3d-3d Correspondence for Seifert Manifolds

Abstract

In this dissertation, we investigate the 3d-3d correspondence for manifolds. This correspondence, originating from string and M-theory, relates the dynamics of three-dimensional quantum field theories with the geometry of three-manifolds. We first start in Chapter II with the simplest cases and demonstrate the extremely rich interplay between geometry and physics even when the manifold is just a direct product. In this particular case, by examining the problem from various vantage points, we generalize the celebrated relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) quantization of Chern-Simons and 4) the of the moduli space of flat connections to a completely new set of relations between 1) the Verlinde for a complex group, 2) the equivariant quantum K-theory of the vortex moduli space, 3) quantization of complex Chern-Simons and 4) the equivariant of the moduli space of Higgs bundles. In Chapter III we move one step up in complexity by looking at the next simplest three-manifolds---lens spaces. We test the 3d-3d correspondence for theories that are labeled by spaces, reaching a full agreement between the of the 3d N=2 lens space theory and the partition function of complex Chern-Simons on the space. The two different types of manifolds studied in the previous two chapters also have interesting interactions. We show in Chapter IV the equivalence between two seemingly distinct 2d TQFTs: one comes from the Coulomb branch index of the class S on a space, the other is the Verlinde formula. We check this relation explicitly for SU(2) and demonstrate that the SU(N) equivariant Verlinde algebra can be derived using field via (generalized) Argyres-Seiberg dualities. In the last chapter, we directly jump to the most general situation, giving a proposal for the 3d-3d correspondence for an arbitrary manifold. We remark on the huge class of novel dualities relating different descriptions of the Seifert theory associated with the same manifold and suggest ways that our proposal could be tested.