Comments on QCD$_3$ and anomalies with fundamental and adjoint matter

TL;DR

This work analyzes 3d SU($N$) gauge theories with a massless Majorana fermion in the adjoint and fundamental matter, uncovering mixed flavor–$T$ parity anomalies and a mod-16 time-reversal obstruction that constrain IR dynamics. The authors develop a framework using the flavor group $F= ext{U}(N_f)/ ext{Z}_N$, define mod-2 indices $I_1$ and $I_2$, and compute mod-16 indices $ u_T$ and $ u_{T'}$ via anomaly inflow to a 4d SPT bulk. They show that both fermionic and scalar fundamental matter yield the same parity anomaly signature $I_1=0$, $I_2=1$, necessitating IR phases that saturate these anomalies, such as free Dirac fermions charged under $F$ plus neutral Majoranas. The results have implications for the IR phase structure of minimally supersymmetric 3d theories, inform large-$N$ behavior, and connect to analogous4d anomaly structures and domain-wall dynamics.

Abstract

't Hooft anomaly matching is powerful for constraining the low energy phases of gauge theories. In 3d one common anomaly is the parity anomaly in a $T$-symmetric theory where one cannot gauge the global symmetry group without breaking the time-reversal symmetry. We find that a $T$-symmetric $\text{SU}(N)$ gauge theory with either fermionic or bosonic matter in the fundamental representation of the gauge group has a parity anomaly between the flavor group and $T$-symmetry provided that there is also a massless Majorana fermion in the adjoint representation of the gauge group. We then analyze the parity anomaly in this theory, together with the more recent mod 16 time-reversal anomaly, and give some free fermion proposals as candidates for the low energy phases consistent with the anomalies. We make brief comments about the large $N$ limit and the $\T$-broken regimes in the conclusion as well as related anomalies in 4d. }