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

In this paper, we construct the dual $Y_{\mathrm{\hslash }}^{\ast }(\mathfrak{d})$ and double $D{Y}_{\mathrm{\hslash }}(\mathfrak{d})$ of the Yangian $Y_{\mathrm{\hslash }}(\mathfrak{d})$ associated with a cotangent Lie algebra $\mathfrak{d}=T^{\ast }\mathfrak{g}$. We define a coherent factorization algebra version of the dual Yangian $Y_{\mathrm{\hslash }}^{\ast }(\mathfrak{d}{)}^{\mathrm{c}\mathrm{o}-\mathrm{o}\mathrm{p}}$ with opposite coproduct. Furthermore, we define a quantum vertex algebra structure on the quantum vacuum module $V_{\mathrm{\hslash },k}(\mathfrak{d})$ of central extensions ${\widehat{DY}}_{\mathrm{\hslash },\mathrm{\ell }}(\mathfrak{d})$ 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 $\mathfrak{g}\mathfrak{l}(1\mathrm{\mid }1)$.