Sheaves on P2 and generalised Appell functions

Abstract

A closed expression is given for the generating function of ( virtual) Poincaré polynomials of moduli spaces of semi-stable sheaves on the p rojective plane P 2 with arbitrary rank r and Chern classes. This generating function is known to equa l the partition function of topologically twisted gauge theory with N = 4 supersymmetry and gauge group U ( r ) , which localizes on the Hermitian Yang-Mills solutions of th e gauge field. To classify and study the novel generating functions, the notion of Appell f unctions with signature ( n + , n − ) is introduced. For n − = 1 , these novel functions reduce to the known class of Appell functions with multiple variables or higher level.