An appetizer for supersymmetry enhancement

We conjecture infrared emergent $mathcal{N}=4$ supersymmetry for a class of three-dimensional $mathcal{N}=2$ $U(1)$ gauge theories coupled with a single chiral multiplet. One example is the case where $U(1)$ gauge group has the Chern-Simons level $-frac{3}2$ and the chiral multiplet has gauge charge $+1$. Other examples are related to this example either by known dualities or rescaling the Abelian gauge field. We give three independent evidences for the conjecture: 1) exact match between the central charges of the $U(1)$ R-symmetry current and the $U(1)$ topological symmetry current, 2) semi-classical construction of the $mathcal{N}=4$ stress-tensor multiplet, and 3) an IR duality between a direct product of the two copies of the 3d theory on the one hand, and an $mathcal{N}=4$ theory obtained by gauging the diagonal $SU(2)$ flavor symmetry of the $T[SU(2)]$ theory, on the other. The duality in 3) follows from geometrical aspects of the 3d-3d correspondence.