Equivariant volumes of non-compact quotients and instanton counting

Details Equivariant volumes of non-compact quotients and instanton counting Equivariant volumes of non-compact quotients and instanton counting

2006-09-29

arxiv.org/abs/math/0609841v3

Commun.Math.Phys.281:827-857,2008

Mathematics

Symplectic Geometry

34 pages, 2 figures; minor typos corrected, to appear in Comm. Math. Phys

Scientific paper

10.1007/s00220-008-0501-x

Motivated by Nekrasov's instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-K\"ahler geometry by means of the Jeffrey-Kirwan residue-formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on $\R^{4}$ and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.

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