Non-perturbative Topological String Partition Function on Twisted Affine Line Bundle over $\mathbb{C}\times T^2$

Using instanton partition function for five dimensional $U(1)$ gauge theory with eight supercharges and a single adjoint massive hypermultiplet on the $\Omega$ background, we give explicit expression for non-perturbative corrections to the topological string theory in the holomorphic limit. It was argued that in this case the theory is compactified on the twisted affine line bundle over $\mathbb{C}\times T^2$. We perform calculations in two ways. First we modify the integration contour by adding poles responsible for non-perturbative physics in accordance with a recent proposal. Then, we compute the genus zero Gopakumar-Vafa invariants for our case and evaluate the non-perturbative corrections to the partition function. We check that both calculations give the same result.