PIRSA:19030107

Twisted 3d supersymmetric gauge theories and the enumerative geometry of quasi-maps

APA

Ferrari, A. (2019). Twisted 3d supersymmetric gauge theories and the enumerative geometry of quasi-maps. Perimeter Institute. https://pirsa.org/19030107

MLA

Ferrari, Andrea. Twisted 3d supersymmetric gauge theories and the enumerative geometry of quasi-maps. Perimeter Institute, Mar. 18, 2019, https://pirsa.org/19030107

BibTex

          @misc{ pirsa_PIRSA:19030107,
            doi = {10.48660/19030107},
            url = {https://pirsa.org/19030107},
            author = {Ferrari, Andrea},
            keywords = {Mathematical physics},
            language = {en},
            title = {Twisted 3d supersymmetric gauge theories and the enumerative geometry of quasi-maps},
            publisher = {Perimeter Institute},
            year = {2019},
            month = {mar},
            note = {PIRSA:19030107 see, \url{https://pirsa.org}}
          }
          

Abstract

I discuss a geometric interpretation of the twisted indexes of 3d (softly broken) $\cN=4$ gauge theories on $S^1 \times \Sigma$ where $\Sigma$ is a closed genus $g$ Riemann surface, mainly focussing on quivers with unitary gauge groups. The path integral localises to a moduli space of solutions to generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. I demonstrate that the twisted indexes computed in previous work reproduce the virtual Euler characteristic of the moduli spaces of twisted quasi-maps. I investigate 3d $\cN=4$ 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. I will conclude with some remarks about how holomorphic Morse theory can be used to access the spaces of supersymmetric ground states in limits where $\cN=4$ supersymmetry is fully restored. These spaces of ground states may be related to the spaces of conformal blocks for the VOAs introduced by Costello and Gaiotto.