Week 10.02.2025 – 15.02.2025

This course will give a technical introduction to the black hole information paradox (BHIP). In the first lecture, we will begin with a review of quantum path integrals, both in Lorentzian and in Euclidean signature. We will review the Euclidean path integral connection with statistical mechanics and thermodynamics while also reviewing the derivation of the first law of thermodynamics in standard equilibrium statistical mechanics. We will then introduce the laws of black hole thermodynamics, and study them in particular examples. The second lecture will be devoted to the Unruh effect. We will study free quantum field theory in Rindler space, which, locally, is the spacetime observed by a uniformly accelerated observer. We will derive that this observer measures a temperature related to the observer proper acceleration. The third lecture will be devoted to classical and quantum information theory including notions of conditional probability, mutual information, and entropy inequalities, in settings with finite numbers of degrees of freedom. We will also introduce the Page curve and its significance. Finally, in the fourth lecture we will set up a toy model of the BHIP in Anti de Sitter space (AdS). Because AdS is believed to have a dual description as a conformal quantum field theory, we will use this duality to our advantage. We end with a broad discussion synthesizing what we have learned, and what is left to understand.

I will discuss a few examples of holographic dual pairs that arise in string theory for quantum theories that are not conformal. In particular, we will focus on Euclidean supersymmetric gauge theories on spheres and compute observables using supergravity and string theory on one side of the duality and supersymmetric localization on the other. This two-pronged approach allows us to study perturbative and non-perturbative corrections to the leading-order holographic result.

I will discuss a few examples of holographic dual pairs that arise in string theory for quantum theories that are not conformal. In particular, we will focus on Euclidean supersymmetric gauge theories on spheres and compute observables using supergravity and string theory on one side of the duality and supersymmetric localization on the other. This two-pronged approach allows us to study perturbative and non-perturbative corrections to the leading-order holographic result.

I'll briefly review the classical double copy, which maps exact solutions of classical gauge theories like electromagnetism, to solutions of general relativity. We will relate several gravitational objects (including horizons, Penrose limits, and asymptotics, and duals to fluid systems) to their gauge theory analogues.