter applications is characterized. Several regions of the $(\gamma,\delta)$ plane can be excluded as the extremal solutions have unacceptable singularities. The classical solutions have generically zero [[0004 Black hole entropy|entropy]] at [[0472 CMT for extremal BH|zero temperature]], except when $\gamma=\delta$ where the entropy at extremality is finite. The general scaling of DC [[0435 Holographic resistivity|resistivity]] with temperature at low temperature, and AC conductivity at low frequency and temperature across the whole $(\gamma,\delta)$ plane, is found. There is a codimension-one region where the DC resistivity is linear in the temperature. For massive carriers, it is shown that when the scalar operator is not the dilaton, the DC resistivity scales as the heat capacity (and entropy) for planar ($3d$) systems. Regions are identified where the theory at finite density is a Mott-like insulator at $T=0$. We also find that at low enough temperatures the entropy due to the charge carriers is generically larger than at zero charge density.\] ## Refs - [[0472 CMT for extremal BH]] ## Class of theories - studies ==Einstein-Maxwell-Dilaton== theories with a ==scalar potential== - $S_g=M^{p-1} \int d^{p+1} x \sqrt{-g}\left[R-\frac{1}{2}(\partial \phi)^2+V(\phi)\right]$ - $S_{M a x}=-M^{p-1} \int d^{p+1} x \sqrt{-g} \frac{Z(\phi)}{4} F_{\mu \nu} F^{\mu \nu}$ - scalar potential - $V(\phi)=-2 \Lambda e^{-\delta \phi}$ - gauge coupling - $Z(\phi)=e^{\gamma \phi}$ # Cohen, Elvang, Kiermaier ## On-shell constructibility of tree amplitudes in general field theories \[Links: [arXiv](https://arxiv.org/abs/1010.0257), [PDF](https://arxiv.org/pdf/1010.0257.pdf)\] \[Abstract: We study "[[0551 On-shell recursion relations|on-shell constructibility]]" of tree amplitudes from recursion relations in general 4-dimensional local field theories with any type of particles, both massless and massive. Our analysis applies to renormalizable as well as non-renormalizable interactions, with or without supersymmetry. We focus on recursion relations that arise from complex deformations of all external momenta. Under certain conditions, these "all-line shift recursion relations" imply the [[0061 Maximally helicity violating amplitudes|MHV]] vertex expansion. We derive a simple sufficient criterion for the validity of the all-line shift recursion relations. It depends only on the mass dimensions of the coupling constants and on the sum of helicities of the external particles. Our proof is strikingly simple since it just relies on dimensional analysis and little-group transformation properties. In particular, the results demonstrate that all tree amplitudes with $n>4$ external states are constructible in any power-counting renormalizable theory. Aspects of all-line shift constructibility are illustrated in numerous examples, ranging from pure scalar theory and the massless Wess-Zumino model to theories with higher-derivative interactions, gluon-Higgs fusion, and Z-boson scattering. We propose a sharp physical interpretation of our constructibility criterion: the all-line shift fails precisely for those classes of $n$-point amplitudes that can receive local contributions from independent gauge-invariant $n$-field operators.\] # Giombi, Yin ## Higher Spins in AdS and Twistorial Holography \[Links: [arXiv](https://arxiv.org/abs/1004.3736), [PDF](https://arxiv.org/pdf/1004.3736)\] \[Abstract: In this paper we simplify and extend previous work on three-point functions in Vasiliev's [[0421 Higher-spin gravity|higher spin]] gauge theory in AdS4. We work in a gauge in which the space-time dependence of Vasiliev's master fields is gauged away completely, leaving only the internal twistor-like variables. The correlation functions of boundary operators can be easily computed in this gauge. We find complete agreement of the tree level three point functions of higher spin currents in Vasiliev's theory with the conjectured dual free $O(N)$ vector theory.\] # Headrick ## Entanglement Renyi entropies in holographic theories \[Links: [arXiv](https://arxiv.org/abs/1006.0047), [PDF](https://arxiv.org/pdf/1006.0047.pdf)\] \[Abstract: Ryu and Takayanagi conjectured a formula for the [[0301 Entanglement entropy|entanglement (von Neumann) entropy]] of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called [[0293 Renyi entropy|Renyi entropies]]. Using Euclidean gravity, Fursaev computed the entanglement Renyi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the [[0007 RT surface|RT formula]]. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional CFTs, considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the [[0300 Mutual information|mutual information]] between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large-$N$ symmetric-product theories have the same EREs as holographic ones.\] ## Summary - first holographic [[0293 Renyi entropy|Renyi]] calculation # Heemskerk, Polchinski ## Holographic and Wilsonian Renormalization Groups \[Links: [arXiv](https://arxiv.org/abs/1010.1264), [PDF](https://arxiv.org/pdf/1010.1264.pdf)\] \[Abstract: \] ## Refs - [[0257 Holographic RG flow]] ## Summary - *studies* consequences of the idea that radial evolution = Wilson flow in the CFT - two expected results: 1. importance of multi-trace operators 2. gravitational path integral need to be done in gauge-fixed rather than Wheeler-DeWitt form # Kirillov Jr, Balsam ## Turaev-Viro invariants as an extended TQFT \[Links: [arXiv](https://arxiv.org/abs/1004.1533), [PDF](https://arxiv.org/pdf/1004.1533)\] \[Abstract: In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category $C$, to [[0097 3d manifolds|3-manifolds]] with corners. We demonstrate that this gives an [[0623 Extended TQFT|extended TQFT]] which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center $Z(C)$. In the present paper we give a partial proof of this statement.\] # Marolf, Rangamani, van Raamsdonk ## Holographic models of de Sitter QFTs \[Links: [arXiv](https://arxiv.org/abs/1007.3996), [PDF](https://arxiv.org/pdf/1007.3996.pdf)\] \[Abstract: We describe the dynamics of strongly coupled field theories in de Sitter spacetime using the [[0001 AdS-CFT|holographic gauge/gravity duality]]. The main motivation for this is to explore the possibility of dynamical phase transitions during cosmological evolution. Specifically, we study two classes of theories: (i) conformal field theories on de Sitter in the static patch which are maintained in equilibrium at temperatures that may differ from the de Sitter temperature and (ii) confining gauge theories on de Sitter spacetime. In the former case we show the such states make sense from the holographic viewpoint in that they have regular bulk gravity solutions. In the latter situation we add to the evidence for a confinement/deconfinement transition for a large $N$ planar gauge theory as the cosmological acceleration is increased past a critical value. For the field theories we study, the critical acceleration corresponds to a de Sitter temperature which is less than the Minkowski space deconfinement transition temperature by a factor of the spacetime dimension.\] ## Refs - [[0207 Euclidean state preparation]] - extension to higher compactification: [[2011#Blackman. McDermott, van Raamsdonk]] ## Summary 1. CFT on dS - boundary is just dS with extra directions - only one bulk solution: hyperbolic BH - no phase transition 2. confining gauge theories on dS - bdry = dS times a [[0448 Scherk-Schwarz compactification|Scherk-Schwarz compactification]] (period $2\pi R$) - dS temperature: $T_{dS}=\frac{H}{2\pi}$ - => HR = radius of $S^1_{SS}$ over size of $S_{d-1}$ - Phase transitions - small $HR$ - dual is [[0168 Bubble of nothing|bubble of nothing]] - this is confining phase - IR end is also dS, corresponding to the bubble wall - large $HR$ - dual is topological BH - this is a deconfined plasma phase, in accordance with the presence of horizon in the bulk ## How to vary $T$ and $T_{dS}$ independently? - use a static patch of dS rather than global dS # Myers, Sinha (Nov) ## Holographic c-theorems in arbitrary dimensions \[Links: [arXiv](https://arxiv.org/abs/1011.5819), [PDF](https://arxiv.org/pdf/1011.5819.pdf)\] \[Abstract: \] ## Refs - [[0350 Holographic c-theorem]] - preliminary report [[MyersSinha201006]] ## Assumptions - the matter theory has a number of extrema $\mathcal{L}_\text{matter}=d(d-1) \alpha_{i} / L^{2}$. At these values, the metric solution is simply global AdS ## Summary - find a certain quantity $a_{d}^{*}$ that satisfies a [[0351 Irreversibility theorems]] in both odd and even dimensions - when the couplings are tuned to remove non-unitary operators (require 2nd order linearised EOM around pure AdS) - identify this quantity with a universal contribution to the [[0301 Entanglement entropy]] (in the boundary field theory) - $S_{\text {univ }} \propto a_{d}^{*}$ - reduces to A-type trace [[0306 Weyl anomaly|anomaly]] in even dimensions - performs a replica calculation in bulk and finds agreement ## Einstein - metric: $ d s^{2}=e^{2 A(r)}\left(-d t^{2}+d \vec{x}_{d-1}^{2}\right)+d r^{2} $ - the monotonic quantity: $a(r) \equiv \frac{\pi^{d / 2}}{\Gamma(d / 2)\left(\ell_{\mathrm{P}} A^{\prime}(r)\right)^{d-1}}$ - $a'(r)\ge0$ by [[0480 Null energy condition|NEC]] => [[0350 Holographic c-theorem]] ## Quasi-topological theory - $I=\frac{1}{2 \ell_{\mathrm{P}}^{d-1}} \int \mathrm{d}^{d+1} x \sqrt{-g}\left[\frac{d(d-1)}{L^{2}} \alpha+R+\frac{\lambda L^{2}}{(d-2)(d-3)} \mathcal{X}_{4}-\frac{8(2 d-1) \mu L^{4}}{(d-5)(d-2)\left(3 d^{2}-21 d+4\right)} \mathcal{Z}_{d+1}\right]$ - at any stationary point, there is an AdS solution with a curvature scale $\tilde{L}^{2}=L^{2} / f_{\infty}$ where $\alpha=f_{\infty}-\lambda f_{\infty}^{2}-\mu f_{\infty}^{2}$ ## Higher derivative theory - more general coefficients, but still need to use unitarity to constrain them - obtains the same conclusion ## Relation to [[0301 Entanglement entropy]] ### Holographic - use Wald formula on a surface cutting global AdS into two halves giving the entanglement entropy between two halves of the sphere for the CFT in the ground state - **conjecture**: Placing a $d$-dimensional CFT on $S^{d−1} \times R$ and calculating the entanglement entropy of the ground state between two halves of the sphere, one finds a universal contribution: $S_{univ} \sim a^∗_d$ (as detailed in eq. (5.27)). Then in RG flows between fixed points, $(a^∗ _d)_{UV} ≥ (a^∗_d)_{IR}$. ### Non-holographic - see sec. 5.4 - entanglement entropy: $S=\lim _{\epsilon \rightarrow 0}\left(\frac{\partial}{\partial \epsilon}+1\right) \log Z_{1-\epsilon} .$ - connection to trace anomaly: - $\tilde{L} \frac{\partial S}{\partial \tilde{L}}=\lim _{\epsilon \rightarrow 0}\left(\frac{\partial}{\partial \epsilon}+1\right) \tilde{L} \frac{\partial}{\partial \tilde{L}} \log Z_{1-\epsilon}$=\lim _{\epsilon \rightarrow 0}\left(\frac{\partial}{\partial \epsilon}+1\right) \int d^{d} x \sqrt{g}\left\langle T_{a}^{a}\right\rangle$ # Nickel, Son ## Deconstructing holographic liquids \[Links: [arXiv](https://arxiv.org/abs/1009.3094), [PDF](https://arxiv.org/pdf/1009.3094.pdf)\] \[Abstract: We argue that there exist simple effective field theories describing the long-distance dynamics of holographic liquids. The degrees of freedom responsible for the transport of charge and energy-momentum are Goldstone modes. These modes are coupled to a strongly coupled infrared sector through emergent gauge and gravitational fields. The IR degrees of freedom are described holographically by the near-horizon part of the metric, while the Goldstone bosons are described by a field-theoretical Lagrangian. In the cases where the holographic dual involves a black hole, this picture allows for a direct connection between the holographic prescription where currents live on the boundary, and the [[0229 Membrane paradigm|membrane paradigm]] where currents live on the horizon. The zero-temperature sound mode in the D3-D7 system is also re-analyzed and re-interpreted within this formalism.\] # Penedones ## Writing CFT correlation functions as AdS scattering amplitudes \[Links: [arXiv](https://arxiv.org/abs/1011.1485), [PDF](https://arxiv.org/pdf/1011.1485.pdf)\] \[Abstract: We explore the Mellin representation of conformal correlation functions recently proposed by Mack. Examples in the [[0001 AdS-CFT|AdS/CFT]] context reinforce the analogy between Mellin amplitudes and scattering amplitudes. We conjecture a simple formula relating the bulk scattering amplitudes to the asymptotic behavior of Mellin amplitudes and show that previous results on the flat space limit of AdS follow from our new formula. We find that the Mellin amplitudes are particularly useful in the case of conformal gauge theories in the planar limit. In this case, the four point Mellin amplitudes are meromorphic functions whose poles and their residues are entirely determined by two and three point functions of single-trace operators. This makes the Mellin amplitudes the ideal objects to attempt the conformal bootstrap program in higher dimensions.\] # Teschner ## Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I \[Links: [arXiv](https://arxiv.org/abs/1005.2846), [PDF](https://arxiv.org/pdf/1005.2846)\] \[Abstract: We discuss the relation between [[0562 Liouville theory|Liouville theory]] and the Hitchin integrable system, which can be seen in two ways as a two step process involving quantization and hyperkaehler rotation. The modular duality of Liouville theory and the relation between Liouville theory and the $SL(2)$-WZNW-model give a new perspective on the geometric Langlands correspondence and on its relation to conformal field theory.\] # van Putten ## Extended black hole cosmologies in de Sitter space \[Links: [arXiv](https://arxiv.org/abs/1003.0604), [PDF](https://arxiv.org/pdf/1003.0604.pdf)\] \[Abstract: \] ## Mistake - see [[0310 Initial data in AdS]] ## Summary - superposition principle for time-symmetric BH spacetimes to (A)dS ([[0310 Initial data in AdS]]) # van Raamsdonk (Essay) ## Building up spacetime with quantum entanglement \[Links: [arXiv](https://arxiv.org/abs/1005.3035), [PDF](https://arxiv.org/pdf/1005.3035.pdf)\] \[Abstract: In this essay, we argue that the emergence of classically connected spacetimes is intimately related to the quantum entanglement of degrees of freedom in a non-perturbative description of quantum gravity. Disentangling the degrees of freedom associated with two regions of spacetime results in these regions pulling apart and pinching off from each other in a way that can be quantified by standard measures of entanglement.\] ## Comments - won first prize in essay contest - conjectures that the entanglement between two regions is related to the distance - some further study of [[0220 ER=EPR]] ## Summary - shows that certain quantum superpositions of states corresponding to disconnected spacetimes give rise to connected spacetimes - gives an example - decreasing entanglement -> increases proper distance & decreases area separating two regions # Wall (Oct) ## GSL implies a quantum singularity theorem \[Links: [arXiv](https://arxiv.org/abs/1010.5513), [PDF](https://arxiv.org/pdf/1010.5513.pdf)\] \[Abstract: The [[0082 Generalised second law|GSL]] can be used to prove a [[0225 Singularity theorems|singularity theorem]], by generalizing the notion of a trapped surface to quantum situations. Like Penrose's original singularity theorem, it implies that spacetime is null geodesically incomplete inside black holes, and to the past of spatially infinite Friedmann--Robertson--Walker cosmologies. If space is finite instead, the generalized second law requires that there only be a finite amount of entropy producing processes in the past, unless there is a reversal of the arrow of time. In asymptotically flat spacetime, the generalized second law also rules out [[0083 Traversable wormhole|traversable wormholes]], negative masses, and other forms of faster-than-light travel between asymptotic regions, as well as closed timelike curves. Furthermore it is impossible to form [[0051 Baby universes|baby universes]] which eventually become independent of the mother universe, or to restart inflation. Since the semiclassical approximation is used only in regions with low curvature, it is argued that the results may hold in full quantum gravity. An introductory section describes the second law and its time-reverse, in ordinary and generalized thermodynamics, using either the fine-grained or the coarse-grained entropy (The fine-grained version is used in all results except those relating to the arrow of time.) A proof of the coarse-grained ordinary second law is given.\] > ## Refs - [[0082 Generalised second law]] - [[0225 Singularity theorems]] ## Summary - *uses* [[0082 Generalised second law|GSL]] to generalise the notion of trapped surfaces to *quantum* situations which then implies a [[0225 Singularity theorems|singularity theorem]] ## Comments - choice of horizon: any future *causal* horizon - following [[JacobsonParentani2003]][](https://arxiv.org/abs/gr-qc/0302099) ## Implications - spacetime is null geodesically incomplete inside black holes, and to the past of spatially infinite Friedmann--Robertson--Walker cosmologies - If space is finite, GSL requires that there only be a finite amount of entropy producing processes in the past, unless there is a reversal of the arrow of time. - In asymptotically flat spacetime, the generalized second law also rules out **traversable wormholes**, **negative masses**, and **other forms of faster-than-light travel between asymptotic regions**, as well as **closed timelike curves**. Furthermore it is impossible to form **baby universes** which eventually become independent of the mother universe, or to restart inflation. # Witten (Jan) ## Analytic Continuation Of Chern-Simons Theory \[Links: [arXiv](https://arxiv.org/abs/1001.2933), [PDF](https://arxiv.org/pdf/1001.2933.pdf)\] \[Abstract: The title of this article refers to analytic continuation of three-dimensional [[0089 Chern-Simons theory|Chern-Simons]] gauge theory away from integer values of the usual coupling parameter $k$, to explore questions such as the volume conjecture, or analytic continuation of three-dimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by rotating the integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in $S^3$). We also find that the space of possible integration cycles for Chern-Simons theory can be interpreted as the "physical Hilbert space" of a twisted version of $\mathcal{N}=4$ super Yang-Mills theory in four dimensions.\] ## Comments - useful to study [[0438 Small black holes in AdS|small BH in AdS]] because small BHs have zero intersection point (the meaning of which is explained in this paper) while thermal AdS and large BHs have one; this is an alternative understanding of the fact that small BHs are local maxima while the other two are minima <!-- comment from donmarolf --> # Zilhao, Witek, Sperhake, Cardoso, Gualtieri, Herdeiro, Nerozzi ## Numerical relativity for D dimensional axially symmetric space-times: formalism and code tests \[Links: [arXiv](https://arxiv.org/abs/1001.2302), [PDF](https://arxiv.org/pdf/1001.2302.pdf)\] \[Abstract: The numerical evolution of Einstein's field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the [[0001 AdS-CFT|gauge-gravity duality]], to modelling black hole production in TeV gravity scenarios, analysis of the stability of exact solutions and tests of [[0221 Weak cosmic censorship|Cosmic Censorship]]. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of $D$ dimensional vacuum space-times with an $SO(D-2)$ isometry group for $D\ge 5$, or $SO(D-3)$ for $D\ge 6$. Performing a dimensional reduction on a $(D-4)$-sphere, the $D$ dimensional vacuum Einstein equations are rewritten as a 3+1 dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata and Nakamura (BSSN) formulation. This allows the use of existing 3+1 dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in $D$ dimensions and a procedure to match them to our 3+1 dimensional evolution equations is given. We have implemented our framework by adapting the LEAN code and perform a variety of simulations of non-spinning black hole space-times. Specifically, we present a modified moving puncture gauge which facilitates long term stable simulations in $D=5$. We further demonstrate the internal consistency of the code by studying convergence and comparing numerical versus analytic results in the case of geodesic slicing for $D=5,6$.\] ## Summary - generalises [[0285 Brill-Lindquist initial data]] to higher $D$ with symmetry