Week 09.10.2023 – 15.10.2023

In these lectures we will provide a basic introduction to Supergravity as it arises in String Theory and M-Theory. We will start by introducing vielbeins and spin connections in order to construct supergravity actions. In the second lecture we will briefly introduce the maximal supergravity theories in ten and eleven-dimensions. We will briefly discuss special holonomy manifolds, explicitly construct BPS p-brane solutions and prove their non-perturbative stability. Time permitting we will discuss toroidal compactifications and U-duality. I will assume basic MSc level material (Riemannian geometry, fermions and rigid supersymmetry). The lecture notes that will be provided are largely self-contained but the text book “Supergravity” by Freedman and van Proeyen contains more details.

We will give an intuitive explanation of why and how matrices (or, more precisely, large-N gauge theories) can describe a black hole, without assuming knowledge of quantum mechanics and holographic duality. Firstly, we explain an intuitive picture inroduced by Witten: diagonal entries of matrices describe particles and off-diagonal entries describe strings connecting particles. When many strings are excited, a lot of energy and entropy are packed in a small region and form black hole. Next, we consider classical dynamics of matrix model. Specifically, we colide two black holes. Using the energy conservation, equipartition law of energy and elementary school math, we show that black hole becomes colder after the merger. Matrices know black hole's negative heat capacity! To gain a little bit more intuition, we will look at ants. Collective behavior of ants has a striking similarity to black hole. The mapping rule is ant -> particle, pheromone -> string, and ant trail -> black hole. Tuning parameters such as temperature or each ant's laziness, we can obtain three kinds of phase diagrams. Each of them has a counterpart in large-N gauge theories. If time permits, I will explain the mechanism applicable to strongly-coupled and highly quantum regime needed for quantitative agreement with Einstein gravity. (This part requires a good understanding of undergraduate-level quantum mechanics.)

Scattering amplitudes in strong background fields provide an arena where perturbative and non-perturbative physics meet, with important applications ranging from laser physics to black holes, but their study is hampered by the cumbersome nature of QFT in the background field formalism. In this talk, I will try to convince you that strong-field scattering amplitudes contain a wealth of physical information which cannot be obtained with standard perturbative techniques, ranging from all-order classical observables to constraints on exact solutions. Furthermore, I will discuss how in chiral strong fields, remarkable progress is possible using methods based on twistor 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 will discuss two classes of line defects in the O(N) critical model at the Wilson-Fisher fixed point. These extended excitations are relevant for condensed matter systems, such as doped quantum antiferromagnets. After reviewing some state-of-the-art analytic bootstrap techniques, we will apply them to compute the correlator of two bulk excitations at first order in the epsilon expansion. From this result we are able to extract an infinite set of defect CFT data.

Characters of representations of chiral algebras are important tools in conformal field theory. They are a special chase of chiral torus 1-point functions (namely those where the vacuum has been inserted) and their modular properties famously give rise to the Verlinde formula. In this talk we will generalise from vacuum insertions to insertions from any irreducible representation in the example of the su2 WZW models at non-negative integral level and explore their modular properties.