The circle quantum group and the infinite root stack of a curve (with an appendix by Tatsuki Kuwagaki)

In the present paper, we give a definition of the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ of the circle $S^1\colon =\mathbb{R}/\mathbb{Z}$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ associated to the rational circle $S^1_\mathbb{Q}\colon= \mathbb{Q}/\mathbb{Z}$ as a direct limit of $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(n))$'s, where the order is given by divisibility of positive integers. The quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $X_\infty$ over a fixed smooth projective curve $X$ defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $g_X$, of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. Moreover, we show that $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(+\infty))$ and $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(\infty))$ are subalgebras of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. As proved by T. Kuwagaki in the appendix, the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $S^1$.