Chiralization of Quiver Varieties

Abstract

Given a quiver Q with gauge dimension $\mathbf{v}$ and framing dimension $\mathbf{w}$, one can define the extended quiver variety $\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w})$, which is a smooth family of deformations of the Nakajima quiver variety $\mathcal{M}(\mathbf{v},\mathbf{w})$. In this paper we discuss two vertex algebras which chiralize the geometry $\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w})$. We construct a sheaf of $\hslash$-adic vertex superalgebras ${\mathcal{D}}_{\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w}),\hslash }^{\mathrm{ch}}$ on $\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w})$ which quantizes the jet bundle of $\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w})$, and define a vertex algebra ${\mathsf{D}}^{\mathrm{ch}}(\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w}))$ to be the $\hslash =1$ specialization of the ${\mathbb{C}}^{\times }$-finite part of the vector space of global sections $\hat{I}“(\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w}),{\mathcal{D}}_{\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w}),\hslash }^{\mathrm{ch}})$. We define another vertex superalgebra $\mathcal{V}(\mathbf{v},\mathbf{w})$ by BRST reduction of the tensor product of the ${\hat{I}}^2{\hat{I}}^{3}bc$-system and Heisenberg VOA associated to the quiver Q, and show that there exists a natural vertex superalgebra map from $\mathcal{V}(\mathbf{v},\mathbf{w})$ to ${\mathsf{D}}^{\mathrm{ch}}(\tilde{\mathcal{M}}(\mathbf{v},\mathbf{w}))$. Under certain technical assumptions, we prove that the negative degree BRST cohomologies of the tensor product of ${\hat{I}}^2{\hat{I}}^{3}bc$-systems and Heisenberg VOA associated to the quiver Q are zero, and under stronger assumptions, that the aforementioned vertex superalgebra map is injective. Physically, the vertex superalgebra $\mathcal{V}(\mathbf{v},\mathbf{w})$ is closely related to the boundary VOA of the H-twisted 3D $\mathcal{N}=4$ quiver gauge theory associated to the quiver Q with gauge and framing dimension vectors $\mathbf{v}$ and $\mathbf{w}$.

94 pages, 4 figures