N=2 quiver gauge theories on A-type ALE spaces

Abstract

We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in $\mathcal{N}=2$ gauge theories on ALE spaces based on quiver varieties and the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity ${\mathbb{C}}^2/{\mathbb{Z}}_k$, in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion free sheaves on a suitable orbifold compactification of $X_k$. We extend these computations to generic $\mathcal{N}=2$ superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg-Witten geometry.

arXiv comment34 pages; v2: minor textual and title changes, reference added

References in corpus (25)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• D-branes, Quivers, and ALE Instantons
• Supersymmetric Yang-Mills Systems And Integrable Systems
• Instantons on noncommutative R^4, and (2,0) superconformal six dimensional theory
• Instanton counting on blowup. I. 4-dimensional pure gauge theory
• Para-Liouville/Toda central charges from M5-branes
• Seiberg-Witten Theory and Random Partitions
• Multi instanton calculus on ALE spaces
• Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories
• Gauge Theories on ALE Space and Super Liouville Correlation Functions
• Instanton moduli spaces and bases in coset conformal field theory
• Instantons and 2d Superconformal field theory
• Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory
• The Nekrasov Conjecture for Toric Surfaces
• Smooth toric DM stacks
• Coset conformal blocks and N=2 gauge theories
• N=2 gauge theories on toric singularities, blow-up formulae and W-algebrae
• A Combinatorial Study on Quiver Varieties
• Scheme dependence of instanton counting in ALE spaces
• Coset conformal field theory and instanton counting on C^2/Z_p
• Crystal melting on toric surfaces
• Instantons and framed sheaves on Kähler Deligne-Mumford stacks
• Orbifold instantons, moment maps and Yang-Mills theory with sources
• On the K-theory of the cyclic quiver variety
• Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

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• On 't Hooft Defects, Monopole Bubbling and Supersymmetric Quantum Mechanics
• Instantons from Blow-up
• A brief review of the 2d/4d correspondences
• ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds
• Playing with the index of M-theory
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• $\mathcal{N}=2$ supersymmetric gauge theories on $S^2\times S^2$ and Liouville Gravity
• Noncommutative Instantons in Diverse Dimensions
• Hilbert schemes of points of ${\mathcal{O}}_{{\mathbb{P}}^1}(-n)$ as quiver varieties
• Elliptic algebra, Frenkel-Kac construction and root of unity limit