Coulomb Branch Operators and Mirror Symmetry in Three Dimensions

Abstract

We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional $\mathcal{N} = 4$ abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a one-dimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the ($n\leq 3$)-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere $HS^3$ with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the $HS^3$ wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on $S^3$ with an arbitrary number of operator insertions. Finally, we show that our results can be recovered by dimensionally reducing the Schur index of 4D $\mathcal{N} = 2$ theories decorated by BPS 't Hooft-Wilson loops.

Figures (2)

• Figure 1: A schematic 2D representation of $S^3$ given by $X_1^2+X_2^2+X_3^2+X_4^2=r^2$. The 1D TQFT lives on the $S^1$ defined by $X_1=X_2=0$ (red) and parametrized by the angle $\varphi$. The $S^3$ can be cut into two hemispheres $HS^3_{\pm}\cong B^3$ whose boundary forms an $S^2=\partial HS^3_{\pm}$ (blue circle) defined by $X_4=0$. The 1D TQFT circle intersects this $S^2$ at two points identified with its North ($N$) and South ($S$) poles.
• Figure 2: The $U(1)^N/U(1)$ gauge theory that is mirror dual to SQED$_N$.