Compactifications of ADE conformal matter on a torus

TL;DR

The paper develops a unified framework to realize torus compactifications of ADE conformal matter on flux backgrounds by passing through 5d affine quiver domains that encode 6d flux as domain walls. It provides explicit Lagrangian 4d theories built from 5d blocks for ADE types A, D, and E, and validates them via anomaly matching and index computations, revealing IR symmetry enhancements and novel dualities. The construction hinges on precise flux assignments tied to permutations of Cartan factors and on robust gluing rules that assemble tubes into tori, with integer total flux ensuring six-dimensional consistency. The results extend ADE compactifications beyond previously analyzed A-type cases and offer a broad, testable dictionary between 6d flux data and 4d quiver dynamics. Overall, the work broadens the landscape of 4d theories arising from higher-dimensional ADE conformal matter and clarifies how flux on a torus shapes their IR physics and dualities.

Abstract

In this paper we study compactifications of ADE type conformal matter, N M5 branes probing ADE singularity, on torus with flux for global symmetry. We systematically construct the four dimensional theories by first going to five dimensions and studying interfaces. We claim that certain interfaces can be associated with turning on flux in six dimensions. The interface models when compactified on a circle comprise building blocks for constructing four dimensional models associated to flux compactifications of six dimensional theories on a torus. The theories in four dimensions turn out to be quiver gauge theories and the construction implies many interesting cases of IR symmetry enhancements and dualities of such theories.

Figures (39)

• Figure 1: Compactification on a torus in six dimensions with flux $F$ for the global symmetry is constructed as a combination of blocks. Each block is associated with a tube and flux $F_j$ such that $\sum F_j =F$. The blocks are obtained by first going to five dimensions, considering then interfaces $B_j$, and then compactifying on an additional circle. This provides a systematic way to construct compactifications.
• Figure 2: Quiver diagrams for the domain walls in the $A_4$ quiver gauge theory. The domain wall on the left is for the boundary condition $\mathcal{B}=\{+,+,+,-,-\}$ and the domain wall on the right is for $\mathcal{B}=\{+,-,+,-,-\}$. The square boxes denote the $SU(N)^5\times SU(N)'{}^5$ gauge symmetries of 5d gauge theories on both sides of the walls. The symbol $\times$ denotes flip fields coupled to the baryonic operator made from $q_i$.
• Figure 3: Domain walls for $\mathcal{B}=\{+,+,+,-,-\}$ (left) and $\mathcal{B}=\{+,-,+,-,-\}$ (right). The $U(1)$ global charges of the 4d chiral fields denoted by their fugacities are fixed by the gauge-global mixed anomaly cancellation and superpotential terms. Here the permutation $\sigma_\beta=(1\;2\;3)(\;4\;5)$ and $\sigma_\gamma=\emptyset$ for the left tube. For the right tube $\sigma_\beta=(\;1\;3\;) (2\;5\;4\;)$ amd $\sigma_\gamma=\emptyset$.
• Figure 4: Gluing two domain walls with $\mathcal{D}_1=\{+,+,+,-,-\}_\beta$ and $\mathcal{D}_2=\{+,-,+,-,-\}_\beta$. The total flux of the final domain wall is $F_{\rm tot}=(3/5,1/5,1/5,-1/5,-4/5)_\beta+(1/5,3/5,-1/5,-1/5,-2/5)_\beta=(4/5,4/5,0,-2/5,-6/5)_\beta$.
• Figure 5: Domain walls of type $\mathcal{T}=\gamma$ related to $SU(k)_\gamma$ fluxes. The left one is related to the flux $F=(-1/4,-1/4,-1/2,1)_\gamma$ and the right one is related to the flux $F=(-1/2,1/2,-1/2,1/2)_\gamma$ in the 6d $SU(k)_\gamma$ symmetry.