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

TL;DR

The paper develops a robust topological vertex framework to compute the Nekrasov partition functions for 5d gauge theories realized by 5-brane webs containing an orientifold O7$^{-}$. By resolving the orientifold, non-toric diagrams are treated with the vertex formalism, enabling explicit results for SU($2N$) with an antisymmetric hypermultiplet, pure USp($2N$), and SU($2N-1$) with antisymmetric matter; the authors establish a precise dictionary between web lengths and gauge theory moduli, masses, and instanton fugacities. They perform detailed perturbative and instanton calculations, including 1-instanton checks, and demonstrate perfect agreement with Nekrasov localization across multiple cases and CS levels. The work also elucidates Higgsing relations between the theories (e.g., SU($2N$) with antisymmetric to USp($2N$), and to SU($2N-1$) with antisymmetric) and shows how decoupled factors cancel to yield the correct 5d physics. Overall, the results extend the reach of the topological vertex to non-toric geometries from orientifold-resolved web diagrams and provide a concrete computational toolkit for these UV-complete 5d 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.

Figures (23)

• Figure 1: The web diagram realising an $SU(2N)_k$ gauge theory with antisymmetric matter in the presence of an O7$^-$-plane.
• Figure 2: The web diagram realising an $SU(2N)_k$ theory with antisymmetric matter after the quantum resolution of the orientifold plane and the extraction of the 7-branes to infinity.
• Figure 3: Examples of $SU(2N)_k$ with antisymmetric matter as a Higgsing of a linear quiver. On the left the case of $N=2$, on the right the case of $N=3$. The 5-branes with the green dot get shrunken for the Higgsing.
• Figure 4: The definition of the parameters entering in the brane web of the 5d $SU(2N)_k$ gauge theory with antisymmetric matter.
• Figure 5: The web diagram of Figure \ref{['fig:SU2Nantisym']} after performing the $T^{-1}$ transformation.