Stringy Instanton Counting and Topological Strings

Abstract

We study the stringy instanton partition function of four dimensional $\mathcal{N}=2$ $U(N)$ supersymmetric gauge theory which was obtained by Bonelli et al in 2013. In type IIB string theory on ${\mathbb{C}}^2\times T^{\ast }{\mathbb{P}}^1\times \mathbb{C}$, the stringy $U(N)$ instantons of charge $k$ are described by $k$ D1-branes wrapping around the ${\mathbb{P}}^1$ bound to $N$ D5-branes on ${\mathbb{C}}^2\times {\mathbb{P}}^1$. The KK corrections induced by compactification of the ${\mathbb{P}}^1$ give the stringy corrections. We find a relation between the stringy instanton partition function whose quantum stringy corrections have been removed and the K-theoretic instanton partition function, or by geometric engineering, the refined topological A-model partition function on a local toric Calabi-Yau threefold. We also study the quantum stringy corrections in the stringy instanton partition function which is not captured by the refined topological strings.

37 pages, 3 figures. v2: minor corrections and references added. v3: minor changes, appendix D added and references added

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