Modular properties of full 5D SYM partition function

Abstract

Abstract: We study properties of the full partition function for the U(1) 5D (Formula presented.) gauge theory with adjoint hypermultiplet of mass M. This theory is ultimately related to abelian 6D (2,0) theory. We construct the full non-perturbative partition function on toric Sasaki-Einstein manifolds by gluing flat copies of the Nekrasov partition function and we express the full partition function in terms of the generalized double elliptic gamma function G2C associated with a certain moment map cone C. The answer exhibits a curious SL(4, ℤ) modular property. Finally, we propose a set of rules to construct the partition function that resembles the calculation of 5d supersymmetric partition function with the insertion of defects of various co-dimensions.