Non-Lagrangian Theories from Brane Junctions

TL;DR

This work develops a comprehensive framework for non-Lagrangian 5D T_N theories using 5-brane junctions, deriving Seiberg–Witten curves and Nekrasov partition functions and validating them against 5D indices. It demonstrates how toric/topological-string techniques yield partition functions that, after removing non-full spin contributions, match the 5D superconformal indices and connect to q-deformed W_N Toda correlators via an extended AGTW correspondence. The results include explicit SW curves for T_N (notably the E_6-enhanced T_3), a proposed normalization scheme to subtract spurious stringy states, and a structured mapping between 5D partition functions and 2D q-Toda three-point functions, with detailed analysis for TN and T_2 cases. The study advances our understanding of the non-Lagrangian landscape, enabling a concrete bridge between brane constructions, topological strings, and higher-rank Toda CFT data, and lays groundwork for future explorations of E7/E8 sectors and full TN correlators.

Abstract

In this article we use 5-brane junctions to study the 5D T_N SCFTs corresponding to the 5D N=1 uplift of the 4D N=2 strongly coupled gauge theories, which are obtained by compactifying N M5 branes on a sphere with three full punctures. Even though these theories have no Lagrangian description, by using the 5-brane junctions proposed by Benini, Benvenuti and Tachikawa, we are able to derive their Seiberg-Witten curves and Nekrasov partition functions. We cross-check our results with the 5D superconformal index proposed by Kim, Kim and Lee. Through the AGTW correspondence, we discuss the relations between 5D superconformal indices and n-point functions of the q-deformed W_N Toda theories.

Figures (23)

• Figure 1: In this figure, the possible ways of constructing $SU(2)$ gauge theory with $N_f=1$ are depicted. The blue circles denote the 7-brane, the solid lines the $(p,q)$ 5-branes and the dashed lines the branch cuts that start from the 7-branes and extend to infinity. In part (a) we have a 5-brane loop with five 7-branes in it. In part (b) we pulled four of the 7-branes out of the 5-brane loop. In part (c) we also pull out the fifth 7-brane. In part (d) we pulled the 7-branes to infinity and are left with a 5-brane web.
• Figure 2: To obtain the $SU(2)$ gauge theory with $N_f=5$, we begin with the pure $SU(2)$ web-toric diagram with five D7 branes inserted. In the first part of the figure all the flavors have the same mass, and the symmetry enhancement is manifest. In the second part of the figure, a mass deformation is performed by moving the 7-branes vertically.
• Figure 3: Then, from the mass deformed configuration, we pull the 7-branes outside of the NS5-branes following the Hanany Witten effect. Lastly, after a flop on the lower right leg of the diagram we arrive at the $E_6$ web-toric diagram suggested by Benini, Benvenuti and Tachikawa.
• Figure 4: Part (a) shows the brane setup for $T_3$, part (b) the setup for the $SU(2)$ gauge theory with five flavors and part (c) the setup for the $SU(3)$ gauge theory with six flavor.
• Figure 5: Part (a) shows the toric diagram and part (b) the brane setup for $E_6$ CFT.