Coulomb branch quantization and abelianized monopole bubbling

Abstract

We develop an approach to the study of Coulomb branch operators in 3D N = 4 gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For "good" and "ugly" theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.

Non-Abelian mirror symmetry beyond the chiral ring

“…Their correlation functions depend only on the ordering of the insertions. The TQM contains nontrivial information about the operator product expansion (OPE) data of the full SCFT, which can be computed systematically from supersymmetric localization after mapping the TQM to a great S 1 on S 3 [31][32][33], and plays an important role in determining the full OPE data of the SCFT using the conformal bootstrap technique [23,[34][35][36][37]. In recent work [38], these TQMs are formalized as noncommutative associative algebras equipped with an even and positive short star product -equivalently, a (twisted) trace or bilinear form.…”

“…The latter is essential for mapping the TQM data to CFT correlators. The action of mirror symmetry in the TQM sectors has been studied to a limited extent in [31][32][33], focusing mainly on the case where the corresponding SCFTs arise from abelian gauge theories such as SQED and abelian quivers. 3 The bootstrap analysis for the particular case of SQED 2 , or equivalently the T [SU (2)] theory, was carried out in [37], where nontrivial evidence for the (self-)mirror symmetry beyond the TQM sector was found.…”

“…Some details of our Higgs branch TQM computations, which tend to be more convoluted than their Coulomb branch counterparts, are gathered in Appendix A. In Appendix B, we consider additional quantizations of the A-and D-type singularities via field theory (outside the context of mirror symmetry), generalizing some examples from [32,33]. 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.…”

“…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 ).…”

“…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 ).…”