Kavli Affiliate: Masahito Yamazaki
| Summary:
Given a quiver Q with gauge dimension $bf v$ and framing dimension $bf w$, one can define the extended quiver variety $widetildemathcal M(mathbf v,mathbf w)$, 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 $widetildemathcal M(mathbf v,mathbf w)$. We construct a sheaf of $hbar$-adic vertex superalgebras $mathscr D^mathrmch_widetildemathcal M(mathbf v,mathbf w),hbar$ on $widetildemathcal M(mathbf v,mathbf w)$ which quantizes the jet bundle of $widetildemathcal M(mathbf v,mathbf w)$, and define a vertex algebra $mathsf D^mathrmch(widetildemathcal M(mathbf v,mathbf w))$ to be the $hbar=1$ specialization of the $mathbb C^times$-finite part of the vector space of global sections $Γ(widetildemathcal M(mathbf v,mathbf w), mathscr D^mathrmch_widetildemathcal M(mathbf v,mathbf w),hbar)$. We define another vertex superalgebra $mathcal V(mathbf v,mathbf w)$ by BRST reduction of the tensor product of the $βγbc$-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 $mathsf D^mathrmch(widetildemathcal M(mathbf v,mathbf w))$. Under certain technical assumptions, we prove that the negative degree BRST cohomologies of the tensor product of $βγbc$-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 $bf v$ and $bf w$.
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