It has been a long-standing conjecture that any CFT with a large central charge and a large gap M in the spectrum of single-trace operators must be dual to a local effective field theory in AdS. In my talk, I will discuss a proof of a sharp form of this conjecture. In particular, I will explain how to derive numerical bounds on bulk Wilson coefficients in terms of M using the conformal bootstrap. The bounds exhibit scaling in M expected from dimensional analysis in the bulk. The main technical tools are dispersive CFT sum rules. These sum rules provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows one to apply recently-developed flat-space methods to construct positive CFT functionals. My talk will be based on https://arxiv.org/pdf/2106.10274.pdf, which is joint work with S. Caron-Huot, L. Rastelli, and D. Simmons-Duffin.

The Ryu-Takayanagi formula and its generalizations have led to a surprising amount of progress in our understanding of quantum gravity in the last fifteen years, culminating in the recent derivation of the Page curve in toy models of evaporating black holes. However, we still don’t understand why these formulas are true from a canonical point of view. In this talk, I will attempt to make progress on this problem by developing an analogy between gravitational entropy formulas in low-dimensional examples of holography and similar-looking formulas that have appeared in the study of entanglement entropy in emergent gauge theories. This talk will be based on 1807.06575, 2107.11872, and 2107.12634.

I will discuss the fundamentals of quantum field theory on a rigid de Sitter space. First, I will show that the perturbative expansion of late-time correlation functions to all orders can be equivalently generated by a non-unitary Lagrangian on a Euclidean AdS geometry. This finding simplifies dramatically perturbative computations, as well as allows us to establish basic properties of these correlators, which comprise a Euclidean CFT. Second, I use this to infer the analytic structure of the spectral density that captures the conformal partial wave expansion of a late-time four-point function, to derive an OPE expansion, and to constrain the operator spectrum. Third, I will prove that unitarity of the de Sitter theory manifests itself as the positivity of the spectral density. This statement does not rely on the use of Euclidean AdS Lagrangians and holds non-perturbatively. Zoom link: https://us06web.zoom.us/j/82867324531?pwd=cXBkMUNpQlE4M3h2UEdabWpOZkNIdz09

I will discuss two topics on the large charge limit and holography. First, I will discuss general features of the large-charge limit of superconformal field theories at large N. In particular, I will point out a simple setup to analyze the large charge expansion of the planar N=4 super Yang-Mills and discuss its holographic interpretation. Second, I will discuss the large-charge limit of the defect CFT on the Wilson line and its relation to the matrix model. Zoom link: https://us06web.zoom.us/j/82867324531?pwd=cXBkMUNpQlE4M3h2UEdabWpOZkNIdz09

Lonti Autumn 2021 Series: Lecture 1. Live Tutorial. Please register at https://lonti.weebly.com/registration.html to receive joining instructions for this live session which will be held via Zoom. Four examples of an anomaly are presented, two from quantum mechanics and two from quantum field theory. The first example is a charged bead on a wire in the presence of a magnetic field. This example of a 't Hooft anomaly is related to the theta angle in Yang-Mills theory. The remaining three examples present scale and conformal anomalies. We will scatter a plane wave off an attractive delta function in two dimensions. We also look at a massless scalar field, both in two dimensions without a boundary and in three dimensions with one.

In this talk, we will consider line defects in d-dimensional CFTs. The ambient CFT places nontrivial constraints on renormalization group flows on such line defects. We will see that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. We will demonstrate this generalization in some concrete examples, including a flow between Wilson loops in 4 dimensions, and an O(3) bosonic theory coupled to impurities with large isospin.

Lonti Autumn 2021 Series: Lecture 1. Release of Recorded Lecture. Available here: https://youtu.be/hiUnq_5iiPM. Four examples of an anomaly are presented, two from quantum mechanics and two from quantum field theory. The first example is a charged bead on a wire in the presence of a magnetic field. This example of a 't Hooft anomaly is related to the theta angle in Yang-Mills theory. The remaining three examples present scale and conformal anomalies. We will scatter a plane wave off an attractive delta function in two dimensions. We also look at a massless scalar field, both in two dimensions without a boundary and in three dimensions with one.

The spectral problem for N=4 Super Yang-Mills can be formulated as a set of quantisation conditions on a handful of functions called Q-functions. Recent analysis suggests that the Q-functions can be used as simple building blocks for 3-point correlation functions. This strongly resembles the situation in integrable spin chains where the wave functions factorise into a simple product of Q-functions in a special basis called Sklyanin’s separation of variables (SoV) basis which is one of the most powerful approaches for solving integrable systems. Unfortunately this framework has only been developed for the simplest integrable spin chains with sl(2) symmetry, far from the psu(2,2|4) needed to describe N=4 SYM. In this talk I will review recent advances in developing the SoV approach for higher rank integrable spin chains. I will explain how to construct the SoV basis in a systematic fashion and how it links to the representation theory of the system. Next, I will discuss a new approach for obtaining the measure in separated variables based on the famous Baxter TQ equation and how the approach naturally provides a large family of correlation functions as very simple determinants in Q-functions. I will briefly discuss how the approach can be applied directly to certain 4d QFTs, in particular the fishnet cousin of N=4 SYM.

Five-dimensional non-abelian gauge theories have a U(1) global symmetry associated with instantonic particles. I will describe a mixed 't Hooft anomaly between this and other global symmetries of the theory, namely the one-form center symmetry or ordinary flavor symmetry for theories with fundamental matter. I will then apply these results to supersymmetric gauge theories, analysing the symmetry enhancement patterns occurring at their conjectured RG fixed points.

Four-dimensional gauge theories can flow in the IR to non-trivial CFTs. By employing Borel resummation techniques both to the ordinary perturbative series and to the Banks-Zaks conformal expansion, we first analyze the conformal window of QCD and find substantial evidence that QCD with n_f=12 flavours flows in the IR to a CFT. We then study UV fixed points for SU(n_c) gauge theories with fundamental fermion matter in 4+2epsilon dimensions. Using resummation techniques similar to those used in the 4d QCD case, we provide evidence for the existence of non-supersymmetric CFTs in d=5 space-time dimensions in a certain range of colors and flavours.

Understanding the fate of semi-classical black hole solutions at very late times is one of the most important open questions in quantum gravity. In this paper, we provide a path integral definition of the volume of the black hole interior and study it at arbitrarily late times for black holes in various models of two-dimensional gravity. Because of a novel universal cancellation between the contributions of the semi-classical black hole spectrum and some of its non-perturbative corrections, we find that, after a linear growth at early times, the length of the interior saturates at a time, and towards a value, that is exponentially large in the entropy of the black hole. This provides a non-perturbative confirmation of the complexity equals volume proposal since complexity is also expected to plateau at the same value and at the same time.

The aim of this talk is to give a mathematical definition of the superconformal index of 3d supersymmetric gauge theories. This can be computed exactly using supersymmetric localisation, leading to an explicit contour integral formula involving infinite q-Pochammer symbols. I will explain how this may be understood as the Witten index of a supersymmetric quantum mechanics, or index of a twisted Dirac operator on a certain infinite-dimensional space closely related to one introduced by Braverman-Finkelberg-Nakajima.

In this talk, I will discuss AdS super gluon scattering amplitudes in various spacetime dimensions. These amplitudes are dual to correlation functions in a variety of non-maximally supersymmetric CFTs, such as the 6d E-string theory, 5d Seiberg exceptional theories, etc. I will introduce a powerful method based on symmetries and consistency conditions, and show that it fixes all the infinitely many four-point amplitudes at tree level. I will also point out many interesting properties and structures of these amplitudes, which include the flat space limit, Parisi-Sourlas-like dimensional reduction, hidden conformal symmetry, and a color-kinematic duality in AdS. Along the way, I will also review some earlier progress and the relation with this work. I will conclude with a brief discussion of various open problems. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]

Renormalization group methods have been used for almost 50 years to obtain results for critical exponents of conformal field theories (CFTs), while relying on assumptions and approximations that are not rigorously justified. The agreement with experiments is good in many cases, e.g. the 3D Ising model, but disagreements between theory and experiment that have remained unresolved for decades also exist. This indicates that our understanding of critical phenomena may be incomplete. More recently, the numerical conformal bootstrap, a fully nonperturbative method, has proven to be very powerful in calculating critical exponents and other physical observables of CFTs. In this talk we will review the numerical conformal bootstrap method and discuss potential resolutions it has suggested for unsettled questions pertaining to critical phenomena in frustrated magnets and structural phase transitions. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]

This talk will discuss higher-derivative corrections to four-dimensional gauged supergravity and their holographic implications. After briefly reviewing the construction of N=2 supersymmetric higher-derivative invariants, I will introduce a four-derivative action parameterized by two real constants. In this theory, one can show that the two-derivative solutions are not modified by the higher-derivative corrections. This fact has important consequences for the regularized on-shell action, as well as for the thermodynamics of black hole solutions. Moreover, in the context of AdS4/CFT3 holography, I will explain how our results lead to an explicit expression for subleading corrections in the large-N expansion of supersymmetric partition functions of a large class of dual field theories arising from M2 and M5 branes. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]

We review the recent progress regarding current operators in integrable models, focusing especially on integrable spin chains. These operators describe the flow of the conserved charges, and they are important for the construction of Generalized Hydrodynamics. They are also connected to long range deformations and TTbar-like deformations of the spin chains, and also to the theory of factorized correlation functions. We argue that these operators are very special, because their mean values can be computed relatively easily even in nested spin chains. This is rather unique because mean values in nested models are rather difficult to compute for generic operators. We review these various connections and also show how to construct current operators using the Quantum Inverse Scattering Approach, the canonical framework developed by the Leningrad school. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]

I will discuss the tools we are developing to calculate correlation functions of primordial inflationary perturbations. In this talk, we will focus on the limit where gravitational excitations are neglected and the cosmological spacetime is assumed to be exactly de Sitter. Even in this simplifying limit, which corresponds to a Quantum Field Theory on a de Sitter background, very few examples of explicit analytic calculations exist and little is known about basic properties of the correlators. I will show that for any dS QFT there exists a theory formulated in a Euclidean Anti-de SItter space and which reproduces all the correlation functions. This leads to major technical simplifications and allows to demonstrate various analytic properties of the cosmological observables. Understanding of these properties has both phenomenological applications in inflation, as well as provides some hints about more fundamental description for cosmological spacetimes. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]

In this talk I will discuss boundary RG flows for a 4d free scalar field coupled to large N vector models on a 3d boundary. These RG flows connect decoupled fixed points with the free and the critical vector model on the boundary, and they enjoy strong-weak dualities. I will also consider adding gauge fields to the setup. [Please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]

An important open question in black hole thermodynamics is about the existence of a "mass gap" between an extremal black hole and the lightest near-extremal state within a sector of fixed charge. In this talk, I will discuss how to reliably compute the partition function of 4d Reissner-Nordstrom near-extremal black holes at temperature scales comparable to the conjectured gap. I will show that the density of states at fixed charge does not exhibit a gap in the simplest gravitational non-supersymmetric theories; rather, at the expected gap energy scale, we see a continuum of states whose meaning we will extensively discuss. Finally, I will present a similar computation for nearly-BPS black holes in 4d N=2 supergravity. As opposed to their non-supersymmetric counterparts, such black holes do in fact exhibit a gap consistent with various string theory predictions. [please email alejandro.cabo_bizet@kcl.ac.uk for the zoom link]

The S-matrix bootstrap can be used to carve out the space of physical theories. What can we say about the space of theories of quantum gravity based on this approach? Based on work with Andrea Guerrieri and Joao Penedones. Zoom link: https://zoom.us/j/94504664165?pwd=c3VmMDNsbkRwWWdoUUxIRDhUcjB4dz09 (for password email dionysios.anninos@kcl.ac.uk)