22nd Geometry and Physics Colloquium

Title 1:  Resurgence and topological strings

Abstract 1: The theory of resurgence provides a precise and unified mathematical formulation of the non-perturbative sectors of a physical theory, based solely on its perturbative expansion. In recent years it has been applied to topological quantum field theories and  topological strings, leading to many insights and results. For example, it has been found that, in many cases, the Stokes constants appearing in resurgent analysis are related to BPS invariants. In this talk I will first review the relevant aspects of the resurgence program, and then I will review recent progress on its implementation in topological string theory. In particular, I will present exact results on multi-instanton amplitudes for the topological string on arbitrary Calabi-Yau manifolds, and applications of these results to asymptotic problems in the theory of Gromov-Witten invariants.

Title 2: The geometry of black hole entropy functions

Abstract 2: One of the major successes of string theory has been a precise microstate counting of black hole entropy, at least for certain classes of supersymmetric black holes in asymptotically flat spacetimes. Recently a growing body of similar work has been developed for black holes in asymptotically Anti-de Sitter (AdS) spacetimes, where the microstate counting uses the AdS/CFT correspondence and a dual field theory description. After introducing the general topic, I will describe a certain Riemannian geometry associated to such black holes in AdS. This involves a new type of PDE for a Kahler metric. While solving this PDE is in general a hard problem, I explain how solutions are critical points of an "entropy function", which when evaluated on a solution is precisely the associated black hole entropy. Geometric techniques then allow one to compute the entropy for various families of black holes, without solving the Einstein equations explicitly. I will conclude by commenting on the relationship between this entropy function and the dual field theory.