Modular properties of full 5D SYM partition function

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We study properties of the full partition function for the U(1) 5D N=2* 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.