Higgs and Coulomb branches from vertex operator algebras

We explore the geometric interpretation of the twisted index of 3d N = 4 gauge theories on S 1 × Σ where Σ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of solutions to generalised vortex equations on Σ, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d N = 4 mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters. arXiv:1812.05567v3 [hep-th] 2 Jul 2019 C Characteristic classes on a fixed locus and their integration 60

Section: Discussionmentioning

confidence: 99%

Twisted indices of 3d $$ \mathcal{N} $$ = 4 gauge theories and enumerative geometry of quasi-maps

Bullimore

Ferrari

Kim

2019

“…If the Higgs branch M H happens to be smoothly resolved, then there is a functor from C B to quasi-coherent sheaves on M H itself, modulo a grading shift, which connects with the sigma-model analysis of [20]. The C A category is also very closely related to the category of modules for boundary VOA's of 3d N = 4 theories studied by [68,69]. (The relation is analogous to that of line operators in Chern-Simons theory and modules for WZW [70,71].)…”

Section: Some General Structurementioning

confidence: 99%

“…A version of the category (4.69) recently appeared in work of Costello-Creutzig-Gaiotto on chiral boundary conditions for 3d N = 4 theories [69]. There, it was the category of modules for a boundary VOA.…”

Section: Categorymentioning

confidence: 99%

See 1 more Smart Citation

Mirror symmetry and line operators

Dimofte

Garner

Geracie

et al. 2020

We study half-BPS line operators in 3d N = 4 gauge theories, focusing in particular on the algebras of local operators at their junctions. It is known that there are two basic types of such line operators, distinguished by the SUSY subalgebras that they preserve; the two types can roughly be called "Wilson lines" and "vortex lines," and are exchanged under 3d mirror symmetry. We describe a large class of vortex lines that can be characterized by basic algebraic data, and propose a mathematical scheme to compute the algebras of local operators at their junctions -including monopole operators -in terms of this data. The computation generalizes mathematical and physical definitions/analyses of the bulk Coulombbranch chiral ring. We fully classify the junctions of half-BPS Wilson lines and of half-BPS vortex lines in abelian gauge theories with sufficient matter. We also test our computational scheme in a non-abelian quiver gauge theory, using a 3d-mirror-map of line operators from work of Assel and Gomis.

“…It would be interesting to examine the difference equations satisfied by the supersymmetric indices [10,42,43,30] which correlate with the half-BPS line or surface operators. From the perspective of the VOAs, including half-BPS line or surface operators can produce non-trivial modules in VOAs [44]. The analysis of these modules may be fruitfully addressed by computing the supersymmetric indices as they can be viewed as promotions of characters of these modules in that they reduce to these characters in the special fugacity limits.…”

Section: Future Workmentioning

confidence: 99%

Abelian dualities of $$ \mathcal{N} $$ = (0, 4) boundary conditions

Okazaki

2019

We propose dual pairs of N = (0, 4) half-BPS boundary conditions for 3d N = 4 Abelian gauge theories related to mirror symmetry and S-duality by showing the matching of boundary 't Hooft anomalies and supersymmetric indices. We find simple N = (0, 4) mirror symmetry between 2d N = (0, 4) Abelian gauge theories and free Fermi multiplets that generalizes N = (0, 2) Abelian duality. We also propose a prescription for computing half-index of enriched Neumann boundary condition including 2d boundary bosonic matters by gauging the 2d boundary flavor symmetry of Dirichlet boundary condition. By coupling N = (0, 4) half-BPS boundary configurations of 3d N = 4 gauge theories to quarter-BPS corner configurations of 4d N = 4 Super Yang-Mills theories, we further obtain a new type of 4d-3d-2d duality that may involve 3d non-Abelian gauge symmetry.