D-type Conformal Matter and SU/USp Quivers

TL;DR

This work studies how to realize 4d ${\cal N}=1$ theories from compactifications of the 6d minimal D-type conformal matter on tori with flux in $SO(4N)$, by first reducing on a circle to 5d and interpreting flux as duality domain walls between distinct 5d realizations. The authors construct explicit 5d flux-domain-wall setups (notably a ${USp}(2N)$–${SU}(N+1)$ duality wall with 1/4 flux) and develop gluing rules to assemble general flux configurations, then derive 4d Lagrangians for tubes and tori and test them via anomaly inflow from the 6d anomaly polynomial and by computing superconformal indices. They find novel ${\cal N}=1$ dualities in 4d that arise from different 5d reductions, including two inequivalent spinor embeddings of $SO(4N{+}12)$ that yield distinct 4d spectra but identical anomalies. The results extend the 6d-to-4d compactification program to D-type conformal matter, provide a coherent framework for flux-domain-wall constructions, and illuminate connections to E-string/A-type setups and potential ADE generalizations.

Abstract

We discuss the four dimensional models obtained by compactifying a single M5 brane probing $D_{N}$ singularity (minimal D-type $(1,0)$ conformal matter in six dimensions) on a torus with flux for abelian subgroups of the $SO(4N)$ flavor symmetry. We derive the resulting quiver field theories in four dimensions by first compactifying on a circle and relating the flux to duality domain walls in five dimensions. This leads to novel ${\cal N}=1$ dualities in 4 dimensions which arise from distinct five dimensional realizations of the circle compactifications of the D-type conformal matter.

Figures (13)

• Figure 1: The $3d$ mirror dual of the SCFT one gets by compactification of the $6d$ SCFT $(D_{N+3},D_{N+3})$ conformal matter on $T^3$ without fluxes.
• Figure 2: Basic flux domain walls. Figure (a) is the $USp(2N)-SU(N\!+\!1)$ type domain wall with flux 1/4 preserving $U(1)\times SU(2N\!+\!6)$ symmetry. $USp(2N)\times SU(N\!+\!1)$ symmetry are gauged by the $5d$ bulk gauge groups. The chiral fields $M$ and $M'$ are from the hypermultiplets with Neumann boundary conditions on the two sides of the wall. Figure (b) is the $SU(N\!+\!1)-SU(N\!+\!1)$ type domain wall with flux 1/2 preserving $U(1)\times SU(2)\times SU(4N+8)$ symmetry. $SU(N\!+\!1)\times SU(N\!+\!1)$ symmetry are gauged by the $5d$ bulk gauge groups. The chiral fields $M,\tilde{M}$ and $M',\tilde{M}'$ are from the $5d$ hypermultiplets with Neumann boundary condition. There is a gauge singlet chiral field denoted by '$X$' which couples to the baryonic operator of the bi-fundamental chiral field $q$.
• Figure 3: Figure (a) is the domain wall connecting two $5d$$USp(2N)$ gauge theories with flux $(\tfrac{1}{2},\tfrac{1}{2},\cdots ,\tfrac{1}{2})$. Figure (b) is the domain wall connecting two $5d$$SU(N+1)$ gauge theories with flux $(\frac{1}{2},\frac{1}{2},\cdots,\frac{1}{2} , 0 , 0 , ... , 0)$.
• Figure 4: $4d$ theories corresponding to compactifications with flux preserving $SU(2N+6)$. Figure (a) shows the theory associated with a sphere with two punctures and flux $\frac{1}{4}$, while (b) shows the theory associated with a torus and flux $\frac{1}{2}$. The arrow from the $SU$ group to itself stands for an antisymmetric chiral field. There are cubic superpotentials for every triangle and for every antisymmetric chiral coupling it to two bifundamentals. The theory has an $U(1)_x \times SU(2N+6)$ global symmetry as well as a $U(1)_R$ symmetry. For the $U(1)_R$ it is convenient to use the $6d$ R-symmetry under which the bifundamentals have charge $0$, the antisymmetrics have charge $2$, and all the others have charge $1$. The charges under $U(1)_x$ are shown using fugacities.
• Figure 5: $4d$ theories corresponding to compactifications with flux preserving $SU(r) \times U(1)\times SO(4N+12-2r)$. All groups are $SU$ except the ones with $2N$ that are symplectic, $USp$. Here $r$ is even so that the $USp(2N)$ gauge group is not anomalous. For bifundamentals between $SU$ groups we adopted a double arrow notation indicating whether the field is in the fundamental or antifundamenal of each $SU$ group. Like before the arrow from the $SU$ group to itself stands for an antisymmetric chiral field. (a) The theory corresponding to a tube with flux $z=-\frac{1}{2}$. There are cubic superpotentials for every triangle and for every antisymmetric chiral coupling it to two bifundamentals. Additionally there is a quartic superpotential for the lower 'triangle'. (b) Connecting two tubes leads to this theory.