Instanton Counting, Quantum Geometry and Algebra

Abstract

The aim of this memoir for "Habilitation à 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.

Habilitation memoir. 308 pages; v2. minor updates, typos corrected; v3. reference updated, typos corrected. This is a preprint of the following work: Taro Kimura, Instanton Counting, Quantum Geometry and Algebra, 2021, Springer, reproduced with permission of Springer Nature Switzerland AG. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-76190-5

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• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• 2-Group Symmetries of 6d Little String Theories and T-duality
• On certain aspects of string theory/gauge theory correspondence
• Quiver Theories and Hilbert Series of Classical Slodowy Intersections
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Cited by in corpus (1)

• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$