Partition functions of web diagrams with an O7$^-$-plane

TL;DR

This paper computes the topological string partition function for 5-brane web diagrams with an O7$^-$-plane, using a method that resolves the orientifold to apply the topological vertex, and confirms results with localization techniques.

Contribution

It introduces a procedure to compute partition functions for web diagrams with orientifolds by resolving the orientifold and applying the topological vertex, extending the computational toolkit for 5d theories.

Findings

Perfect agreement between topological vertex and localization results.

Established parameter and moduli correspondence for 5d theories.

Applied method to $SU(N)$ and $USp(2N)$ theories.

Abstract

We consider the computation of the topological string partition function for 5-brane web diagrams with an O7$^-$-plane. Since upon quantum resolution of the orientifold plane these diagrams become non-toric web diagrams without the orientifold we are able to apply the topological vertex to obtain the Nekrasov partition function of the corresponding 5d theory. We apply this procedure to the case of 5d $SU(N)$ theories with one hypermultiplet in the antisymmetric representation and to the case of 5d pure $USp(2N)$ theories. For these cases we discuss the dictionary between parameters and moduli of the 5d gauge theory and lengths of 5-branes in the web diagram and moreover we perform comparison of the results obtained via application of the topological vertex and the one obtained via localisation techniques, finding in all instances we consider perfect agreement.