# Aharony, Marsano, Minwalla, Papadodimas, van Raamsdonk ## The Hagedorn/deconfinement phase transition in weakly coupled large N gauge theories \[Links: [arXiv](https://arxiv.org/abs/hep-th/0310285), [PDF](https://arxiv.org/pdf/hep-th/0310285.pdf)\] \[Abstract: We demonstrate that weakly coupled, large $N$, $d$-dimensional $SU(N)$ gauge theories on a class of compact spatial manifolds (including $S^{d-1} \times$ time) undergo deconfinement phase transitions at temperatures proportional to the inverse length scale of the manifold in question. The low temperature phase has a free energy of order one, and is characterized by a stringy (Hagedorn) growth in its density of states. The high temperature phase has a free energy of order $N^2$. These phases are separated either by a single first order transition that generically occurs below the Hagedorn temperature or by two continuous phase transitions, the first of which occurs at the Hagedorn temperature. These phase transitions could perhaps be continuously connected to the usual flat space deconfinement transition in the case of confining gauge theories, and to the Hawking-Page nucleation of AdS$_5$ black holes in the case of the $N=4$ supersymmetric Yang-Mills theory. We suggest that deconfinement transitions may generally be interpreted in terms of black hole formation in a dual string theory. Our analysis proceeds by first reducing the Yang-Mills partition function to a (0+0)-dimensional integral over a unitary matrix $U$, which is the holonomy (Wilson loop) of the gauge field around the thermal time circle in Euclidean space; deconfinement transitions are large $N$ transitions in this matrix integral.\] ## Refs - dual to [[0012 Hawking-Page transition]] ## Partition function as an integral over holonomy - $Z(\beta)=e^{-\beta F}=\int \mathcal{D} U e^{-S[U, \beta]}$ - where $U=e^{i\beta\alpha}$ is the holonomy around the Euclidean time circle - the explicit form of $S[U,\beta]$ is known for free theories and depends only on the single particle spectrum on manifold $\mathcal{M}$ # Bray, Chrusciel (Review) ## The Penrose Inequality \[Links: [arXiv](https://arxiv.org/abs/gr-qc/0312047), [PDF](https://arxiv.org/pdf/gr-qc/0312047)\] \[Abstract: In 1973, R. Penrose presented an argument that the total mass of a space-time which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16\pi}$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian [[0476 Penrose inequality|Penrose inequality]]. This inequality was first established by G. Huisken and T. Ilmanen in 1997 for a single black hole and then by one of the authors (H.B.) in 1999 for any number of black holes. The two approaches use two different geometric flow techniques and are described here. We further present some background material concerning the problem at hand, discuss some applications of Penrose-type inequalities, as well as the open questions remaining.\] # de Boer, Solodukhin ## A holographic reduction of Minkowski space-time \[Links: [arXiv](https://arxiv.org/abs/hep-th/0303006), [PDF](https://arxiv.org/pdf/hep-th/0303006.pdf)\] \[Abstract: Minkowski space can be sliced, outside the lightcone, in terms of Euclidean Anti-de Sitter and Lorentzian de Sitter slices. In this paper we investigate what happens when we apply holography to each slice separately. This yields a dual description living on two spheres, which can be interpreted as the boundary of the light cone. The infinite number of slices gives rise to a continuum family of operators on the two spheres for each separate bulk field. For a free field we explain how the Green's function and (trivial) S-matrix in Minkowski space can be reconstructed in terms of two-point functions of some putative conformal field theory on the two spheres. Based on this we propose a Minkowski/CFT correspondence which can also be applied to interacting fields. We comment on the interpretation of the conformal symmetry of the CFT, and on generalizations to curved space.\] ## Comments - early proposal of [[0010 Celestial holography|flat holography]] by [[0454 Flat holography from AdS-CFT|getting it from AdS/CFT]] - realised that one can do a slicing and treat the non-compact directions as non-compact "compact" directions - the reduction gives continuous KK modes rather than discrete # Friedan, Konechny ## On the Boundary Entropy of One-dimensional Quantum Systems at Low Temperature \[Links: [arXiv](https://arxiv.org/abs/hep-th/0312197), [PDF](https://arxiv.org/pdf/hep-th/0312197.pdf)\] \[Abstract: The boundary beta-function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta-function, expressing it as the gradient of the boundary entropy $s$ at fixed non-zero temperature. The gradient formula implies that $s$ decreases under renormalization except at critical points (where it stays constant). At a critical point, the number exp(s) is the ''ground-state degeneracy'', $g$, of [[1991#Affleck, Ludwig|Affleck and Ludwig]], so we have proved their long-standing conjecture that $g$ decreases under renormalization, from critical point to critical point. The gradient formula also implies that $s$ decreases with temperature except at critical points, where it is independent of temperature. The boundary thermodynamic energy $u$ then also decreases with temperature. It remains open whether the boundary entropy of a 1-d quantum system is always bounded below. If s is bounded below, then u is also bounded below.\] ## Refs - [[0351 Irreversibility theorems]] # Fuchs, Runkel, Schweigert ## TFT construction of RCFT correlators II: Unoriented world sheets \[Links: [arXiv](https://arxiv.org/abs/hep-th/0306164), [PDF](https://arxiv.org/pdf/hep-th/0306164)\] \[Abstract: A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category $C$, and the system of chiral blocks, once we select a symmetric special Frobenius algebra $A$ in the category $C$ [I]. Here we show that the construction of [I] can be extended to unoriented world sheets by specifying one additional datum: a reversion on $A$ - an isomorphism from the opposed algebra of $A$ to $A$ that squares to the twist. A given full CFT on oriented surfaces can admit inequivalent reversions, which give rise to different amplitudes on unoriented surfaces, in particular to different Klein bottle amplitudes. We study the classification of reversions, work out the construction of the annulus, Moebius strip and Klein bottle partition functions, and discuss properties of defect lines on non-orientable world sheets. As an illustration, the Ising model is treated in detail.\] # Hartnoll ## Wheeler-DeWitt states of the AdS-Schwarzschild interior \[Links: [arXiv](https://arxiv.org/abs/2208.04348), [PDF](https://arxiv.org/pdf/2208.04348.pdf)\] \[Abstract: \] ## Summary - uses [[0227 Hamilton-Jacobi]] theory: no explicit *time* coordinate; this rewrites the metric in terms of just Hamilton-Jacobi quantities without time, but can recover the full metric by substituting back - works with a [[0254 Minisuperspace]] metric ansatz so that the Lagrangian simplifies - the [[0345 Wheeler-DeWitt (WdW) equation|WDW wavefunction]] evolves outwards to the partition function of the Lorentzian CFT on the boundary - in particular, the Gaussian wavepacket evolves into a micro canonical partition function with an energy window ## Assumptions - Einstein gravity in 4D - a specific ansatz for minisuperspace metric - $\mathbb{Z}_2$ symmetric between two asymptotic boundaries ## Where does time go - can choose what to use as the clock, e.g. $k$ or $g_{tt}$ - with $g_{tt}$ as the clock, it goes to the boundary as $g_{tt}\to - \infty$ and goes to the singularity of BH as it goes to positive infinity # Kachru, Kallosh, Linde, Trivedi ## de Sitter Vacua in String Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0301240), [PDF](https://arxiv.org/pdf/hep-th/0301240)\] \[Abstract: We outline the construction of metastable de Sitter vacua of type IIB string theory. Our starting point is highly warped IIB compactifications with nontrivial NS and RR three-form fluxes. By incorporating known corrections to the superpotential from Euclidean D-brane instantons or gaugino condensation, one can make models with all moduli fixed, yielding a supersymmetric AdS vacuum. Inclusion of a small number of anti-D3 branes in the resulting warped geometry allows one to uplift the AdS minimum and make it a metastable de Sitter ground state. The lifetime of our metastable de Sitter vacua is much greater than the cosmological timescale of 10^10 years. We also prove, under certain conditions, that the lifetime of dS space in string theory will always be shorter than the recurrence time.\] # Klebanov, Maldacena, Seiberg ## D-brane Decay in Two-Dimensional String Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0305159), [PDF](https://arxiv.org/pdf/hep-th/0305159)\] \[Abstract: We consider unstable D0-branes of two dimensional string theory, described by the boundary state of [[2001#Zamolodchikov, Zamolodchikov|Zamolodchikov and Zamolodchikov]] multiplied by the Neumann boundary state for the time coordinate $t$. In the dual description in terms of the $c=1$ [[0197 Matrix model|matrix model]], this D0-brane is described by a matrix eigenvalue on top of the upside down harmonic oscillator potential. As suggested by [[2003#McGreevy, Verlinde|McGreevy and Verlinde]], an eigenvalue rolling down the potential describes D-brane decay. As the eigenvalue moves down the potential to the asymptotic region it can be described as a free relativistic fermion. Bosonizing this fermion we get a description of the state in terms of a coherent state of the tachyon field in the asymptotic region, up to a non-local linear field redefinition by an energy-dependent phase. This coherent state agrees with the exponential of the closed string one-point function on a disk with Sen's marginal boundary interaction for $t$ which describes D0-brane decay.\] # Kovtun, Son, Starinets ## Holography and hydrodynamics: diffusion on stretched horizons \[Links: [arXiv](https://arxiv.org/abs/hep-th/0309213), [PDF](https://arxiv.org/pdf/hep-th/0309213.pdf)\] \[Abstract: \] ## Summary - *obtains* both [[0434 Diffusivity]] constant and [[0430 Holographic shear viscosity]] in terms of the metric # Marolf, Minic, Ross ## Notes on Spacetime Thermodynamics and the Observer-dependence of Entropy \[Links: [arXiv](https://arxiv.org/abs/hep-th/0310022), [PDF](https://arxiv.org/pdf/hep-th/0310022.pdf)\] \[Abstract: \] - says that entropy is observer dependent # Martinec ## The Annular Report on Non-Critical String Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0305148), [PDF](https://arxiv.org/pdf/hep-th/0305148)\] \[Abstract: Recent results on the annulus partition function in [[0562 Liouville theory|Liouville field theory]] are applied to non-critical string theory, both below and above the critical dimension. Liouville gravity coupled to $c\le 1$ matter has a dual formulation as a [[0197 Matrix model|matrix model]]. Two well-known matrix model results are reproduced precisely using the worldsheet formulation: (1) the correlation function of two macroscopic loops, and (2) the leading non-perturbative effects. The latter identifies the eigenvalue instanton amplitudes of the matrix approach with disk instantons of the worldsheet approach, thus demonstrating that the matrix model is the effective dynamics of a D-brane realization of $d\le 1$ non-critical string theory. In the context of string theory above the critical dimension, i.e. $d\ge 25$, Liouville field theory realizes two-dimensional de Sitter gravity on the worldsheet. In this case, appropriate D-brane boundary conditions on the annulus realize the S-matrix for two-dimensional de Sitter gravity.\] # Matteucci (Thesis) ## Gravity, spinors and gauge-natural bundles \[Links: [PDF](https://eprints.soton.ac.uk/50610/1/00247964.pdf)\] # McGreevy, Verlinde ## Strings from Tachyons \[Links: [arXiv](https://arxiv.org/abs/hep-th/0304224), [PDF](https://arxiv.org/pdf/hep-th/0304224)\] \[Abstract: We propose a new interpretation of the $c=1$ matrix model as the world-line theory of $N$ unstable D-particles, in which the hermitian matrix is provided by the non- abelian open string tachyon. For D-particles in 1+1-d string theory, we find a direct quantitative match between the closed string emission due to a rolling tachyon and that due to a rolling eigenvalue in the [[0197 Matrix model|matrix model]]. We explain the origin of the double-scaling limit, and interpret it as an extreme representative of a large equivalence class of dual theories. Finally, we define a concrete decoupling limit of unstable D-particles in IIB string theory that reduces to the c=1 matrix model, suggesting that 1+1-d string theory represents the near-horizon limit of an ultra-dense gas of IIB D-particles.\] # Padilla ## Surface terms and the Gauss-Bonnet Hamiltonian \[Links: [arXiv](https://arxiv.org/abs/gr-qc/0303082), [PDF](https://arxiv.org/pdf/gr-qc/0303082.pdf)\] \[Abstract: We derive the [[0592 Gravitational energy|gravitational Hamiltonian]] starting from the [[0425 Gauss-Bonnet gravity|Gauss-Bonnet]] action, keeping track of all surface terms. This is done using the language of orthonormal frames and forms to keep things as tidy as possible. The surface terms in the Hamiltonian give a remarkably simple expression for the total energy of a spacetime. This expression is consistent with energy expressions found in [hep-th/0212292](https://arxiv.org/abs/hep-th/0212292). However, we can apply our results whatever the choice of background and whatever the symmetries of the spacetime.\] # Ponsot ## Liouville Theory on the Pseudosphere: Bulk-Boundary Structure Constant \[Links: [arXiv](https://arxiv.org/abs/hep-th/0309211), [PDF](https://arxiv.org/pdf/hep-th/0309211.pdf)\] \[Abstract: [[0562 Liouville theory|Liouville field theory]] on the pseudosphere is considered (Dirichlet conditions). We compute explicitely the bulk-boundary structure constant with two different methods: first we use a suggestion made by [[2001#Hosomichi|Hosomichi]] in JHEP 0111 (2001) that relates this quantity directly to the bulk-boundary structure constant with Neumann conditions, then we do a direct computation. Agreement is found.\] # Seiberg, Shih ## Branes, Rings and Matrix Models in Minimal (Super)string Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0312170), [PDF](https://arxiv.org/pdf/hep-th/0312170.pdf)\] \[Abstract: We study both bosonic and supersymmetric $(p,q)$ minimal models coupled to Liouville theory using the ground ring and the various branes of the theory. From the FZZT brane partition function, there emerges a unified, geometric description of all these theories in terms of an auxiliary Riemann surface $M_{p,q}$ and the corresponding matrix model. In terms of this geometric description, both the FZZT and ZZ branes correspond to line integrals of a certain one-form on $M_{p,q}$. Moreover, we argue that there are a finite number of distinct $(m,n)$ ZZ branes, and we show that these ZZ branes are located at the singularities of $M_{p,q}$. Finally, we discuss the possibility that the bosonic and supersymmetric theories with $(p,q)$ odd and relatively prime are identical, as is suggested by the unified treatment of these models.\] # Strominger, Takayanagi ## Correlators in Timelike Bulk Liouville Theory \[Links: [arXiv](https://arxiv.org/abs/hep-th/0303221), [PDF](https://arxiv.org/pdf/hep-th/0303221.pdf)\] \[Abstract: [[0562 Liouville theory|Liouville theory]] with a negative norm boson and no screening charge corresponds to an exact classical solution of closed bosonic string theory describing time-dependent bulk tachyon condensation. A simple expression for the two point function is proposed based on renormalization/analytic continuation of the known results for the ordinary (positive-norm) Liouville theory. The expression agrees exactly with the minisuperspace result for the closed string pair-production rate (which diverges at finite time). Puzzles concerning the three-point function are presented and discussed.\] # Teschner (Mar) ## On the relation between quantum Liouville theory and the quantized Teichm"uller spaces \[Links: [arXiv](https://arxiv.org/abs/hep-th/0303149), [PDF](https://arxiv.org/pdf/hep-th/0303149)\] \[Abstract: We review both the construction of conformal blocks in quantum [[0562 Liouville theory|Liouville theory]] and the quantization of [[0626 Teichmuller TQFT|Teichmüller spaces]] as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture.\] # Teschner (Aug) ## From Liouville Theory to the Quantum Geometry of Riemann Surfaces \[Links: [arXiv](https://arxiv.org/abs/hep-th/0308031), [PDF](https://arxiv.org/pdf/hep-th/0308031)\] \[Abstract: The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the [[0562 Liouville theory|Liouville conformal field theory]] within the context of the quantization of spaces of Riemann surfaces.\]