## Mini-courses

2D gravity and matrix models

by C. Johnson

This series of lectures begins with early motivations for studying 2D quantum gravity as a route to understanding the path integral definition of string theory, and then moves on to the effective 2D quantum gravity that arises when studying near-extreme charged black holes. Both applications emphasize the two primary (and intersecting) interpretations of the random matrix model: as a ’t Hooftian means of defining the sum over surfaces through tessellations, and as a Wignerian means of statistically characterising the spectrum of the (holographic dual) Hamiltonian of the theory. Perturbative and non-perturbative aspects are uncovered in detail.

Lecture notes

A-model topological string theory

by C.-C. M. Liu

We will start with an introduction to Gromov–Witten theory, which can be viewed as a mathematical theory of A-model topological strings on Kähler manifolds. We will then describe mathematical theories and generalisations of two algorithms for computing all-genus A-model open and closed topological strings on toric Calabi–Yau threefolds: (1) the topological vertex, proposed by Aganagic–Klemm–Mariño–Vafa, based on the large N duality relating topological strings to Chern–Simons theory (to be discussed in S. Gukov's course in the third week); and (2) the remodeling conjecture, proposed by Bouchard–Klemm–Mariño–Pasquetti, based on mirror symmetry and topological recursion (defined in V. Bouchard's course in the first week).

Lecture notes

Jackiw–Teitelboim gravity

by G. J. Turiaci

This review covers the derivation of certain holographic dualities between solvable models of two-dimensional gravity and ensembles of random matrices. I will focus on the all-order perturbative (in the topological expansion) considerations that lead to these dualities. Due to time constraints, I will have to skip some important topics, such as the Sachdev–Ye–Kitaev model and the relevance of Jackiw–Teitelboim gravity to higher-dimensional near-extremal black holes. Several suggested calculations or intermediate steps to fill in are marked with red boxes. Some parts of these lectures overlap with the review coauthored with T. Mertens for Living Reviews in Relativity. Given the mathematical background from the first week, I have decided to focus on more technical aspects that were not covered in that review.

Lecture notes

## Seminar

Remodeling conjecture with descendants

by B. Fang

For a toric Calabi–Yau threefold, I will explain the correspondence between an equivariant line bundle supported on a toric subvariety and a relative homology cycle on the covering space of the mirror curve. The Laplace transform along this cycle gives genus-zero descendant Gromov–Witten invariants with a certain Gamma class of that bundle. Hence, the Laplace transform of the topological recursion produces all-genus descendant Gromov–Witten invariants with Gamma classes. This talk is based on ongoing joint work with Melissa Liu, Song Yu, and Zhengyu Zong.