Topological strings and 5d T_N partition functions

TL;DR

This work establishes a robust bridge between refined topological string theory and five-dimensional Nekrasov partition functions for theories engineered by brane webs, including non-Lagrangian $T_N$ theories. By identifying and stripping decoupled M2-brane contributions (the $U(1)$-factor), the refined vertex calculations on toric Calabi–Yau threefolds reproduce accurate SU(N) Nekrasov results and, via Higgsing, access IR theories with enhanced global symmetries such as $E_6$ and $E_7$. The authors provide explicit checks for $T_3$ and low-rank Sp$(1)$ theories with $N_f=2,3,4$, and extend the framework to Higgsed variants and non-toric diagrams, offering prescriptions to compute partition functions in these cases. The work also clarifies the relation between 5d gauge theories and their Higgs-branch descendants, and suggests a path toward applying refined topological vertex methods to non-toric web diagrams, with potential implications for higher-rank $T_N$ theories and 5d/4d dualities.

Abstract

We evaluate the Nekrasov partition function of 5d gauge theories engineered by webs of 5-branes, using the refined topological vertex on the dual Calabi-Yau threefolds. The theories include certain non-Lagrangian theories such as the T_N theory. The refined topological vertex computation generically contains contributions from decoupled M2-branes which are not charged under the 5d gauge symmetry engineered. We argue that, after eliminating them, the refined topological string partition function agrees with the 5d Nekrasov partition function. We explicitly check this for the T_3 theory as well as Sp(1) gauge theories with N_f = 2, 3, 4 flavors. In particular, our method leads to a new expression of the Sp(1) Nekrasov partition functions without any contour integrals. We also develop prescriptions to calculate the partition functions of theories obtained by Higgsing the T_N theory. We compute the partition function of the E_7 theory via this prescription, and find the E_7 global symmetry enhancement. We finally discuss a potential application of the refined topological vertex to non-toric web diagrams.

Figures (27)

• Figure 1: The black solid lines represent the toric fan on the two-dimensional hyperplane for a local $\mathbb{P}^1_f \times \mathbb{P}^1_b$. The red dotted lines denote the dual diagram which can be viewed as a $(p, q)$ 5-brane web in type IIB string theory.
• Figure 2: The toric fans on the two-dimensional hyperplane for toric Calabi--Yau threefolds whose bases are the blow up of $\mathbb{P}^1_f \times \mathbb{P}^1_b$ at $(a): N_f = 1, (b): N_f = 2, (c): N_f = 3, (d): N_f = 4$ points. $H_i, i = 1, 2, 3, 4$ are the blow up divisors. We chose a particular triangulation which will be used in the later computation of the partition functions. M2-branes wrapping $\beta_{\nu_i}$ give fundamental hypermultiplets.
• Figure 3: The red dotted lines represent a web of $(p, q)$ 5-branes for $T_3$ theory. The black solid lines stand for the corresponding toric fan of a particular triangulation.
• Figure 4: The deformation of pieces of 5-branes between 7-branes. One black solid line represents a 5-brane and one $\otimes$ represents a 7-brane. The dotted line stands for the direction where 7-branes extend but the 5-brane does not extend.
• Figure 5: The toric fan of $T_3$ under a particular triangulation. On the two-dimensional hyperplane, the original toric fan can be thought as a two-dimensional toric fan for the base manifold $D$. Then, the two-dimensional toric fan stands for the blow up of $\mathbb{P}^1 \times \mathbb{P}^1$ at five points. The five blow up divisors are depicted by five blue dashed arrows.