Domain Walls in 4d N=1 Supersymmetric Yang-Mills

TL;DR

This paper determines the low-energy topological quantum field theories on domain walls in 4d N=1 super Yang-Mills for various gauge groups, providing explicit proposals and evidence through partition function matching.

Contribution

It explicitly identifies the TQFTs describing domain walls in 4d N=1 SYM for multiple gauge groups, extending previous conjectures with detailed constructions and evidence.

Findings

Proposes specific TQFTs for domain walls in various gauge groups.

Matches UV and IR partition functions to support the TQFT proposals.

Constructs the Hilbert space of spin TQFTs with fermionic states.

Abstract

$4d$ ${\mathcal N}=1$ super Yang-Mills (SYM) with simply connected gauge group $G$ has $h$ gapped vacua arising from the spontaneously broken discrete $R$-symmetry, where $h$ is the dual Coxeter number of $G$. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for $G=SU(N),Sp(N), Spin(N)$ and $G_2$, and for the minimal domain wall connecting neighboring vacua for arbitrary $G$. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for $SU(N)$. We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states -- a subject we delve into in some detail.