Geometric aspects of twisted 3d supersymmetric gauge theories

Thesis

We study geometric aspects of 3d N = 2 and N = 4 supersymmetric gauge theories on the product of a line and a Riemann surface. Performing the topological twist on the Riemann surface, the theories preserve a supersymmetric quantum mechanics on the line. The quantum mechanics has an effective description where its target space is a moduli space of configurations that satisfy generalized vortex equations on the Riemann surface. We propose a construction of the space of supersymmetric ground states of selected N = 2 theories as a graded vector space in terms of a certain cohomology of the moduli spaces. This exhibits a rich dependence on deformation parameters compatible with the topological twist, including superpotentials, real mass parameters, and background vector bundles associated to flavour symmetries. By matching spaces of supersymmetric ground states, we perform new checks of 3d abelian mirror symmetry. We go on to the study of the twisted indices of a 3dN = 4 quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. These can be viewed as virtual Euler characteristics of the moduli spaces of generalized vortex equations, which in this case can be understood algebraically as quasi-maps to the Higgs branch. We demonstrate that this description agrees with the contour integral representation introduced in previous work. We then 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.

Authors: Ferrari, A

• DOI: 10.5287/ora-mzvvrddxj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: 3d; Topological; Hilbert space; Quasi-maps; Riemann surface; Vortex equations
• Subjects: Mathematical physics -- supersymmetry and geometry