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

Abstract

In this paper, we construct the dual $Y_{\hslash }^{\ast }(\mathfrak{d})$ and double $D{Y}_{\hslash }(\mathfrak{d})$ of the Yangian $Y_{\hslash }(\mathfrak{d})$ associated with a cotangent Lie algebra $\mathfrak{d}=T^{\ast }\mathfrak{g}$. We define a coherent factorization algebra version of the dual Yangian $Y_{\hslash }^{\ast }(\mathfrak{d}{)}^{\mathrm{co}-\mathrm{op}}$ with opposite coproduct. Furthermore, we define a quantum vertex algebra structure on the quantum vacuum module $V_{\hslash ,k}(\mathfrak{d})$ of central extensions ${\widehat{DY}}_{\hslash ,\ell }(\mathfrak{d})$ 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 $T^{\ast }\mathfrak{g}$, whose affine VOA is a large subalgebra of the chiral differential operator algebra of $G$, as well as the smallest type-A Lie superalgebra $\mathfrak{gl}(1|1)$.