A new 5d description of 6d D-type minimal conformal matter

TL;DR

The paper proposes a 5d UV completion for circle-compactified 6d D-type minimal conformal matter, realized as an $SU(N+2)$ gauge theory with $N_f=2N+8$ flavors and zero Chern-Simons level. It establishes two complementary routes—the brane construction with Wilson line and an M5-brane on a $D$-type orbifold yielding an affine quiver—to show that the 5d theory flows to the 6d UV fixed point, with a global symmetry enhancement to $SO(4N+16)$. A key insight is identifying the circle radius with the instanton fugacity via Tao diagrams, and decoupling flavors recovers a family of 5d UV fixed points with various CS levels. The analysis of 7-brane monodromy across $SU(n)$ theories provides a detailed map of enhanced non-abelian symmetries, aligning 5d fixed points with the 6d origin and offering a framework for further exact computations of indices and partition functions.

Abstract

We propose a new 5d description of the circle-compactified 6d $(D_{N+4}, D_{N+4})$ minimal conformal matter theory which can be approached by the 6d $\mathcal{N}=(1,0)$ $Sp(N)$ gauge theory with $N_f=2N+8$ flavors and one tensor multiplet. Compactifying the brane set-up for the 6d theory, we arrive at a 5-brane Tao diagram for 5d $\mathcal{N}=1$ $SU(N+2)$ theory of the vanishing Chern-Simons level with $2N+8$ flavors. We conjecture that the 6d theory is recovered as the UV fixed point of this 5d theory. We show that the global symmetry of this 5d theory is $SO(4N+16)$ identical to that of the 6d theory by analyzing the 7-brane monodromy. By using the Tao diagram, we also find the instanton fugacity is exactly given by the circle radius. By decoupling flavors in this 5d theory, one can obtain all the 5d $SU(N+2)$ gauge theories of various Chern-Simons levels and corresponding enhanced global symmetries at the 5d UV fixed point.

Figures (12)

• Figure 1: Left: Type IIA brane realization of the $(D_{N+4}, D_{N+4})$ minimal conformal matter in the tensor branch. Middle: Another brane realization of the same 6d theory. Right: The quiver diagram of the 6d theory.
• Figure 2: The T-dual description of the type IIA brane setup for 6d $\mathcal{N}=(1,0)$$Sp(N)$ gauge theory with $2N+8$ flavors and one tensor multiplet. (For simplicity, the brane description for the $N=1$ case is drawn.) We denote $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ for $[1,0]$ 7-brane (or D7 brane), $[1,-1]$ 7-brane, and $[1,1]$ 7-brane, respectively. The two $O7$-planes are replaced by the pair of $\mathbf{B}$ and $\mathbf{C}$. The branch cuts due to 7-branes are denoted by the dotted lines.
• Figure 3: A $(p,q)$-brane setup for 5d $SU(N+2)$ gauge theory with $2N+8$ flavors. For concreteness, the $N=1$ case is drawn. The horizontal dotted line indicates the center of mass position of $N+2$ color branes. To measure the gauge coupling, one extends the upper and lower $(1,1)$ and $(1,-1)$ 5-branes to the horizontal dotted line, which gives rise to two asymptotic distances between $\mathbf{P_1}$ and $\mathbf{P_2}$, and between $\mathbf{P_3}$ and $\mathbf{P_4}$. The inverse gauge coupling is then given by the average of these two distances.
• Figure 4: The quiver diagram of the affine $\hat{D}_5$ quiver theory.
• Figure 5: The web diagram which realizes an affine $\hat{D}_5$ Dynkin quiver theory. Q and $\Delta$ assigned to a 5-brane segment in the figure represent $e^{-L}$ where $L$ is the length of the corresponding 5-brane. This can be also seen as a diagram S-dual to the web diagram for $SU(3)$ gauge theory with $10$ flavors.