The Non-Perturbative Partition Function of Refined Topological Strings on a Twisted Affine Line Bundle over CxT2.

We compute the non-perturbative partition function of refined topological string theory on a twisted affine line bundle over (\mathbb{C}\times T^2), thereby extending recent result obtained in the unrefined setting to the more general case with two independent deformation parameters. Our analysis builds upon the correspondence between topological string theory and supersymmetric gauge theories and makes use of the instanton partition function of five-dimensional (U(1)) gauge theory with eight supercharges and a single adjoint hypermultiplet, commonly referred to as the $N=2^*$ theory. The gauge theory is placed on the (\Omega)-background with both deformation parameters turned on, providing a natural framework for the refined topological string.

Starting from the exact instanton partition function, we derive an integral representation suitable for investigating its non-perturbative structure. We then perform a detailed analysis of the associated integration contour. By deforming the original contour into one that surrounds the positive real axis while excluding the origin, we identify the relevant contributions arising from poles and obtain a non-perturbative completion of the refined partition function. This contour deformation provides a transparent interpretation of the non-perturbative sectors and their relation to the underlying geometry.

Furthermore, we discuss parallels between the resulting non-perturbative contributions and the Schwinger pair-production mechanism in quantum field theory.