Quantum groupoids from moduli spaces of $G$-bundles

Abstract

In a previous work, we have constructed the Yangian $Y_\hbar (\mathfrak{d})$ of the cotangent Lie algebra $\mathfrak{d}=T^*\mathfrak{g}$ for a simple Lie algebra $\mathfrak{g}$, from the geometry of the equivariant affine Grassmanian associated to $G$ with $\mathfrak{g}=\mathrm{Lie}(G)$. In this paper, we construct a quantum groupoid $\Upsilon_\hbar^\sigma (\mathfrak{d})$ associated to $\mathfrak{d}$ over a formal neighbourhood of the moduli space of $G$-bundles and show that it is a dynamical twist of $Y_\hbar(\mathfrak{d})$. Using this dynamical twist, we construct a dynamical quantum spectral $R$-matrix, which essentially controls the meromorphic braiding of $\Upsilon_\hbar^\sigma (\mathfrak{d})$. This construction is motivated by the Hecke action of the equivariant affine Grassmanian on the moduli space of $G$-bundles in the setting of coherent sheaves. Heuristically speaking, the quantum groupoid $\Upsilon_\hbar^\sigma (\mathfrak{d})$ controls this action at a formal neighbourhood of a regularly stable $G$-bundle. From the work of Costello-Witten-Yamazaki, it is expected that this Hecke action should give rise to a dynamical integrable system. Our result gives a mathematical confirmation of this and an explicit $R$-matrix underlying the integrability.