5d/6d DE instantons from trivalent gluing of web diagrams

TL;DR

The work develops a trivalent SU(2) gluing prescription that computes Nekrasov partition functions for 5d theories with DE-type gauge groups by treating three UV SCFT blocks as SU(2) matter and summing over Young diagrams with an SU(2) vector factor. It provides explicit dual descriptions, using SU(2) trivalent gauging of blocks like $ ext{hat}D_p( ext{SU}(2))$, to realize pure $SO(2N+4)$ and $E_{6,7,8}$ theories, and it extends to 6d uplifts via circle compactifications (including non-Higgsable clusters). The method yields perturbative and instanton data that agree with localization and known elliptic-genus results in multiple cases, and it is shown to extend to the refined topological vertex with consistent flop relations. This approach offers a unified framework for computing higher-instanton Nekrasov functions in DE-type theories and connects 5d/6d physics with non-Lagrangian UV completions. The results pave the way for broader applications, including matter in higher representations and potential Higgsing-based reductions to other gauge families.

Abstract

We propose a new prescription for computing the Nekrasov partition functions of five-dimensional theories with eight supercharges realized by gauging non-perturbative flavor symmetries of three five-dimensional superconformal field theories. The topological vertex formalism gives a way to compute the partition functions of the matter theories with flavor instanton backgrounds, and the gauging is achieved by summing over Young diagrams. We apply the prescription to calculate the Nekrasov partition functions of various five-dimensional gauge theories such as $\mathrm{SO}(2N)$ gauge theories with or without hypermultiplets in the vector representation and also pure $E_6, E_7, E_8$ gauge theories. Furthermore, the technique can be applied to computations of the Nekrasov partition functions of five-dimensional theories which arise from circle compactifications of six-dimensional minimal superconformal field theories characterized by the gauge groups $\mathrm{SU}(3), \mathrm{SO}(8), E_6, E_7, E_8$. We exemplify our method by comparing some of the obtained partition functions with known results and find perfect agreement. We also present a prescription of extending the gluing rule to the refined topological vertex.

Figures (28)

• Figure 1: Left: The 5-brane web for the pure $\mathrm{SU}(p)$ gauge theory with the $\pm p$ CS level. We have $p$ D5-branes which lie in the horizontal direction. The parallel two external NS5-branes imply the non-perturbative $\mathrm{SU}(2)$ flavor symmetry. Right: The S-dual configuration to the 5-brane web on the left. Namely the 5-brane web for the $\widehat{D}_{p}(\mathrm{SU}(2))$ theory.
• Figure 2: A 5-brane web--like description of the theory \ref{['SO2Np4']} which is dual to the pure $\mathrm{SO}(2N+4)$ gauge theory. The prescription for the "trivalent $\mathrm{SU}(2)$ gauging" is going to be given in the next section. Three webs actually does not live in the same plane, and thus do not cross each other in the cases we will deal with in this paper.
• Figure 3: A transition from a 5-brane web with an $O5$--plane to a web--like diagram with trivalent gluing. The left figure represents a 5-brane web of the pure $\mathrm{SO}(2N+4)$ gauge theory using an $O5$--plane. The right figure is a web--like description by replacing the 5-brane for the $\mathrm{SO}(4)$ gauge theory part in the left figure with the two 5-branes webs of the pure $\mathrm{SU}(2)$ gauge theory with no discrete theta angle. Now the three 5-brane webs are connected by the trivalent gluing.
• Figure 4: A 5-brane web for the $\mathrm{SO}(2N+4)$ gauge theory with $M_1 +M_2$ hypermultiplets in the vector representation.
• Figure 5: A 5-brane web--like description for the $\mathrm{SO}(2N+4)$ gauge theory with $M_1 +M_2$ hypermultiplets in the vector representation by replacing the web for the $SO(4)$ part with the two webs for the pure $\mathrm{SU}(2)$. The three 5-brane webs are connected by the trivalent gluing.