Instanton Counting, Quantum Geometry and Algebra

TL;DR

This paper explores the quantum geometric and algebraic structures in supersymmetric gauge theories, focusing on instanton counting, Seiberg-Witten geometry, and the emergence of quiver W-algebras through various quantization methods.

Contribution

It introduces the construction of quiver W-algebras via double quantization of Seiberg-Witten geometry, expanding the understanding of quantum algebraic structures in gauge theories.

Findings

Derived instanton partition functions using multiple methods.

Connected gauge theory to integrable systems and string theory perspectives.

Defined and analyzed quiver W-algebras and their elliptic deformations.

Abstract

The aim of this memoir for "Habilitation \`a Diriger des Recherches" is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how to obtain the instanton partition function and its generalization to quiver gauge theory and supergroup gauge theory in three ways: the equivariant index formula, the contour integral formula, and the combinatorial formula. We then explore the geometric description of $\mathcal{N} = 2$ gauge theory based on Seiberg-Witten geometry together with its string/M-theory perspective. Through its relation to integrable systems, we show how to quantize such a geometric structure via the $\Omega$-deformation of gauge theory. We also discuss the underlying quantum algebraic structure arising from the supersymmetric vacua. We introduce the notion of quiver W-algebra constructed through double quantization of Seiberg-Witten geometry, and show its specific features: affine quiver W-algebras, fractional quiver W-algebras, and their elliptic deformations.