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

TL;DR

The paper addresses how to obtain non-perturbative corrections to the topological string partition function for a non-compact Calabi–Yau described as a twisted affine line bundle over $\mathbb{C}\times T^2$, realized by a 5D $U(1)$ ${\cal N}=2^*$ theory on an $\Omega$-background with a single deformation parameter. It applies two complementary approaches: (i) extracting non-perturbative terms from the Schwinger-type integral via the Hattab–Palti pole-residue prescription using the exact 5D instanton partition function, and (ii) constructing a closed-form, holomorphic non-perturbative partition function from genus-zero GV invariants following Alim et al.'s framework. The main result is that the residues reproduce the universal form built from genus-zero GV data, with explicit GV invariants ${[GV]}_{\ell E+M,0}=-1$, ${[GV]}_{\ell E-M,0}=-1$, ${[GV]}_{\ell E,0}=2$, ${[GV]}_{\ell E,1}=-1$, confirming agreement between the two methods. This provides a concrete non-perturbative completion for this geometry and supports the broader claim that genus-zero GV invariants suffice in the holomorphic limit; the work also clarifies the field-theory limit and the interpretation in terms of M2–M5 brane realizations on the twisted affine line bundle.

Abstract

Using instanton partition function for five dimensional $U(1)$ gauge theory with eight supercharges and a single adjoint massive hypermultiplet on the $Ω$ 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.

Figures (1)

• Figure 1: Toric diagram for ${\cal N}=2^*$$U(1)$ theory. We use the cycle basis $E=B+M$ and $M$.