Twisted Hilbert spaces of 3d supersymmetric gauge theories

Abstract

We study aspects of 3d N = 2 supersymmetric gauge theories on the product of a line and a Riemann surface. Performing a topological twist along the Riemann surface leads to an effective supersymmetric quantum mechanics on the line. We propose a construction of the space of supersymmetric ground states as a graded vector space in terms of a certain cohomology of generalized vortex moduli spaces on the Riemann surface. 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.

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

“…Let us first consider the cohomology of the two complexes on the left hand side. The maps α 0 and α 1 are injective and surjective respectively and therefore the cohomology can be written as 13) which corresponds to the tangent space of the symmetric product at the point D. It follows that some of the massless fermionic fluctuations at the point D encoded by the complexes span the tangent space to the bosonic moduli space, as expected [42]. By Serre duality, it is then easy to see that the combination of the two complexes on the right of (5.12) define the contangent space T * M m over the moduli space.…”

“…A second interesting extension would be to introduce background vector bundles for the flavour symmetries G H and G C on Σ. This is relatively straightforward for G H , but in the case of G C these are expected to induce vector bundles on the moduli spaces of solutions themselves [42], and it would be of great mathematical appeal to uncover generalisations of this phenomenon. It is also possible to study the inclusion of line operators wrapping S 1 , which are similar to the above.…”

“…Finally, it is possible to fully embrace the Hamiltonian point of view and consider the theories on R × Σ as quantum mechanics on R. The space of supersymmetric ground states of these quantum mechanics would provide a categorification of the enumerative invariants we have studied here. First steps in this program were taken in [42]. In the limit t → 1, we expect a close connection to conformal blocks for the vertex operator algebras introduced in [45,46] including in the presence of line operators [47].…”

Factorisation of 3d $$ \mathcal{N} $$ = 4 twisted indices and the geometry of vortex moduli space

“…Protected indices in supersymmetric QFT often have an interpretation in terms of the geometry of an appropriate moduli space of classical solutions. In recent work, Bullimore et al [5,6], have suggested such an interpretation for the twisted index. In this paper we will confirm this proposal and extend it in several ways.…”

3d-3d correspondence for mapping tori

“…Using modern terminology, one might call H(F g ) a categorification of the A-model. A chiral multiplet of R-charge R contributes to this Q A -cohomology a boson φ and a fermion ψ, which after the twist transform as holomorphic sections of K [82,91,93,94]:…”