N=2 gauge theories, instanton moduli spaces and geometric representation theory

Abstract

We survey some of the AGT relations between N=2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces. We treat the cases of gauge theories on both flat space and ALE spaces in some detail, and with emphasis on the implications arising from embedding them into supersymmetric theories in six dimensions. Along the way we construct new toric noncommutative ALE spaces using the general theory of complex algebraic deformations of toric varieties, and indicate how to generalise the construction of instanton moduli spaces. We also compute the equivariant partition functions of topologically twisted six-dimensional Yang-Mills theory with maximal supersymmetry in a general Omega-background, and use the construction to obtain novel reductions to theories in four dimensions.

55 pages; v2: typos corrected and reference added; Final version to appear in the Special Issue "Instanton Counting: Moduli Spaces, Representation Theory and Integrable Systems" of the Journal of Geometry and Physics, eds. U. Bruzzo and F. Sala

References in corpus (14)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Asymptotically free N=2 theories and irregular conformal blocks
• Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory
• Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory
• Notes on Ding-Iohara algebra and AGT conjecture
• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• N=2 quiver gauge theories on A-type ALE spaces
• Polynomials associated with fixed points on the instanton moduli space
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Instantons and framed sheaves on Kähler Deligne-Mumford stacks
• Crystals, instantons and quantum toric geometry
• Noncommutative Deformations of Type A Kleinian Singularities and Hilbert Schemes
• A review on instanton counting and W-algebras
• Isospectral Deformations of Eguchi-Hanson Spaces as Nonunital Spectral Triples

Cited by in corpus (8)

• A slow review of the AGT correspondence
• Higher rank K-theoretic Donaldson-Thomas theory of points
• A geometric recipe for twisted superpotentials
• Elliptic non-Abelian Donaldson-Thomas invariants of ${\mathbb{C}}^3$
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Mayer expansion of the Nekrasov pre potential: the subleading ${\epsilon }_2$-order
• Noncommutative Instantons in Diverse Dimensions
• AGT basis in SCFT for c=3/2 and Uglov Polynomials