Stringy instanton counting and topological strings

A bstract We study the stringy instanton partition function of four dimensional N = 2 U( N ) supersymmetric gauge theory which was obtained by Bonelli et al. in 2013. In type IIB string theory on ℂ 2 × T * ℙ 1 × ℂ , the stringy U( N ) instantons of charge k are described by k D1-branes wrapping around the ℙ 1 bound to N D5-branes on ℂ 2 × ℙ 1 . The KK corrections induced by compactification of the 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.