Mass-deformed $T_N$ as a linear quiver

TL;DR

The authors establish that mass deformations diagonalizing two SU(N) flavor factors of the TN theory yield an IR description as TN−1 coupled to an SU(N−1) vector with N fundamentals, and recursively applying this deformation produces a full linear SU(n) quiver with zero Chern-Simons levels. They substantiate the claim through multiple independent checks: 5d Nekrasov partition functions from brane webs, operator and state matching during the TN→TN−1 flow, moduli-space and Higgs-branch analyses via field theory and 6d/Seiberg–Witten methods, and Seiberg–Witten curves in 4d that reproduce the quiver structure; they also extend the framework to general punctures and to 3d mirrors. The results connect non-Lagrangian TN theories to conventional Lagrangian linear quivers, clarifying how mass deformations reorganize flavor symmetries, gauge groups, and moduli, with implications for class S constructions and dualities. Overall, the work provides a coherent, multi-faceted picture tying TN dynamics under mass deformations to familiar linear-quiver dynamics, enriching the understanding of UV completions and IR phases in higher-dimensional SCFTs.

Abstract

The $T_N$ theory is a non-Lagrangian theory with SU(N) flavor symmetry. We argue that when mass terms are given so that two of SU(N)'s are both broken to SU(N-1) x U(1), it becomes $T_{N-1}$ theory coupled to an SU(N-1) vector multiplet together with N fundamentals. This implies that when two of SU(N)'s are both broken to U(1)$^{N-1}$, the theory becomes a linear quiver. We perform various checks of this statement, by using the 5d partition function, the structure of the coupling constants, the Higgs branch, and the Seiberg-Witten curve. We also study the case with more general punctures.

Figures (5)

• Figure 1: The web diagram for the 5d $T_N$ theory. In our convention, a horizontal, vertical and diagonal line denotes a D5-brane, an NS5-brane and a $(1,1)$ 5-brane respectively. $P_k^{(n)}, Q_k^{(n)}, R_k^{(n)}, 1 \leq k \leq n$ for $1 \leq n \leq N{-}1$ are in a form $e^{iL}$ where $L$ is the length of the corresponding internal line. They can be regarded as fugacities which appear in the computation of the partition function the 5d $T_N$ theory.
• Figure 2: Upper row: the web diagram when the masses for two $\mathrm{SU}(N)$s are equal and far larger than that for the third $\mathrm{SU}(N)$. Lower row: the web diagram looks as this brane configuration, if one squints the eyes. The figures are shown for $N=5$.
• Figure 3: The local deformation of 5-branes between 7-branes when we turn off some mass deformations. The red $\otimes$ represent a 7-brane. The dotted lines indicate another three directions where the 7-branes are extended.
• Figure 4: The web diagram where the puncture associated with the parallel external D5-branes is $[3, 1, 1]$. The red dotted line denotes the branch cut for the D7-brane.
• Figure 5: Left: Young diagram for $Y=[3,2,2,1]$. The numbers inside the boxes are the $p_k$ defined in \ref{['eq:defpk']}. Right: Removing the leftmost column. The numbers inside the boxes represent the $\ell=1,2,\cdots, K$.