Recurrence relation for instanton partition function in SU(N) gauge theory

Abstract

We derive a residue formula and as a consequence a recurrence relation for the instanton partition function in mathcal{N} = 2 supersymmetric theory on ℂ2 with SU(N) gauge group.The particular cases of SU(2) and SU(3) gauge groups were considered in the literature before. The recurrence relation with SU(2) gauge group is long well known and was found as the Alday-Gaiotto-Tachikawa (AGT) counterpart of the Zamolodchikov relation for the Virasoro conformal blocks. In the SU(3) case a residue formula for the term with the minimal number of instantons was found and basing on it a recurrence relation was conjectured.We give a complete proof of the residue formula in all instanton orders in presence of any number of matter hypermultiplets in the adjoint and fundamental representations. The recurrence relation however describes only theories with not too much matter hypermultiplets so that the behaviour at infinity is moderate. The guideline of the proof is an algebro-geometric interpretation of the mathcal{N} = 2 supersymmetric gauge theory partition function in terms of the framed torsion-free sheaves. Lead by this interpretation we formulate a refined version of the residue formula and prove it by direct algebraic manipulations.