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Vortices and Vermas

Bullimore, Mathew; Dimofte, Tudor; Gaiotto, Davide; Hilburn, Justin; Kim, Hee-Cheol

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Tudor Dimofte

Davide Gaiotto

Justin Hilburn

Hee-Cheol Kim


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.


Bullimore, M., Dimofte, T., Gaiotto, D., Hilburn, J., & Kim, H. (2018). Vortices and Vermas. Advances in Theoretical and Mathematical Physics, 22(4), 803-917.

Journal Article Type Article
Acceptance Date Jun 14, 2018
Online Publication Date Dec 5, 2018
Publication Date 2018
Deposit Date Jul 2, 2018
Publicly Available Date May 1, 2019
Journal Advances in Theoretical and Mathematical Physics
Print ISSN 1095-0761
Electronic ISSN 1095-0753
Publisher International Press
Peer Reviewed Peer Reviewed
Volume 22
Issue 4
Pages 803-917
Related Public URLs


Accepted Journal Article (1.2 Mb)

Copyright 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.

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