Double Yangian, Factorization, and qKZ-equation for Cotangent Lie Algebras

Abstract

In this paper, we construct the dual $Y^*_\hbar(\mathfrak d)$ and double $DY_\hbar (\mathfrak d)$ of the Yangian $Y_\hbar (\mathfrak d)$ associated with a cotangent Lie algebra $\mathfrak d=T^*\mathfrak g$. We define a coherent factorization algebra version of the dual Yangian $Y_\hbar^*(\mathfrak d)^{\mathrm{co-op}}$ with opposite coproduct. Furthermore, we define a quantum vertex algebra structure on the quantum vacuum module $V_{\hbar,k}(\mathfrak d)$ of central extensions $\widehat{DY}_{\hbar,\ell} (\mathfrak d)$ of this double Yangian and show that its conformal blocks satisfy quantum KZ equations. We discuss examples of $\mathfrak d$ that arise from 3d $N=4$ gauge theories via the work of Costello-Gaiotto. These examples include Takiff Lie algebras $T^*\mathfrak g$, whose affine VOA is a large subalgebra of the chiral differential operator algebra of $G$, as well as the smallest type-A Lie superalgebra $\mathfrak{gl} (1|1)$.