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# Advances in Theoretical and Mathematical Physics

## Volume 22 (2018)

### Number 4

### Vortices and Vermas

Pages: 803 – 917

DOI: http://dx.doi.org/10.4310/ATMP.2018.v22.n4.a1

#### Authors

#### Abstract

In three-dimensional gauge theories, monopole operators create and destroy vortices. We explore this idea in the context of 3d $\mathcal{N} = 4$ gauge theories in the presence of an $\Omega$-background. In this case, monopole operators generate a non-commutative algebra that quantizes the Coulomb-branch chiral ring. The monopole operators act naturally on a Hilbert space, which is realized concretely as the equivariant cohomology of a moduli space of vortices. The action furnishes the space with the structure of a Verma module for the Coulomb-branch algebra. This leads to a new mathematical definition of the Coulomb-branch algebra itself, related to that of Braverman–Finkelberg–Nakajima. By introducing additional boundary conditions, we find a construction of vortex partition functions of 2d $\mathcal{N} = (2, 2)$ theories as overlaps of coherent states (Whittaker vectors) for Coulomb-branch algebras, generalizing work of Braverman–Feigin–Finkelberg–Rybnikov on a finite version of the AGT correspondence. In the case of 3d linear quiver gauge theories, we use brane constructions to exhibit vortex moduli spaces as handsaw quiver varieties, and realize monopole operators as interfaces between handsaw-quiver quantum mechanics, generalizing work of Nakajima.

We are grateful to many friends and colleagues for stimulating discussions on the subjects of this paper, in particular David Ben-Zvi, Alexander Braverman, Kevin Costello, Michael Finkelberg, Sergei Gukov, Lotte Hollands, Joel Kamnitzer, Gregory Moore, Hiraku Nakajima, Daniel Park, Ben Webster, and Alex Weekes. This paper also benefited greatly from interaction at the workshop “Symplectic Duality and Gauge Theory” at the Perimeter Institute, supported in part by the John Templeton Foundation.

The work of M.B. was supported by ERC Starting Grant no. 306260 ‘Dualities in Supersymmetric Gauge Theories, String Theory and Conformal Field Theories’. The work of T.D. was supported in part by ERC Starting Grant no. 335739 “Quantum fields and knot homologies”, funded by the European Research Council under the European Union’s Seventh Framework Programme, and by the Perimeter Institute for Theoretical Physics. The work of D.G. was supported by the Perimeter Institute for Theoretical Physics. The work of J.H. was supported in part by the Visiting Graduate Fellowship Program at the Perimeter Institute for Theoretical Physics. The work of H.K. was supported in part by NSF grant PHY-1067976 and by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.