# Derived Geometry in Twists of Gauge Theories 1 of 4

Date: Mon, Aug 23, 2021

Location: PIMS, University of Saskatchewan, Online, Zoom

Conference: 2nd PIMS Summer School on Algebraic Geometry in High Energy Physics

Subject: Mathematics, Algebraic Geometry, Physics, Particle Physics and Quantum Field Theory

Class: Scientific

### Abstract:

These lectures will review and develop methods in algebraic geometry (in particular, derived algebraic geometry) to describe topological and holomorphic sectors of quantum field theories. A recurring theme will be the interaction of local and extended operators, and of QFT's in different dimensions. The main examples will come from twists of supersymmetric gauge theories, and will connect to a large body of recent and ongoing work on 3d Coulomb branches, 3d mirror symmetry (and geometric Langlands), logarithmic VOA's and non-semisimple TQFT's, and categorified cluster algebras.

The basic plan for the lectures is:

- Lecture 1 (2d warmup): categories of boundary conditions, interfaces, and Koszul duality
- Lectures 2 and 3 (3d): twists of 3d N=2 and N=4 gauge theories; vertex algebras, chiral categories, and braided tensor categories; d mirror symmetry; quantum groups at roots of unity and derived non-semisimple 3d TQFT's (compared and contrasted with Chern-Simons theory)
- Lecture 4 (4d): line and surface operators in 4d N=2 gauge theory, the coherent Satake category, and relations to Schur indices and 4d N=2 vertex algebras