Non-Abelian mirror symmetry beyond the chiral ring

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 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.2 A variation of the Ω-background [25][26][27] leads to related deformation quantizations of the Higgs and Coulomb branches [28][29][30]. It would be interesting to spell out the explicit relation to the TQM sector.3 Specifically, the abelian A-type mirror symmetries were analyzed in [32], and a simple N = 8 nonabelian A-type mirror symmetry was analyzed in [33]. The D 3 case was also discussed in Appendix F.2 of [33], but this belongs to the A-series (D 3 ∼ = A 3 ).

Three dimensional mirror symmetry beyond ADE quivers and Argyres-Douglas theories

“…In the literature, mirror symmetry for quiver gauge theories of the ADE type (and their affine cousins) has been studied in a lot of detail [17][18][19][20][21][22][23]. This is essentially due to the fact that these theories have well-known realization in Type IIB String Theory [24][25][26][27] and M-Theory [17,28,29].…”

“…The first set involves gauging operations, while the second involves flavoring-gauging operations. The resultant dual pairs have appeared in the literature of 3d mirror symmetry [18,19,23], and were studied from a Type IIB point of view in [25,27]. We present them here to illustrate how the Abelian S-type operations work in simple examples involving non-Abelian gauge groups.…”

“…23) where R k are irreducible representations labelled by the Dynkin labels k, and χ[R k (G)](µ i ) denotes the associated character as a function of the fugacities {µ i }. Therefore, a series expansion of the Coulomb and the Higgs indices in terms of x and x respectively can be used to read off the global symmetries of the respective branches.B.2 3d N = 4 mirror symmetryThe basic example of 3d N = 4 mirror symmetry involves an SQED with a single hypermultiplet on one side and a twisted hypermultiplet on the other.…”

Algebras, traces, and boundary correlators in $$ \mathcal{N} $$ = 4 SYM

“…The algebras themselves can be identified either in the SCFT context [4,5], or using the Omega-background [12][13][14] applied to 3d N = 4 theories [24][25][26][27][28] (which uses Omega-deformations of the A and B models [24,29,30]), or from the S 3 supersymmetric background [6][7][8]. For some recent progress on the latter approach see [9,[31][32][33][34][35][36][37][38]. In principle, this list may continue, as any background that is "equivariant" in the appropriate sense can be used for this purpose: for example, the 1d sector construction was recently extended to other backgrounds, such as S 2 × S 1 , see [39].…”

Topological correlators and surface defects from equivariant cohomology

“…The same approach was later extended to the Coulomb-branch sector through mirror symmetry [6,7], and to five dimensions [8]. For other recent developments, see also [9][10][11].…”