Dualities of Adjoint QCD$_3$ from Branes

Abstract: We consider an 'electric' $U(N)$ level $k$ QCD$3$ theory with one adjoint Majorana fermion. Inspired by brane dynamics, we suggest that for $k \ge N/2$ the massive $m<0$ theory, in the vicinity of the supersymmetric point, admits a $U(k-\frac{N}{2}){-(\frac{1}{2}k+\frac{3}{4}N),-(k+\frac{N}{2})}$ 'magnetic' dual with one adjoint Majorana fermion. The magnetic theory flows in the IR to a topological $U(k-\frac{N}{2}){-N,-(k+\frac{N}{2})}$ pure Chern-Simons theory in agreement with the dynamics of the electric theory. When $k<N/2$ the magnetic dual is $U(\frac{N}{2}-k){\frac{1}{2}k+\frac{3}{4}N,N}$ with one adjoint Majorana fermion. Depending on the sign of the fermion mass, the magnetic theory flows to either $U(\frac{N}{2}-k){N,N}$ or $U(\frac{N}{2}-k){\frac{1}{2}N+k,N}$ TQFT. A second magnetic theory, $U(N/2+k){\frac{1}{2}k-\frac{3}{4}N,-N}$, flows to either $U(\frac{N}{2}+k){-N,-N}$ or $U(\frac{N}{2}+k)_{-(\frac{1}{2}N-k),-N}$ TQFT. Dualities for $SO$ and $USp$ theories with one adjoint fermion are also discussed.