Intrinsic non-perturbative topological strings

TL;DR

This paper derives and solves difference equations from topological string theory on various Calabi-Yau geometries, revealing non-perturbative effects, strong coupling behaviors, and relations to Donaldson-Thomas invariants.

Contribution

It introduces exact solutions to difference equations in topological string theory, connecting non-perturbative content with known invariants and providing explicit expansions near singularities.

Findings

Solutions expressed by special functions enable non-perturbative analysis.

Strong coupling expansions reveal D-brane and NS5-brane contributions.

Explicit relations established between topological strings and Donaldson-Thomas invariants.

Abstract

We study difference equations which are obtained from the asymptotic expansion of topological string theory on the deformed and the resolved conifold geometries as well as for topological string theory on arbitrary families of Calabi-Yau manifolds near generic singularities at finite distance in the moduli space. Analytic solutions in the topological string coupling to these equations are found. The solutions are given by known special functions and can be used to extract the strong coupling expansion as well as the non-perturbative content. The strong coupling expansions show the characteristics of D-brane and NS5-brane contributions, this is illustrated for the quintic Calabi-Yau threefold. For the resolved conifold, an expression involving both the Gopakumar-Vafa resummation as well as the refined topological string in the Nekrasov-Shatashvili limit is obtained and compared to expected results in the literature. Furthermore, a precise relation between the non-perturbative partition function of topological strings and the generating function of non-commutative Donaldson-Thomas invariants is given. Moreover, the expansion of the topological string on the resolved conifold near its singular small volume locus is studied. Exact expressions for the leading singular term as well as the regular terms in this expansion are provided and proved. The constant term of this expansion turns out to be the known Gromov-Witten constant map contribution.