I will discuss a connection between harmonic analysis on hyperbolic n-manifolds and conformal field theory in n-1 dimensions. Used in one direction, this connection leads to new spectral bounds on hyperbolic manifolds. Used in the other direction, it offers a new viewpoint on the spectra data of conformal field theories.

We will consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy's variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this talk we will discuss the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA. In particular we analyse the stability of Cardy's ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. Based on arXiv:2306.13719.

**Send email to jeremy.mann@kcl.ac.uk for link to online seminar.** In general relativity, spacetime metrics satisfy the Einstein equations, which are wave equations in the harmonic gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the work of Choquet-Bruhat. For an initial data set satisfying the vacuum constraint equations, there exists a solution to the vacuum Einstein equations and it is geometrically unique in the domain of dependence of the initial surface. On contrast, the initial boundary value problem (IBVP) has been much less understood. To solve for an vacuum metric in a region with time-like boundary, one needs to impose boundary conditions to guarantee geometric uniqueness of the solution. However, due to gauge issues occurring on the boundary, there has not been a satisfying choice of boundary conditions. In this talk I will discuss obstacles in establishing a well-defined IBVP for vacuum Einstein equations and the geometric uniqueness problem. Then I will talk about an existence and geometric uniqueness result in a joint work with Michael Anderson.

A ’t Hooft loop is a typical disorder operator defined in any gauge theory that can be studied by a combination of holography, localization and integrability. After reviewing the quantum mechanics magnetic monopoles, I will describe how integrability and Bethe ansatz can help to study ’t Hooft loops in the N=4 super-Yang-Mills theory.

The boundary correlation functions for a quantum field theory (QFT) in a fixed anti–de Sitter (AdS) background should reduce to S-matrix elements in the flat-space limit. We consider this procedure in detail for four-point functions. With minimal assumptions we rigorously show that the resulting S-matrix element obeys a dispersion relation, the nonlinear unitarity conditions, and the Froissart-Martin bound. QFT in AdS thus provides an alternative route to fundamental QFT results that normally rely on the LSZ axioms.

We consider the renormalization group flow of a quantum field theory (QFT) in Anti-de Sitter (AdS) space. We derive sum rules that express UV data and the energy of a chosen eigenstate in terms of the spectral densities and certain correlation functions of the theory. In two dimensions, this leads to a bootstrap setup that involves the UV central charge and may allow us to follow a Renormalization Group (RG) flow non-perturbatively by continuously varying the AdS radius. Along the way, we establish the convergence properties of the newly discovered local block decomposition, which applies to three-point functions involving one bulk and two boundary operators.

Most realizations of holography in String Theory involve space-times of the form of AdS X Y, where Y is an internal space which geometrizes R symmetries. The initimate connection of bulk reconstruction in AdS with the entanglement of subregions of the base space of the dual field theory suggests the possibility that a reconstruction of the internal space could be related to entanglement among internal degrees of freedom. We initiate an investigation into this issue by studying properties of RT surfaces which are smeared along the AdS directions, and anchored on the boundary of a subregion of the internal space Y. In cases where the product space appears as an IR geometry of a higher dimensional asymptotically AdS space we identify the area of the RT surface with the entanglement entropy associated with an operator subalgebra, and speculate on stand alone instances of product spaces.

For a distant observer with finite lifetime the main characteristic a black hole is trapping of light. Semiclassical description of black holes and especially the logical basis for construction of exotic horizonless models are based on two common but usually implicit assumptions. The first is a consequence of the cosmic censorship conjecture, namely that curvature scalars are finite at apparent horizons. The second is that horizons form in finite asymptotic time (i.e. according to distant observers), a property implicitly assumed in conventional descriptions of black hole formation and evaporation. On the other hand, traversable wormholes are required to form in finite time and to be sufficiently regular by their design specifications. Taking these as the only requirements within the semiclassical framework, one finds that in spherical symmetry only two classes of black/white hole solutions are admissible: each describing only evaporating black holes and expanding white holes. I review their properties and present the implications. For example, the null energy condition is violated in the vicinity of the outer and satisfied in the vicinity of the inner apparent/anti-trapping horizon. A test particle falls into a black hole in a finite time (according to a distant clock), and it is possible to be swallowed by a white hole. Kerr-Vaidya black holes share these qualitative features. I conclude by discussing how the recent observation suggest that black holes are horizonless objects, and why some potential models of such objects, like wormholes, are ruled out.

4D N=2 theories exhibit a rich structure of RG flows between UV and IR SCFTs. With G. Elliot, M.J. Kang and C. Lawrie, we have been this structure of RG flows from a variety of perspectives. In this talk, I would like to focus on the Vertex Operator Algebra (VOA) that can be associated to a 4D N=2 SCFT, and what these VOAs can tell us about the Higgs Branch RG flow between these theories.

We analyze the coefficients of partition functions of Vafa-Witten theory for the complex projective plane CP^2. We experimentally study the growth of the coefficients for gauge group SU(2) and SU(3), which are examples of mock modular forms of depth 1 and 2 respectively. We also introduce the notion of "mock cusp form'', and study an example of weight 3 related to the SU(3) partition function. Numerical experiments on the first 200 coefficients suggest that the coefficients of a mock modular form of weight k grow as the coefficients of a modular form of weight k, that is to say as n^{k-1}. On the other hand the coefficients of the mock cusp form appear to grow as n^{3/2}, which exceeds the growth of classical cusp forms of weight 3. We provide bounds using saddle point analysis, which however largely exceed the experimental observation.

Based on 2306.05551 with Ankur and D. Carmi. Studying QFT in AdS allows to translate phenomena in massive QFT in the bulk to properties of the boundary conformal correlators. I will illustrate this in the example of a strongly coupled gauge theory, namely scalar QED in dimension D<4. The tool that I will use to compute is the large N expansion, where N is the number of flavors. I will show that the four-point function of the charged operator dual to the scalar electrons can be computed exactly in the coupling at leading order at large N, both in the Coulomb and in the Higgs phase, and explain its salient properties. Finally I will discuss an IR divergence present in integer dimension D=3 that signals the breaking of the AdS isometries due to a boundary running coupling.

Recently, a relation was introduced connecting codes of various types with the space of abelian (Narain) 2d CFTs. We extend this relation to provide holographic description of code CFTs in terms of abelian Chern-Simons theory in the bulk. For codes over the alphabet Z_p corresponding bulk theory is, schematically, U(1)_p times U(1)_{-p} where p stands for the level. Furthermore, CFT partition function averaged over all code theories for the codes of a given type is holographically given by the Chern-Simons partition function summed over all possible 3d geometries. This provides an explicit and controllable example of holographic correspondence where a finite ensemble of CFTs is dual to "topological/CS gravity" in the bulk. The parameter p controls the size of the ensemble and "how topological" the bulk theory is. Say, for p=1 any given Narain CFT is described holographically in terms of U(1)_1^n times U(1)_{-1}^n Chern-Simons, which does not distinguish between different 3d geometries (and hence can be evaluated on any of them). When p approaches infinity, the ensemble of code theories covers the whole Narain moduli space with the bulk theory becoming "U(1)-gravity" proposed by Maloney-Witten and Afkhami-Jeddi et al.

I will introduce the partial Bondi gauge for 4-dimensional spacetimes. This gauge includes the usual Bondi gauge and Newman-Unti gauge and is designed to approach asymptotic boundaries along null rays. The new gauge is defined by three conditions on the metric (g_{rr}=0=g_{rA}) and relaxes the condition on the radial coordinate. I will discuss the solution space and asymptotic symmetries. Most importantly, by relaxing the gauge, we uncover new large symmetries that characterize asymptotically flat spacetimes.

A manifestation of the black hole information loss problem is that the two-point function of probe operators in an eternal AdS black hole decays exponentially fast in time, whereas, on the boundary, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. In the thermodynamic limit of infinite entropy (infinite volume or large N), the operators that cluster in time are expected to form an algebra. We prove that this algebra is a unique and very special infinite dimensional algebra called the III_1 factor. This has implications for the emergence of a local bulk in holography. An important example is \mathcal{N}=4 SYM, above the Hawking-Page phase transition. The clustering of the single trace operators implies that the algebra is a type III_1 factor. We prove a generalization of a conjecture of Leutheusser and Liu to arbitrary out-of-equilibrium states. We explicitly construct the C^*-algebra and von Neumann subalgebras associated with time bands and more generally, arbitrary open sets of the bulk spacetime in the strict N\to \infty limit. The emergence of time algebras is intimately tied to the second law of thermodynamics and the emergence of an arrow of time.

I will discuss the eikonal scattering of two gravitationally interacting bodies, showing that exponentiation of the scattering phase matrix is a direct consequence of the group contraction $SU(2) \rightarrow ISO(2)$, in the large angular momentum limit. The emergence of the classical limit is understood in terms of the continuous-spin representations admitted by $ISO(2)$. We will compare the competing classical and quantum corrections to the leading classical eikonal scattering in the transplanckian regime and discuss how observables are extracted from the scattering phase matrix.

We will discuss BPS objects in M theory compactified on a Calabi-Yau three fold X. From the microscopic point of view such degeneracies are encoded in the partition function of the topological strings on X through the Gopakumar-Vafa formula. For the first part of the talk, as an example we will focus on quintic, and discuss how Gopakumar-Vafa invariants can be calculated systematically from the knowledge of boundary condition on the moduli space together with holomorphic ambiguity equation and mirror symmetry. When the entropy thus obtained is plotted against the left moving angular momentum for fixed M2 brane charge, there is a clear transition point at a critical angular momentum. Comparison of the the curve with leading order results from supergravity in 5d shows a large deviation. We will explain the conceptual origin of such deviations using Ooguri-Strominger-Vafa conjecture in 4d string theory though 4d-5d lift. In particular we will observe that the curve is well approximated by the (suitably corrected) entropy of BMPV blackhole for smaller angular momentum and for larger angular momentum by the (suitably corrected) entropy of a particular EEMR blackring. We will show that these observations remain valid on a class of one parameter Calabi-Yau three folds.

Recently, there have been some very impressive advances in generative models for sound, text and images. In this talk, I will look into applications of generative models to Lattice QCD. The models I will consider are flows, which are families of diffeomorphisms transforming simple base distributions into complicated target distributions. Traditional ML flows are on vector spaces, which is different from our setup where we need to deal with products of SU(N). I will give details on how we built these flows, and explain how known symmetries of LQCD can be incorporated into them.

Matrix models offer non-perturbative descriptions of quantum gravity in simple settings, allowing us to study large-N dualities between gauge and string theories. However, the large-N expansions of matrix models lead to divergent series, only defined as asymptotic series. By fine-tuning the couplings of the matrix model we obtain models of pure gravity coupled to minimal conformal field theories. The free energy for the simplest of these "minimal models" is 2d gravity also admits an asymptotic expansion which formally satisfies the Painlevé I equation. These asymptotic properties are connected to the existence of exponentially small contributions not captured by a perturbative analysis and whose physical interpretation can be elusive. The emerging structure can be accurately described by means of a resurgent transseries, capturing this perturbative/non-perturbative connection and its consequences. This talk will focus on the essential role of this resurgent transseries for the cases of Painlevé I and the quartic matrix model: together with exponentially accurate numerical and summation methods, one can show how to go beyond the asymptotic results and obtain (analytically and numerically) non-perturbative data. If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.

For a weakly nonlinear classical system, the kinetic equation for waves governs the evolution of the occupation number of a given wavevector. It is like the Boltzmann equation, but for waves instead of particles. As has been known for half a century, in addition to thermal equilibrium, the kinetic equation has another stationary solution: a turbulent state, describing a cascade of energy. Wave turbulence is observed in a wide range of physical contexts, most notably in surface gravity waves in the ocean. Higher order terms in the kinetic equation, going beyond leading order in the nonlinearity, have never been computed. We describe a method, based on quantum field theory, for computing such terms. We show that higher order terms can exhibit UV divergences. We sum the most divergent diagrams (bubble diagrams) to derive a kind of renormalized kinetic equation. Based on 2203.08168, 2212.02555, and work in progress with G. Falkovich. If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.

I give an overview of work characterizing radiation in generic four-dimensional conformal field theories. I argue that for theories with conformal scalars, the radiated energy is not positive definite and the radiated power is not Lorentz invariant. I then determine the coupling dependence of radiation, for N=2 superconformal field theories in the planar limit. This involves a purely combinatorial solution of certain matrix models, in terms of tree graphs. If you are planning to attend, please send and email to pietro.benetti_genolini@kcl.ac.uk or alan.rios_fukelman@kcl.ac.uk so your name is added to the participants list in order to grant you access to the building.