## Mini-courses

Chern–Simons theory

by S. Gukov

This series of lectures provides an introduction to complex Chern–Simons theory, a rather simple-looking quantum field theory that has attracted the attention of both physicists and mathematicians in the past decades. After a brief overview of the perturbative theory, we discuss aspects of non-perturbative Chern–Simons theory with an eye towards quantum topology. In particular, we define the non-perturbative partition function Ẑ(M₃) as a topological invariant of plumbed 3-manifolds and describe its content in terms of lattice models and spin TQFTs in 3 dimensions. Finally, time permitting, we comment on the relations with characters of log-VOAs, knot invariants and resurgence.

References: 1302.0015, 1701.06567, and slides.
Codes: Ẑ for Seifert manifolds and for plumbed manifolds.

BPS geometry

by A. Neitzke

These talks explore the geometry associated with 𝒩 = 2 supersymmetric field theories, focusing on the notion of BPS states. The series will cover several key topics: an introduction to the Coulomb branches of 𝒩 = 2 theories and a discussion of the wall-crossing formula, with particular emphasis on class 𝒮 theories. We will also interpret the wall-crossing formula in the context of algebras of line defects and explore how line defect vacuum expectation values serve as distinguished cluster-type coordinates. The talks will further discuss the thermodynamic Bethe ansatz for line defect vevs and its implications for hyperkähler metrics. Time permitting, we will also discuss q-deformations and conformal blocks.

Lecture notes

Gauge theory invitation to quantum geometry

by N. Nekrasov

The non-perturbative nature of the vacuum structure induced by instantons in supersymmetric gauge theories gives rise to quantum geometric objects of great interest in both physics and mathematics. These lectures serve as an introduction to such quantum geometric aspects in the context of 𝒩 = 2 gauge theories in four dimensions. After discussing the geometry of classical 4d Yang–Mills theory, we introduce the moduli spaces of instantons in dimension two and four and describe aspects of their compactifications. We then construct an interesting class of 𝒩 = 2 supersymmetric quiver gauge theories in four dimensions and consider the problem of instanton counting in this context, describing the Ω-deformation, Seiberg–Witten theory and the non-perturbative Dyson–Schwinger equations along the way.

Slides

## Seminar

(Quantum) topological recursion for 𝒩 = 2 gauge theory in Ω-background

by G. Borot

From the AGT correspondence, the instanton equivariant counting function in 4d 𝒩 = 2 pure gauge theory (Nekrasov partition function) has a simple algebraic description: it is the squarenorm of a Whittaker vector for the 𝒲-algebra (I will explain what this is). The talk aims at explaining how to solve these 𝒲-constraints to all orders in a topological expansion, in terms of the geometry of a (quantum) spectral curve. Whether the spectral curve is quantum or not does not affect the structure of the recursion, but it affects the way the recursion operator is constructed. If time allows I will give an overview of what can be learned (or what we would like to learn) from this connection to topological recursion.