Exact partition functions of Higgsed 5d $T_N$ theories

TL;DR

This work develops a general, constructive procedure to compute exact partition functions for five-dimensional class $ ext{S}$ theories realized on Higgs branches of the $T_N$ theories, including non-toric brane-webs. The authors leverage the refined topological vertex and a systematic tuning of external branes to implement Higgsing in the partition function, carefully removing decoupled singlet contributions. They apply the method to the $T_6$ diagram to produce the partition function of the $E_8$ theory, demonstrating agreement with the $Sp(1)$ gauge theory with seven fundamentals plus one antisymmetric hypermultiplet and showing explicit global symmetry enhancement to $E_8$ via the superconformal index. This provides a practical, non-Lagrangian-amenable route to exact 5d partition functions in Higgs branches, offering a path to analyze higher-rank and non-toric theories and to test dualities and symmetry enhancements in richly structured 5d SCFTs.

Abstract

We present a general prescription by which we can systematically compute exact partition functions of five-dimensional supersymmetric theories which arise in Higgs branches of the $T_N$ theory. The theories may be realized by webs of 5-branes whose dual geometries are non-toric. We have checked our method by calculating the partition functions of the theories realized in various Higgs branches of the $T_3$ theory. A particularly interesting example is the $E_8$ theory which can be obtained by Higgsing the $T_6$ theory. We explicitly compute the partition function of the $E_8$ theory and find the agreement with the field theory result as well as the enhancement of the global symmetry to $E_8$.

Figures (13)

• Figure 1: The left figure shows a $(p, q)$ 5-brane web that appears as a part of the web diagram of the $T_N$ theory. The process of going to the right figure represents putting parallel horizontal external 5-branes on one 7-brane. $||$ denotes the choice of the preferred direction in the computation of the refined topological vertex. $\otimes$ represents a 7-brane.
• Figure 2: Left: The process of putting the parallel vertical external 5-branes on one 7-brane with the particular choice of the preferred direction correlated with the one in figure \ref{['fig:Higgs1']}. Right: The same process as the left figure but with a difference choice of the preferred directions.
• Figure 3: The web diagram for the $T_3$ theory. $Q_i, (i=1, 2, 3, 4, 5)$ and $Q_b, Q_f$ parameterise the lengths of the corresponding internal 5-branes.
• Figure 4: The Dynkin diagram of the affine $E_6$ Lie algebra. The nodes in the dotted line represent the Dynkin diagram of $SO(10)$. The nodes in the solid lines denote the Dynkin diagram of $SU(3) \times SU(3) \times SU(3)$.
• Figure 5: Left: The web diagram of the first kind of the Higgsed $T_3$ theory. Right: The dot diagram corresponding to the web on the left. The red line shows the new external leg.