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Vortices and Vermas.

Bullimore, Mathew and Dimofte, Tudor and Gaiotto, Davide and Hilburn, Justin and Kim, Hee-Cheol (2018) 'Vortices and Vermas.', Advances in theoretical and mathematical physics., 22 (4). pp. 803-917.


In three-dimensional gauge theories, monopole operators create and destroy vortices. We explore this idea in the context of 3d N=4 gauge theories in the presence of an Ω-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 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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Publisher statement:Copyright © International Press. First published in Bullimore, Mathew, Dimofte, Tudor, Gaiotto, Davide, Hilburn, Justin & Kim, Hee-Cheol (2018). Vortices and Vermas. Advances in Theoretical and Mathematical Physics 22(4): 803-917 published by International Press.
Date accepted:14 June 2018
Date deposited:02 July 2018
Date of first online publication:05 December 2018
Date first made open access:01 May 2019

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