Abstract
Mirror symmetry is a type of infrared duality in 3D quantum field theory that relates the low-energy dynamics of two distinct ultraviolet descriptions. Though first discovered in the supersymmetric context, it has far-reaching implications for understanding nonperturbative physics in general 3D quantum field theories. We study mirror symmetry in 3D $\mathcal{N}=4$ supersymmetric field theories whose Higgs or Coulomb branches realize $D$- and $E$-type Kleinian singularities in the $ADE$ classification, generalizing previous work on the $A$-type case. Such theories include the $SU(2)$ gauge theory coupled to fundamental matter in the $D$-type case and non-Lagrangian generalizations thereof in the $E$-type case. In these cases, the mirror description is given by a quiver gauge theory of affine $D$- or $E$-type. We investigate the mirror map at the level of the recently identified 1D protected subsector described by topological quantum mechanics, which implements a deformation quantization of the corresponding $ADE$ singularity. We give an explicit dictionary between the monopole operators and their dual mesonic operators in the $D$-type case. Along the way, we extract various operator product expansion (OPE) coefficients for the quantized Higgs and Coulomb branches. We conclude by offering some perspectives on how the topological subsectors of the $E$-type quivers might shed light on their non-Lagrangian duals.
Highlights
Three-dimensional gauge theories are strongly coupled at low energies due to the positive mass dimension of the Yang-Mills coupling
When the superconformal field theory (SCFT) arises from an renormalization group (RG) flow with a Lagrangian description in the UV, the Higgs branch sector is directly accessible by supersymmetric localization [31], but the Coulomb branch sector includes monopole operators, which are disorder operators that cannot be represented in terms of the Lagrangian fields [32,33]
In [32,33], a method for computing all observables within the Coulomb branch topological quantum mechanics (TQM) was obtained for 3D N 1⁄4 4 gauge theories of cotangent-type by constructing a set of “shift operators,” acting on functions of σ ∈ t and B ∈ Λ∨W, whose algebra is a representation of the 1D operator product expansion (OPE).7
Summary
Three-dimensional gauge theories are strongly coupled at low energies due to the positive mass dimension of the Yang-Mills coupling. Supersymmetry (SUSY), especially the localization method, allows us to extract nontrivial dynamical data from quantum field theories, such as their protected operator spectrum, low-energy effective action, and supersymmetric partition functions, which all play important roles in testing and refining the duality maps. The primary goal of this paper is to carry out the analysis of mirror symmetry at the level of the TQM for the D- and E-type cases, where the relevant (mirror) gauge theories are non-Abelian. We carry out a similar deformation quantization of E-type singularities in Sec. VI and, in Sec. VII, present some motivating remarks toward understanding the nonLagrangian theories whose Coulomb branches realize (1.1) with g 1⁄4 En through the lens of the TQM. In Appendix C, we give a self-contained exposition of the Higgs branch chiral rings of the affine D- and E-type quivers, filling some gaps in the literature
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