Domain walls in 4d $$ \mathcal{N} $$ = 1 SYM

Abstract

4d$$ \mathcal{N} $$ 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 G2, 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.

Global anomalies on the Hilbert space

“…Chern-Simons theory. Using the formulae in [61] for the number of bosonic and fermionic states in H R-R of this Chern-Simons theory 11 we find that the spectrum is indeed supersymmetric! Our argument applies to other gauge theories with higher rank real representations and is a nontrivial prediction of their spectrum.…”

“…Ultimately, we want to have a procedure to associate a partition function to a manifold equipped with appropriate structures. First, we can triangulate the manifold, equipping it with a discretization of the various structures we want to endow it with: a flat connection along the edges of the triangulation, some discrete version of the spin structure and 11 In [61] it was shown that SO(2n+1) 2k+1 Chern-Simons theory has n+k−1 k−1 bosons and n+k−1 k fermions in the Hilbert space H R-R . The spectrum is supersymmetric for n = k, when the theory is time-reversal invariant.…”

“…This SPT has a unique state on any spin structure Hilbert space H XY . The fermion parity of this state is known to be (−1) F = (−1) Arf(T 2 )ν/2 (see [21,61] and appendix B.2). Therefore, the T-invariant combined system ψ ν × SO(ν/2) −1 has a time-reversal algebra…”

Mildly flavoring domain walls in Sp(N) SQCD

“…Acharya and Vafa [5] studied the domain walls of pure SYM for SU(N ) gauge group, proposing an appropriate TQFT as the 3d effective description of the k-wall. SYM's with other gauge groups were considered in [7,26,27]. [7] studied domain walls in SQCD with flavors, with SU(N ) and Sp(N ) gauge group and number of flavors less than h, the dual Coxeter number of the gauge algebra.…”

“…See table 1. (See also [7,27] for computation of Witten Indexes in TQFT's and in NLSM's on Grassmannians. )…”