Vortices and Vermas

TL;DR

This work builds a detailed bridge between three-dimensional $\mathcal{N}=4$ gauge theories in an $\Omega$-background and two-dimensional vortex dynamics by treating monopole operators as interfaces that connect vortex quantum-mechanics sectors. The authors show that the Hilbert space of supersymmetric ground states, realized as the equivariant cohomology of vortex moduli spaces, carries a Verma-module structure for the quantized Coulomb-branch algebra $\mathbb{C}_\epsilon[\mathcal{M}_C]$, providing a new, intrinsic construction of the Coulomb-branch algebra via correspondences. By incorporating half-BPS boundary conditions, they construct vortex partition functions as overlaps of generalized Whittaker vectors for these algebras, presenting a finite-version of the AGT correspondence and connecting to Braverman–Feigin–Finkelberg–Nakajima’s framework. The paper also extends these ideas to 3d linear quivers with handsaw-quiver realizations of vortex moduli spaces and interprets monopole operators as interfaces between handsaw-quiver quantum mechanics, thereby unifying geometric, algebraic, and physical perspectives on Coulomb-branch quantization and vortex dynamics. The abelian case and triangular/quiver examples illustrate explicit realizations of the Coulomb-branch algebra, its Verma-module action, and the boundary-state/Wittaker structures, highlighting the role of symplectic duality and the connection to finite AGT through Whittaker overlaps.

Abstract

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 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 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.

Figures (14)

• Figure 1: A 3d $\mathcal{N}=4$ theory in the $\Omega$-background, with a fixed vacuum $\nu$ at spatial infinity.
• Figure 2: A 3d $\mathcal{N}=4$ theory in the $\Omega$-background, with a boundary condition.
• Figure 3: A 3d $\mathcal{N}=4$ theory in the $\Omega$-background, sandwiched between half-BPS boundary conditions.
• Figure 4: A triangular, linear quiver.
• Figure 5: Modification of the holomorphic data at $z=p$ and $t=t_1$.