Topological vertex for Higgsed 5d $T_N$ theories

TL;DR

This work develops a direct topological-vertex method to compute Nekrasov partition functions for Higgs branches of 5d $T_N$ theories, even when the Higgsed webs are not toric. By formulating rules for applying the (refined) topological vertex to Higgsed diagrams and identifying decoupled factors directly from the Higgsed geometry, the authors compute exact partition functions for higher-rank $E_6$, $E_7$, and $E_8$ theories and validate them against corresponding $Sp(N)$ gauge theories with fundamental and antisymmetric matter. The method yields a product structure for rank-$N$ theories and substantially simplifies previous UV-tuning approaches, with extensions to refined vertices in select Higgsing configurations. Overall, the paper provides a practical, checkable framework for extracting IR Nekrasov partition functions from non-toric Higgsed webs, with implications for dualities, symmetry enhancements, and potential generalizations to defects and six-dimensional elliptic genera.

Abstract

We analyse the computation of the partition function of 5d $T_N$ theories in Higgs branches using the topological vertex. The theories are realised by a web of $(p,q)$ 5-branes whose dual description may be given by an M-theory compactification on a certain local non-toric Calabi-Yau threefold. We explicitly show how it is possible to directly apply the topological vertex to the non-toric geometry. Using this novel technique, which considerably simplifies the computation by the existing method, we are able to compute the partition function of the higher rank $E_6$, $E_7$ and $E_8$ theories. Moreover we show how in some specific cases similar results can be extended to the computation of the partition function of 5d $T_N$ theories in the Higgs branch using the refined topological vertex. These cases require a modification of the refined topological vertex.

Figures (23)

• Figure 1: The web diagram for the $T_3$ theory. Each $\otimes$ represents a 7-brane.
• Figure 2: An example of Higgsing by tuning the length of two 5-branes labelled as $Q_1, Q_2$. The broken lines represent the directions along $x^7, x^8, x^9$.
• Figure 3: Higgsing of parallel horizontal external legs in a $T_N$ diagram. The orange dots indicate the lines that are shrunk to zero length, and hence we use $Q =1$ in the computation of the topological string partition function.
• Figure 4: Higgsing of parallel horizontal legs in a $T_N$ diagram. The orange dots indicate the lines that are shrunk to zero length.
• Figure 5: Higgsing of parallel horizontal legs in a $T_N$ diagram with some additional parts of the diagram glued. The orange dots indicate the lines that are shrunk to zero length.