We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group G. Topological order in 2+1d is known to be characterised in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of G a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the centre of G. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realising all MTCs (unitary or non-unitary) with rank r<=5 in terms of Seifert manifolds and a choice of Lie group G.

In this talk, I provide an improved definition of new conserved quantities derived from the energy-momentum tensor in curved spacetime by introducing an additional scalar function. I find that the conserved current and the associated conserved charge become geometric under a certain initial condition of the scalar function, and show that such a conserved geometric current generally exists in curved spacetime. Furthermore, I demonstrate that the geometric conserved current agrees with the entropy current for the perfect fluid, thus the conserved charge is the total entropy of the system.

Effective Field Theories (EFTs) allow us to describe low energy physics without knowing the specific UV completion. This comes at the cost of having free parameters (Wilson coefficients) whose values encode the UV physics, but are not constraint from a standard EFT point of view. It is well known that some values can lead to unphysical properties of these theories. In this talk, I will present a low energy technique to put bounds on these coefficients by requiring causal propagation. I will show how these bounds can be obtained in flat space and then move on to how apply these techniques in cosmological spacetimes. Throughout the talk I will present bounds on scalar and photon EFTs.

Recent years have seen an upsurge of interest in deformations of two-dimensional sigma-models which preserve classical integrability. Integrability is known to offer powerful techniques for solving such models exactly, even in complex scenarios such as at strong coupling. This talk introduces classical integrability, and the role played by worldsheet dualities in the development of a large family of integrable deformations. The second part of the talk focuses on the application of these deformations within the AdS/CFT correspondence, in order to obtain exact methods for addressing gauge and gravity theories with reduced Noether (super)symmetries. However, current "AdS/CFT integrability" methods are mostly restricted to the undeformed, maximally (super)symmetric instances. To enhance their applicability to a broader range of theoretical models, the concept of “twisted” AdS/CFT integrability is introduced, specifically targeting the “Jordanian” subclass of integrable deformations. Recent and ongoing work in this area will be discussed.

I will show how to construct a Lagrangian based on a notion of minimal coupling that includes classical spin effects that is relevant to describe Kerr binaries in the post-Minkowski (PM) regime. Using such Lagrangian, I will derive expressions for the classical amplitude for the elastic 2—>2 process at 1PM and 2PM. I will then consider radiation reaction effects and their connection to the imaginary part of the 3PM spinning eikonal phase.

From the perspective of classical gravity, a black hole is the simplest object we know of. At the same time, it possesses huge entropy, hinting at an incredibly complex microstructure: understanding this fact falls in the realm of quantum gravity. In this talk I will review recent results concerning the microscopics and the thermodynamics of the fast spinning black holes, and I will describe how recently developed techniques allow to compute the quantum corrections to the entropy of near-extremal Kerr black holes. The quantum-corrected near-extremal entropy exhibits 3/2logT behavior characteristic of the Schwarzian model and predicts a lifting of the ground state degeneracy for the extremal Kerr black hole.

The quest to understand the microscopic origins of Bekenstein-Hawking entropy has been a longstanding challenge for physicists. In the context of AdS black holes, this entropy is expected to be explicable in terms of the states of the holographic dual quantum field theory, as per the AdS/CFT framework. In my talk, I will introduce the idea of gluing gravitational blocks for supersymmetric AdS black holes in String/M-theory with arbitrary rotation and generic electric and magnetic charges. This approach provides insights into the large N behavior of the partition function of the corresponding holographic dual field theory. Time permitting, I will also delve into the partition function of the 6d (2, 0) theory on S^2 x M_3, where M_3 is either a three-sphere or S^2 x S^1, and analyse its large N behaviour.

Recent developments on Feynman integrals and string amplitudes greatly benefitted from multiple polylogarithms and their elliptic analogues — iterated integrals on the sphere and the torus, respectively. In this talk, I will review the Brown-Levin construction of elliptic polylogarithms and propose a generalization to Riemann surfaces of arbitrary genus. In particular, iterated integrals on a higher-genus surface will be derived from a flat connection. The integration kernels of our flat connection consist of modular tensors, built from convolutions of Arakelov Green functions and their derivatives with holomorphic Abelian differentials. At genus one, these convolutions reproduce the Kronecker-Eisenstein kernels of elliptic polylogarithms and modular graph forms. I will conclude with an outlook on open problems and work in progress.

I will describe how various vertices and scattering amplitudes, involving background fields, probe trace anomaly coefficients in a four-dimensional (4D) renormalization group (RG) flow. Specifically, I will explain how to couple dilaton and graviton fields to the degrees of freedom of 4D QFT, ensuring the conservation of the Weyl anomaly along the RG flow for the coupled system. By providing dynamics to the dilaton and graviton fields, I will demonstrate that the graviton-dilaton scattering amplitude receives a universal contribution, exhibiting helicity flipping and being proportional to (Δc−Δa) along any RG flow. Here, Δc and Δa represent the differences in the UV and IR CFT c- and a-trace anomalies, respectively. Using a dispersion relation, (Δc−Δa) can be related to spinning massive states in the spectrum of the QFT. We test our proposal through various perturbative examples. Finally, as an application of the proposal of probing the trace anomalies using scattering amplitude, we have derived a non-perturbative bound on the UV CFT a-anomaly coefficient using numerical S-matrix bootstrap program for massive RG flow.

I will discuss our recent calculations of the observables involved in the scattering of two black holes or neutron stars at fourth post-Minkowskian order (three loops) using the Worldline Quantum Field Theory (WQFT) framework. These 4PM observables now include both spin-orbit and adiabatic tidal corrections — inclusion of the latter necessitates a renormalization of the underlying classical effective field theory (EFT). I will also explain how the Effective-One-Body (EOB) may be used to resum the observables, and provide input data for future-generation gravitational waveform models.

We present a new formulation of string and particle amplitudes that emerges from simple one-dimensional models. The key is a new way to parametrize the positive part of Teichmüller space. The formulation works at all orders in the perturbation series, including non-planar contributions to the amplitudes. The relationship between string and particle amplitudes is made manifest as a "tropical limit". The results are well adapted to studying the scattering of large numbers of particles or amplitudes at high loop order. The talk will in part cover results from arXiv:2309.15913, 2311.09284.

We propose a theoretical framework that explains how the mass of simple and higher-order networks emerges from their topology and geometry. We use the discrete topological Dirac operator to define an action for a massless self-interacting topological Dirac field inspired by the Nambu–Jona-Lasinio model. The mass of the network is strictly speaking the mass of this topological Dirac field defined on the network; it results from the chiral symmetry breaking of the model and satisfies a self-consistent gap equation. Interestingly, it is shown that the mass of a network depends on its spectral properties, topology, and geometry. Due to the breaking of the matter–antimatter symmetry observed for the harmonic modes of the discrete topological Dirac operator, two possible definitions of the network mass can be given. For both possible definitions, the mass of the network comes from a gap equation with the difference among the two definitions encoded in the value of the bare mass. Indeed, the bare mass can be determined either by the Betti number β0 or by the Betti number β1 of the network. We provide numerical results on the mass of different networks, including random graphs, scale-free, and real weighted collaboration networks. We also discuss the generalization of these results to higher-order networks, defining the mass of simplicial complexes. The observed dependence of the mass of the considered topological Dirac field with the topology and geometry of the network could lead to interesting physics in the scenario in which the considered Dirac field is coupled with a dynamical evolution of the underlying network structure.

In the 2nd lecture, we dig into the underlying logic of entanglement bootstrap. Illustrative examples are aimed to be simple but nontrivial. The following will be included: (1) We explain a few basic uses of strong subadditivity and quantum Markov states; we explain why axiom A0 is crucial to protect coherence. We derive the information convex set of the sphere as an application. (2) Related to the information convex set of the annulus, we explain the definition of quantum dimensions, why the vacuum has the smallest entropy, and why a certain "merged state" has the maximum entropy. (3) We classify immersed annuli on a sphere and explain why some puzzles of figure-8 annulus are not solved in naive ways. (4) In the context the reference state has a 0-form symmetry, we sketch a way to create a symmetry defect line, which (in some models) permutes anyons. This lecture is given using an ipad and is, thus, flexible. We also discuss topics from questions (feedbacks) during the 1st (and 2nd) lecture. See https://www.london-tqft.co.uk for details.

In the calculation of the perturbative amplitude of superstring theory at one loop, modular graph functions (MGFs) emerge as notable mathematical constructs. These MGFs, representing Feynman diagrams on the surface of a torus, must be integrated over the fundamental domain. My talk will introduce MGFs, elucidate their generating series, and delve into the concept of equivariance, playing a key role in classifying MGFs. Additionally, I will cover recent advancements in understanding the generating series of MGFs and the integration of MGFs over the fundamental domain.

We analyze a model of qubits which we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model is generic in that it includes all possible q-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. We will discuss implications for Hamiltonian complexity, the factorization problem in quantum gravity, and quantum simulations of holography. Based on 2311.01516 with Mike Winer.

Topological quantum field theory can emerge in gapped many-body quantum systems at low energies. In 2+1D systems, anyons can emerge, and in 3+1D, emergent excitations, including point-particles and loops-like excitations, possibly knotted or linked. In this lecture, we introduce an ongoing effort to understand (in fact, derive) laws of the emergent theory in 2+1D, 3+1D, (and higher D) gapped systems from a few axioms about the entanglement of a many-body ground state wave function. This research program, referred to as entanglement bootstrap, is an approach independent of quantum field theory, and it uses nontrivial quantum information and topology ideas. We explain the axioms and key concepts. We sketch the proof of several main theorems, including the definition of superselection sectors (anyons in 2+1D, point and loop excitations in 3+1D), the fusion spaces, and their constraints. We explain why immersion (i.e., local embedding) is valuable for, e.g., putting systems on closed space manifolds and what we hope to learn next. (see https://www.london-tqft.co.uk for more details)

We discuss the initial boundary value problem in general relativity (with vanishing cosmological constant). 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 these results with analogous results for the Dirichlet problem both in Lorentzian and Euclidean signature, where a notion of conformal black hole thermodynamics will be developed. Time permitting, we will discuss implications for holography and de Sitter space.

There has recently been a lot of activity in the field of generalised symmetries. In the context of supersymmetric gauge theories with interesting moduli spaces of vacua, such as 3d N=4 theories, global symmetries may enjoy a geometric interpretation: they act as isometries of the moduli space. In this talk I will informally put forward the idea that by enhancing the notion of moduli space one can in a similar fashion geometrise generalised symmetries. I will focus on simple 3d N=4 abelian example and talk about various things including what I'll call 0- and 1-form resolutions as well as automorphism 2-groups.

I will present the problem of building local AdS bulk observables from boundary CFT data. Focusing on QFTs coupled to a rigid AdS background, we study the analyticity constraints that bulk locality imposes on bulk-boundary-boundary 3-point functions ("AdS form factors"). We reformulate these constraints as a complete, non-perturbative set of sum rules. These sum rules lead to additional constraints on the boundary CFT on top of crossing, and can be implemented numerically in the bootstrap. We study the flat limit when these "AdS form factors" become form factors. (Based on 2305.07078)

The TTbar deformation is an irrelevant deformation of 2D field theories associated with nonlocal UV behaviour. Despite its apparent solvability, many aspects of the deformation remain mysterious. In this talk, I will present exact results for the TTbar deformation of 2D U(N) Yang-Mills theory. Carrying out the analysis at the level of each instanton sector, we can determine the nonperturbative contributions to the partition function and prove that the spectrum undergoes a truncation (a property only conjectured for other TTbar-deformed theories). We then derive the large-N limit by studying the relevant flow equation, uncovering a rich phase diagram where phase transitions are driven by instanton condensation.