Academic Year 2023/2024

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

One of the major insights gained from holographic duality is the relation between the physics of black holes and quantum chaotic systems. This relation is made precise in the duality between two dimensional JT gravity and random matrix theory. In this work, we generalize this to a duality between AdS3 gravity and a random ensemble of approximate CFT's. The latter is described by a combined tensor and matrix model, describing the OPE coefficients and spectrum of a theory that approximately satisfies the bootstrap constraints. We show that the Feynman diagrams of the random ensemble produce a sum over 3 manifolds that agrees with the partition function of 3d gravity. A crucial element of this dictionary is the Virasoro TQFT, which defines the bulk gravitational path integral via the cutting and sewing relations of topological field theory. This TQFT has gravitational edge modes degrees of freedom whose entanglement gives rise to gravitational entropy.

In this talk I will provide an overview of our current research at the interface of quantum field theory (QFT), Lorentzian geometry and higher categorical structures. I will present operads which encode the rich algebraic structure of QFTs on Lorentzian manifolds and show that in low dimensions their algebras relate to familiar algebraic structures. Our operads share certain similarities with the little disk operads from topology, in particular they involve a homotopical localization at geometric embeddings related to ‘time evolution’. I will show that, in contrast to the topological context, this homotopical localization can be strictified in many important classes of examples, which is loosely speaking due to the 1-dimensional nature of time evolution in Lorentzian geometry. I will conclude by explaining how simple examples of such Lorentzian QFTs can be constructed from a homotopical generalization of the concept of Green’s operators for hyperbolic partial differential equations, which we call Green hyperbolic complexes. Throughout this talk, I will frequently comment on the similarities and differences between our approach, factorization algebras and functorial field theories.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

Quantum error-correcting codes play a pivotal role in enabling fault-tolerant quantum computation. In quantum error-correcting codes, the quantum information is encoded globally via quantum entanglements: the knowledge of individual subsystems, even when combined, reveals nothing about the overall state. In this talk, we explore the quantification of how entangled quantum error-correcting codes are, via a quantity we term "product overlap", the maximal fidelity between any code state and any product state. We will show that the product overlap of a quantum error-correcting code must be exponentially small in the code distance if it (1) is a low-density parity check (LPDC) code, or (2) is a stabilizer code, or (3) has high code rate. On the opposite side, for fixing code distance, we construct a class of codes where the product overlap reaches one as the code length increases.

The ring of symmetric functions plays a central role in representation theory. It connects with exactly solvable lattice models of statistical mechanics and quantum many-body systems by observing that the eigenfunctions of the transfer matrices or Hamiltonian (the Bethe wave functions) are symmetric polynomials. For infinite lattices with suitable boundary conditions, one can use the transfer matrices of exactly solvable lattice models to obtain combinatorial formulae for vertex operators of symmetric functions. This links the area of statistical lattice models and quantum spin-chains (via the boson-fermion correspondence) with integrable hierarchies of PDEs such as the Kadomtsev-Petiashvili equation where it is known that particular solutions, tau-functions, are given by symmetric functions.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

Consistently combining quantum effects and gravitational dynamics is a notoriously difficult problem, but relevant if you want to understand, e.g., black hole interiors or early universe cosmology (or if you are just curious!). A simpler version of this problem is addressing quantum effects in curved spacetime by treating spacetime as classical, while quantising the matter content: semi-classical gravity. In general this remains a difficult problem due to the non-linear nature of general relativity. A context in which one does have analytic control over semi-classical gravity is in two-dimensional dilaton (2d) gravity. For example, 2d gravity is rich enough to provide a toy model of black hole evaporation. In this talk I will highlight the role of 2d gravity in gaining insight into the black hole information paradox using the 'island formula'. I will furthermore provide a derivation of some semi-classical 2d gravity models through an approach called braneworld holography, based on upcoming work, and make a connection to Liouville gravity.

The talk will outline some recent progress in identifying realistic models of particle physics in heterotic string theory, supported by several mathematical and computational advancements which include: analytic expressions for bundle valued cohomology dimensions on complex projective varieties, heuristic methods of discrete optimisation such as reinforcement learning and genetic algorithms, as well as efficient neural-network approaches for the computation of Ricci-flat metrics on Calabi-Yau manifolds, hermitian Yang-Mills connections on holomorphic vector bundles and bundle valued harmonic forms. I will present a proof of concept computation of quark masses in a string model that recovers the exact standard model spectrum.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

There is a class of 2d sigma-models, known as integrable deformations, which are exactly solvable. Insights into the origin of this special property are provided by two 4d gauge theoretic descriptions -- 4d Chern-Simons theory and self-dual Yang-Mills theory. Recently, both of these 4d models have been realised as children of a parent theory: 6d holomorphic Chern-Simons theory on twistor space. After a review of these topics, I will discuss our work extending this formalism, beyond the WZW model.

The partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of a Calabi-Yau threefold. I will discuss how the resurgence analysis of the partition function allows one to extract BPS invariants of the same underlying geometry. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. This S-duality leads to a new modular structure in the topological string coupling. I will also discuss relations to difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

A class of three-dimensional N=4 SCFTs has recently been introduced that by definition have zero-dimensional Higgs and Coulomb branches. It is expected that their topological twists correspond to a remarkable class of 3d TQFTs described by non-unitary modular tensor categories. In this talk I will introduce boundary Vertex Operator Algebras (VOAs) for this class of TQFTs. On physical grounds, I will explain why it is reasonable to expect these VOAs to be rational. I will then explain how to explicitly construct the VOAs, and talk about novel level-rank dualities enjoyed by Virasoro minimal models that emerge from the simplest examples.

This talk is about my recent work with Camille Laurent-Gengoux. I will present our results about classifying singular foliations admitting a given leaf $L$ in a manifold $M$ and a given transverse model $(\mathbb{R}^d, \tau)$, where $\mathbb{R}^d$ is the fibre of a normal bundle of $L$ in $M$, and $\tau$ is a singular foliation in $\mathbb{R}^d$ admitting 0 as a leaf. Such a classification is motivated by the fact that every foliation $\mathcal{F}$ induces a singular foliation in the fibres of a normal bundle, the transverse (singular) foliation, and these transverse foliations at each point in $L$ are canonically isomorphic. These isomorphisms are given by the parallel transport of what one calls $\mathcal{F}$-connections. The idea of this talk is to recover $\mathcal{F}$ given $(\mathbb{R}^d, \tau)$, and we will see that in a local neighbourhood around $L$ every foliation admitting $(\mathbb{R}^d, \tau)$ as transverse model is given by a certain infinite-dimensional principal bundle.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

Applying the Yang-Baxter (YB) deformations to the famously integrable AdS5 x S5 string give rise to a variety of new integrable models. In the context of the AdS/CFT correspondence, these models are conjectured to be dual to gauge theories on various noncommutative spacetimes obtained via Drinfel’d twists. To date, however, it was unclear how to formulate such noncommutative gauge theories precisely beyond the simplest case of constant noncommutativity. In my talk, I will show how to construct gauge invariant noncommutative Yang-Mills actions for a broad class of noncommutative structures, relying on a deformed version of the Hodge star operation. I will also show how to include matter fields and hence how to construct noncommutative versions of N=4 SYM which give promising candidates for the dual theory to YB deformations of the AdS5 x S5 string. I will construct gauge invariant operators for the deformed models. Finally, I will comment on the connection to a possible spin chain picture for the one loop anomalous dimension of such operators.

A major open problem in General Relativity is the classification of higher dimensional black holes, including supersymmetric black holes. In this talk I will present my recent work on supersymmetric black holes of five-dimensional minimal supergravity. In particular, assuming only a single axial symmetry that `commutes' with supersymmetry, I provide a classification of asymptotically flat and asymptotically Kaluza-Klein black holes. These are the first examples of higher dimensional black holes with only two Killing fields. I demonstrate that these solutions are globally determined by a set of harmonic functions on $R^3$ with simple poles corresponding to connected components of the horizon or fixed points of the axial symmetry. The allowed horizon topologies are $S^3$, $L(p,1)$ and $S^2\times S^1$. Furthermore, by dimensional reduction, the Kaluza-Klein black holes naturally reduce to, and provide a classification for four-dimensional supersymmetric black holes first described by Denef.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

String theory and holography provide important points of view on the topological nature of symmetries acting on quantum systems. In this talk I will discuss how symmetry structure can be described from the bulk in AdS/CFT, both from top down, in string theory, and bottom up perspectives. I will discuss how branes can realize non-invertible symmetry operators, and the various properties associated to them such as fusion rules.

I will discuss two-dimensional CFTs that exhibit a hallmark feature of quantum chaos: universal repulsion of energy levels as described by a regime of linear growth of the spectral form factor. This physical input together with modular invariance strongly constrains the spectral correlations and the subleading corrections to the linear growth. I will demonstrate how these are determined by a trace formula, which highlights an interplay of universal physical properties of chaotic CFTs and analytic number theory. The trace formula manifests the fact that the simplest possible CFT correlations consistent with quantum chaos are those described by a Euclidean wormhole in AdS3 gravity with [torus]×[interval] topology. While the trace formula immediately provides these results, I will also discuss its mechanism in more detail; in particular, it is interesting to see how a decomposition of the spectral form factor into a set of modular invariant Maass cusp forms encodes quantum chaos.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

We review the holographic duality between the free O(N) vector model and higher spin gravity. Conserved spinning primary currents of the conformal field theory (CFT) are dual to spinning gauge fields in the gravity theory. Reducing to independent components of the conserved CFT currents one finds two components at each spin. After gauge fixing the gravity and then reducing to independent components, one finds two components of the gauge field at each spin. Collective field theory provides a systematic way to map between these two sets of degrees of freedom, providing a complete and explicit identification between the dynamical degrees of freedom of the CFT and the dual gravity. The resulting map exhibits many features expected of holographic duality: it provides a valid bulk reconstruction, it reproduces insights expected from the holography of information and it provides a microscopic derivation of entanglement wedge reconstruction.

In recent decades, the field of physical mathematics has undergone remarkable developments, leading to many powerful results. Among them is the discovery of connections between quantum spectral problems on the one hand, and supersymmetric gauge theory or topological string theory on the other. In this talk I will review some aspects of this interplay and present a new connection between N = 2 supersymmetric gauge theories in four dimensions and operator theory. In particular, I will focus on one example of an integral operator related to Painlevé equations and whose spectral traces compute correlation functions of the 2d Ising model. Adopting the approach of Tracy and Widom, I will provide an explicit expression for its eigenfunctions via an O(2) matrix model. I will then show that these eigenfunctions are computed by surface defects in SU(2) super Yang-Mills in the topological string phase of the Ω-background. This result also gives a strong coupling expression for such defects, which resums the instanton expansion. Even though I will focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations. This is based on joint work with M. François.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

The Kyiv formula gives the generic tau function of Painlevé equation (and more generally isomonodromy deformation equations) in terms of conformal blocks. Equivalently due to AGT correspondence the formula presents tau function in terms of Nekrasov partition function of 4d supersymmetric gauge theory. There are different approaches to the proof of this formula using different properties of the corresponding theories: degenerate fields in CFT, blowup equations in gauge theories, localization formulas. I will explain the statement, examples, different approaches to the proof, and (mainly conjectural) generalizations of this formula.

In the past few years there has been substantial progress in computing correlation functions in integrable systems of high rank. One of the main driving forces has been a recently developed tool known as Functional Separation of Variables (FSoV). This technique, first developed in the context of 4D gauge theory, is applicable in both integrable QFTs and spin chains alike. In this talk I will review this construction and explain how it can be used in spin chains to compute all off-diagonal matrix elements of a complete set of observables. The matrix elements take the form of highly compact determinants in Baxter Q-functions which contain all state-dependent information which contain all of the state-dependent information. The approach is highly general and works for both finite and infinite-dimensional representations alike.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

The conformal bootstrap aims to solve CFTs using elementary assumptions. In this talk I will describe the status of this program in its simplest possible setting.

There is a graphene-like boundary conformal field theory which consists of charged conformal degrees of freedom confined to a surface interacting with a photon in the bulk. A long introduction will develop several features of this theory: its relation to graphene and three dimensional QED; ways to introduce supersymmetry; its behavior under the action of SL(2,Z). Then I will talk about recent unpublished work describing our efforts to subject this theory to the numerical conformal bootstrap.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

In the late 60s, D. Atkinson proved in a series of papers the existence of functions satisfying rigorously the constraints of the S-matrix bootstrap for the 2-to-2 S-matrix of scalar, gapped theories, following an approach suggested by Mandelstam. Beyond the mathematical results themselves, the proof, based on establishing the existence of a fixed point of a certain map, also suggests a procedure to be implemented numerically and which would produce fully consistent S-matrix functions via iterating dispersion relations, and using as an input a quantity related to the inelasticity of a given scattering process. In this talk, I will present the results of a paper in collaboration with A. Zhiboedov, about the first implementation of this scheme, and I’ll present some work in progress about going beyond this scheme using machine-learning inspired numerical solvers. I will first review some basic concepts of the S-matrix program, and state our working assumptions. I will then present our numerical non-perturbative S-matrices, and discuss some of their properties. They correspond to scalar, massive phi^4-like S-matrices in 3 and 4 dimensions, and have interesting and non-trivial high energy and near-threshold behaviour. They also allow to make contact with the running of the coupling constant. I will also compare to other approaches to the S-matrix bootstrap in the literature.

Motivated by the static patch of de Sitter space, we discuss timelike surfaces in general relativity and the initial boundary value problem. We consider a non-standard set of boundary conditions, known as conformal boundary conditions, where the conformal class of the induced metric and the trace of the extrinsic curvature are fixed at the boundary. We compare those results with analogous results for the Dirichlet problem both in Lorentzian and Euclidean signature.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

Carrollian Conformal Field Theories (CFTs) have been proposed as co-dimension one holographic duals to asymptotically flat spacetimes as opposed to Celestial CFTs which are co-dimension two. In this talk, drawing inspiration from Celestial holography, we show by a suitable generalisation of the flat space limit of AdS that keeps track of the previously disregarded null direction, one can reproduce Carrollian CFT correlation functions from AdS Witten diagrams. In particular, considering Witten diagrams in AdS_4, we reproduce two and three-point correlation functions for three dimensional Carrollian CFTs in the so called delta-function branch. This establishes a direct link between AdS holography and flat space holography. We then show how this construction can be straightforwardly generalized to spinning particles. We also obtain a generalised anti-podal matching condition that now depends on the retarded time direction. Based on arXiv:2303.07388 [hep-th].

Dissipative relativistic hydrodynamics can be used to describe the thermalised behaviour of strongly coupled fluids such as super Yang-Mills plasma at late times. This late-time behaviour, given by a hydrodynamic series expansion in small gradients, approaches a hydrodynamic attractor solution. Surprisingly, this dissipative evolution towards the attractor is accurate even when the system is still quite anisotropic. This early success is intimately related to the asymptotic nature of the hydrodynamic expansion. Its resurgent properties explicitly show how transient non hydrodynamic modes are encoded in the late-time expansion, and that including these modes one can obtain information about the early non-equilibrium behaviour of the system. In this talk I will review and discuss the role of resurgence and summations in simple models of relativistic hydrodynamics as well as applications for the SYM plasma and the fluid-gravity correspondence.

A session during which speakers will give a short introduction to their talks for PhD students and postdocs.

I will introduce Penrose tilings ("PTs") and quantum error correcting codes ("QECCs"). A PT is a remarkable, intrinsically non-periodic way of tiling the plane whose many beautiful and unexpected properties have fascinated physicists, mathematicians, and geometry lovers of all sorts, ever since its discovery in the 1970s. A QECC is a clever way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Such codes play an increasingly important role in physics: in quantum computing (where they protect the delicate quantum state of the computer); in condensed matter physics (where they underpin the notion of topologically-ordered phases); and even in quantum gravity (where the "holographic" or "gauge/gravity" duality may be understood as such a code). Although PTs and QECCs might seem unrelated, I will explain how a PT gives rise to (or, in a sense, *is*) a new type of QECC in which any local errors or erasures in any finite region of the code space, no matter how large, may be diagnosed and corrected. (Joint work with Zhi Li.)

The starting point for this talk is joint work with Tom Bridgeland, where we showed that the tangent bundle TM to the space of stability conditions M is naturally a hyperKahler manifold. This then connects the theory of DT invariants to the theory of integrable systems and twistor theory. The idea was also extended to quantum DT invariants, via a Moyal-deformation of self-duality. This then makes a connection to my old work, and to recent work of others. However, the deformation is of the field equations/integrable systems, not the associated twistor theory. The talks address how an appropriate deformation of twistor space could be achieved by using an appropriate deformation quantization.