Title:   Action of SLE and its holographic expression as a renormalized volume

Title:   Action of SLE and its holographic expression as a renormalized volume

Title:   Critical collapse in 2+1 gravity

Abstract:   I will discuss joint work with S. Cicortas addressing existence of naked singularities in 2+1 dimensional Einstein equations with negative cosmological constant which turn out to describe a transition from a non-collapsing to a black hole forming regime.

Title:   SLE and Random Geometry

Title:   SLE and Random Geometry

Speaker:   Nicolas Curien
Title:   Survey about random hyperbolic surfaces
Abstract:   I will survey recent developments concerning the geometric properties of
random hyperbolic surfaces. While in low genus a clear connection with random
planar maps has recently emerged, the geometry of random hyperbolic surfaces in
large genus remains largely mysterious, despite significant progress on the
spectral side. Starting from the breakthrough work of Maryam Mirzakhani on Weil–
Petersson random surfaces, I will discuss what is known about typical geometric
features such as the distribution of lengths of closed geodesics, the behavior of
the systole, and diameter estimates.
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Speaker:   Frank Ferrari
Title:   JT Gravity on Finite Geometries: finite cut-off holography, self-overlapping loops and the JT CFT
Abstract:   We will outline the construction of the finite cut-off JT theory, both in the discrete and the continuum point of view, emphasizing the latter. The usual quantization, based on the Wheeler-DeWitt equation, is shown to break down. It is replaced by a conformal field theory description, which is akin to the Liouville CFT description of ordinary 2d quantum gravity. The possibility to recover the usual Schwarzian description, which governs the model with asymptotically hyperbolic boundary conditions, as a limit of the finite cut-off theory, involves some interesting conceptual issues related to the emergence of time.
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Speaker:   Hao Geng
Title:   It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-cBCFT Ensemble
Abstract:   We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D
gravity for generic boundary subregion--including RT surface phase transitions--directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large-c behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFT's with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: 1) A simple derivation of the multi-interval entanglement
entropy for the vacuum state of a single CFT; 2) A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large-c limi--a paradigm we refer to as It from ETH.
References: https://arxiv.org/abs/2405.14872, https://arxiv.org/abs/2504.12388
and https://arxiv.org/abs/2505.20385
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Speaker:   Ewain Gwynne
Title:   Directed distances in random planar maps
Abstract:   I will discuss several recent results concerning directed distances in random planar maps. Such distances are discretizations of (hypothetical) directed versions of the Liouville quantum gravity (LQG) metric.
I will first explain our results in the setting of the uniform infinite bipolar-oriented triangulation (UIBOT), which belongs to the $\sqrt{4/3}$-LQG universality class. We construct the Busemann function which measures directed distance to $\infty$ along a natural interface in the UIBOT. We show that in the case of longest (resp. shortest) directed paths, this Busemann function converges in the scaling limit to a $2/3$-stable Lévy process (resp. a $4/3$-stable Lévy process). We also show that in a typical subset of the UIBOT with $n$ edges, longest directed path lengths are of order $n^{3/4}$ and shortest directed path lengths are of order $n^{3/8}$.
I will then discuss several other settings in which we can obtain similar results, including biased bipolar-oriented maps in the $\gamma$-LQG universality class for $\gamma \in (0,\sqrt 2)$, spanning tree decorated maps, critical Fortuin-Kasteleyn decorated maps, and longest increasing subsequences in various models of random permutations.
Based on joint works with Jeonghyun Ahn, Jacopo Borga, Yuyang Feng, Oriol Sole Pi, and Yuanzheng Wang.
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Speaker:   Klaus Richter
Title:   Quantized Chaotic Dynamics on High-dimensional Hyperbolic Manifolds and JT gravity
Abstract:   Jackiw-Teitelboim (JT) gravity, as an exactly solvable model of two-dimensional quantum gravity, has found remarkable application in the study of holography in recent years. I will address JT gravity from a complementary perspective, namely through the lense of quantum chaos. I consider a single prototypical quantum chaotic system, a high-dimensional variant of the Hadamard-Gutzwiller model. Using semiclassical path-integral techniques for chaotic dynamics I will discuss three, intertwined topics: (i) How, in the infinite-dimensional limit, the system's quantum Lyapunov exponent, quantifying scrambling and the growth of out-of-time-order correlators, saturates the Maldacena-Shenker-Stanford bound on chaos, supporting a possible duality with gravity. (ii) How, in view of the factorization problem in quantum gravity, that single dynamical system can still reproduce the leading-topology one- and two-point correlation functions of JT gravity. To this end I will use a semiclassical, but exact calculation based on Selberg’s trace formula. (iii) How subtle correlations between classical periodic orbits provide the key to the correct first topology correction to the two-point function of unorientable topological gravity.
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Speaker:   Yilin Wang
Title:   SLE, Loewner energy, and renormalized volume
Abstract:   In probability theory, universality is the phenomenon where random processes converge to a common limit despite microscopic differences. This phenomenon underlies the emergence of the random simple curve, called SLE loop measure, as the universal scaling limit of interfaces in conformally invariant 2D systems such as critical Ising model. SLE plays a central role in 2D random conformal geometry and a probabilistic approach to 2D quantum gravity and CFT. In particular, considering appropriate variational formulas of SLE gives rise to representation of Virasoro algebra, and its action, called Loewner energy, connects to the Kahler geometry of the universal Teichmuller space, determinants of Laplacians, Coulomb gases, etc.I will give an introductory overview of the link, and discuss the applications and further development in exploring this link. In particular, we will mention the more recent result showing the relation between Loewner energy and the renormalized volume in hyperbolic 3-space (motivated by the AdS/CFT holographic principle), etc..
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Speaker:   Daniel Jafferis
Title:   Tensor models, triangulations, and ensembles of BCFTs
Abstract:   I will formulate a tensor/matrix model for an ensemble of 2d BCFT data, associated to a purely open version of the bootstrap. I will explain how the resulting topological expansion is related pure 3d gravity by gluing tetrahedra, and show how the matrix model captures certain off-shell 3d gravity amplitudes.
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Speaker:   Jérémie Bouttier
Title:   The slice decomposition of planar maps
Abstract:   Random planar maps, also known as dynamical tesselations, are simple yet rich models of 2D random geometries. Over the last decades, their understanding has been greatly improved by combinatorial methods. In this talk I will present such a method, the slice decomposition, which consists in cutting surfaces along geodesics. Based on joint works with Emmanuel Guitter, Marie Albenque, Grégory Miermont, Hugo Manet, Thomas Lejeune and Bertrand Eynard (in chronological order).
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Speaker:   Cynthia Yan
Title:   Positivity constraints for the gravitational path integral
Abstract:   Does the gravitational path integral define a consistent Hilbert space of quantum-gravitational states? In this talk we formulate general positivity conditions that the path integral must satisfy for overlaps between states—prepared by Euclidean manifolds with specified boundaries—to behave as genuine inner products, ensuring nonnegative norms and a probabilistic interpretation for both open- and closed-universe sectors. While these constraints look highly restrictive, we show that they are nevertheless satisfied in a range of nontrivial examples, including settings with wormhole contributions. This yields a well-defined Hilbert-space framework in which one can meaningfully discuss amplitudes and transition probabilities between quantum-gravity states, and it provides a sharp diagnostic of when semiclassical approximations can or cannot define consistent states.
Hashtag: #workshop

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Speaker:   Herman Verlinde
Title:   Complex Liouville theory and SYK holography
Abstract:   TBA
Hashtag: #workshop

Speaker:   Anton Alekseev
Title:   Virasoro coadjoint orbits and hyperbolic metrics
Abstract:   Virasoro coadjoint orbits are infinite dimensional symplectic spaces which admit classification due to Lazutkin-Pankratova, Segal, Kirillov, Witten etc. 
Inspired by works on Jackiw-Teitelboim (JT) gravity, we consider elliptic and exceptional Virasoro orbits, and we establish their relation to moduli spaces of singular hyperbolic metrics on the disk. 
The talk is based on a joint work with Eckhard Meinrenken, and on a work in
progress with Rea Dalipi and Samson Shatashvili.
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Speaker:   Eveliina Peltola
Title:   TBA
Abstract:   TBA
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Speaker:   Catherine Wolfram
Title:   Epstein curves and holography of the Schwarzian action
Abstract:   The circle can be seen as the boundary at infinity of the hyperbolic plane. I will explain a construction from hyperbolic geometry, due to Epstein, to construct a curve in the disk from a diffeomorphism of the circle. It turns out that the Schwarzian action (a function of a diffeomorphism of the circle, and the action of Schwarzian field theory) can be computed in various ways from geometric data about this Epstein curve. While this construction is completely deterministic, from a mathematical physics perspective this is motivated by the proposed holographic duality between Schwarzian field theory on the circle and JT gravity in the disk. I’ll
explain how to construct the Epstein curve, how it is related to the Schwarzian action, how the bi-local observables of Schwarzian field theory can be interpreted as renormalized length using the same Epstein construction, and time permitting mention what we know about the relationship to the Loewner energy. This talk is based on joint work with Franco Vargas Pallete and Yilin Wang.
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Speaker:   Alex Frenkel
Title:   Signatures of Bulk Topology from the 't Hooft Worldsheet
Abstract:   Nonperturbative string theory requires two notions of random geometry: the geometry of the worldsheet, and the geometry of the target space the worldsheet fluctuates in. Part of the randomness is fluctuation over different choices of topology. Somehow, the proposed non-perturbative formulation of string theory in terms of large-N matrix degrees of freedom must describe how both notions of random geometry emerge: both string theory and the semiclassical gravitational path integral must be recovered, as well as the relationship between the two. In this talk I will describe how phase transitions in the large-N matrix theory are related to fluctuations in bulk topology, and provide evidence that this topology may be read off directly from the 't Hooft diagram combinatorics. I will
focus on the Hawking-Page transition (and its well-known description in terms of the deconfinement transition), and touch on how the matrix degrees of freedom give a non-perturbative description of Susskind-Uglum edge modes on the bifurcate horizon.
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Speaker:   Timothy Budd
Title:   Random hyperbolic geometry and Schwarzian field theory
Abstract:   Schwarzian field theory has played an important role in recent studies of the SYK model and JT gravity. A rigorous construction in terms of random circle reparametrisations was recently put forward by Bauerschmidt, Losev and Wildemann. In this talk I will show that it is also obtained as a scaling limit of a natural model, introduced by Chekhov, of random ideal polygons in the hyperbolic plane. Finally, I will comment on its role within the broader landscape of random planar geometry.
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Speaker:   Scott Collier
Title:   Recent developments in minimal string theory
Abstract:   I will review recently established dualities between solvable two-
dimensional string theories and double-scaled matrix integrals. These theories may be thought of as irrational cousins of the older (p,q) minimal string theories. 
Throughout I will emphasize unsolved open problems.
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Speaker:   Antoine Song
Title:   Harmonic maps into high-dimensional spheres
Abstract:   Harmonic maps from surfaces to manifolds are nonlinear analogues of harmonic functions. These maps have been extensively studied in Differential Geometry. I will survey what's (un)known about them, and some recent developments related to harmonic maps from surfaces to high-dimensional Euclidean spheres. Given a closed Riemann surface and a unitary representation of its fundamental group, classical variational theory produces a corresponding equivariant harmonic map from the Poincaré disk to a Euclidean sphere. In general, not much can be said about the geometry of such maps. However, I will explain that by leveraging the theory of random matrices, if the representation is sufficiently generic, then the corresponding harmonic map is actually very close to an immersed hyperbolic plane inside a sphere. Moreover, the asymptotic limit map is essentially unique. Partly joint with Riccardo Caniato and Xingzhe Li.
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Speaker:   Victor Rodriguez
Title:   c=1 strings as a matrix integral
Abstract:   We study the perturbative S-matrix of the c=1 string and show that it admits a description in terms of a double-scaled matrix integral. Together with the well-known duality to matrix quantum mechanics, this leads to a triality between worldsheet string theory, matrix quantum mechanics, and a matrix integral. 
Starting from the complex Liouville string and its dual matrix integral, we derive closed-form Feynman rules for c=1 amplitudes. These naturally describe a discretized target space, with the physical S-matrix recovered by analytic continuation. We show that the amplitudes satisfy perturbative unitarity and a Mirzakhani-type recursion, and we find detailed agreement with matrix quantum mechanics.
Based on work with S. Collier and L. Eberhardt.
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Speaker:   Scott Sheffield
Title:   Yang-Mills and the surprising implications of 1+1=2 and 2+2=4
Abstract:   The fact that 1+1=2 is somehow the heart of conformal probability. It implies that two non-parallel lines in the plane (co-dimension 1) meet at a point (co-dimension 2). It leads to similar results for curves (e.g. the Jordan curve theorem) and to useful crossing dualities for percolation and spanning tree models. It also implies the conformal invariance of the Dirichlet energy and (hence) harmonic functions, Brownian motion and the Gaussian free field.
The fact that 2+2=4 implies analogous results in 4D. Two fully non-parallel 2D planes in 4D meet at a point: just imagine two moving lightsabers colliding at a point of space-time. Similar ideas lead to knot theory, Chern-Simons theory, crossing dualities for various random surface and plaquette spanning tree models in 4D, and special symmetries for 4D Gaussian fields. I will discuss some recent efforts to make productive use of these symmetries in the context of Yang-Mills gauge theory. I will also show some recent simulations of 2D random surfaces embedded in 4D and discuss related open problems.
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Speaker:   Zhenbin Yang
Title:   A convergent genus expansion for the plateau
Abstract:   We conjecture a formula for the spectral form factor of a double-scaled matrix integral in the limit of large time, large density of states, and fixed temperature. The formula has a genus expansion with a nonzero radius of convergence. To understand the origin of this series, we compare to the
semiclassical theory of “encounters” in periodic orbits. In Jackiw-Teitelboim (JT) gravity, encounters correspond to portions of the moduli space integral that mutually cancel (in the orientable case) but individually grow at low energies. At genus one we show how the full moduli space integral resolves the low energy region and gives a finite nonzero answer.
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Speaker:   Xin Sun
Title:   The b-6j Symbol: from Liouville CFT to Virasoro TQFT
Abstract:   The b-6j symbol is an explicit special function coming from a quantum group related to SL(2,R). In this talk, we describe the triple role of the b-6j symbol: as the fusion kernel of the Virasoro conformal blocks, as the boundary three-point structure constant for Liouville conformal field theory, and as the fundamental building block for the Virasoro topological quantum field theory.
Based on three projects, one joint with Ang, Remy, Zhu, one joint with Ghosal, Remy, Sun, Wu, and one joint with Ming, Liu, Wu, Yang.
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Speaker:   Julian Sonner
Title:   From phase space to Krylov space: complexity and the geometry of chaos
Abstract:   We develop a classical counterpart of the Krylov complexity framework by running the Lanczos algorithm directly on the algebra of observables of a Hamiltonian system on a compact symplectic manifold. This construction arises naturally as the semiclassical limit of the quantum Lanczos algorithm.
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Speaker:   Colin Guillarmou
Title:   The WZW H^3 model, a probabilistic construction
Abstract:   We construct the path integral and correlation functions for the WZW model with values in the hyperbolic 3 space H^3. We then prove the correspondence with Liouville correlations discovered in physics by Ribault, Teschner, Schomerus, Hikida...
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Speaker:   Roland Bauerschmidt
Title:   
Abstract:   
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