Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory

Abstract

We study the relationship between instanton counting in N=4 Yang-Mills theory on a generic four-dimensional toric orbifold and the semi-classical expansion of q-deformed Yang-Mills theory on the blowups of the minimal resolution of the orbifold singularity, with an eye to clarifying the recent proposal of using two-dimensional gauge theories to count microstates of black holes in four dimensions. We describe explicitly the instanton contributions to the counting of D-brane bound states which are captured by the two-dimensional gauge theory. We derive an intimate relationship between the two-dimensional Yang-Mills theory and Chern-Simons theory on generic Lens spaces, and use it to show that the correct instanton counting is only reproduced when the Chern-Simons contributions are treated as non-dynamical boundary conditions in the D4-brane gauge theory. We also use this correspondence to discuss the counting of instantons on higher genus ruled Riemann surfaces.

27 pages

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• Black Holes in Type IIA String on Calabi-Yau Threefolds with Affine ADE Geometries and q-Deformed 2d Quiver Gauge Theories
• $T\overline{T}$-deformation of $q$-Yang-Mills theory
• Plethystics and instantons on ALE spaces
• Noncommutative Instantons in Diverse Dimensions
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Exact Lorentz-violating $q$-deformed O($N$) universality class
• Instanton counting on Hirzebruch surfaces
• Notes On U(1) Instanton Counting On $A_{l-1}$ ALE Spaces