Geometric Representation Theory and Gauge Theory
In the last 30 years a new pattern of interaction between mathematics and physics
emerged, in which the latter catalyzed the creation of new mathematical theories.
Most notable examples of this kind of interaction can be found in the theory of
moduli spaces. In algebraic geometry the theory of moduli spaces goes back at
least to Riemann, but they were first rigorously constructed by Mumford only in the
1960s. The theory has experienced an extraordinary development in recent decades,
finding an increasing number of connections with other fields of mathematics
and physics. In particular, moduli spaces of different objects (sheaves, instantons,
curves, stable maps, etc.) have been used to construct invariants (such as Donaldson,
Seiberg-Witten, Gromov-Witten, Donaldson-Thomas invariants) that solve longstanding,
difficult enumerative problems. These invariants are related to the partition
functions and expectation values of quantum field and string theories. In recent
years, developments in both fields have led to an unprecedented cross-fertilization
between geometry and physics.
These striking interactions between geometry and physics were the theme of the
CIME School Geometric Representation Theory and Gauge Theory. The School
took place at the Grand Hotel San Michele, Cetraro, Italy, in June, Monday 25
to Friday 29, 2018. The present volume is a collection of notes of the lectures
delivered at the school. It consists of three articles from Alexander Braverman and
Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively.
These are (somewhat informal) lecture notes for the CIME summer school “Geometric Representation Theory and Gauge Theory” in June 2018. In these notes we review the constructions and results of Braverman et al. (Adv Theor Math Phys 22(5):1017–1147, 2018; Adv Theor Math Phys 23(1):75–166, 2019; Adv Theor Math Phys 23(2):253–344, 2019) where a mathematical definition of Coulomb branches of 3d N = 4 quantum gauge theories (of cotangent type) is given, and also present a framework for studying some further mathematical structures (e.g. categories of line operators in the corresponding topologically twisted theories) related to these theories.