Higher dimensional formal loop spaces

Abstract

If $M$ is a symplectic manifold then the space of smooth loops $\mathrm C^{\infty}(\mathrm S^1,M)$ inherits of a quasi-symplectic form. We will focus in this article on an algebraic analogue of that result. In 2004, Kapranov and Vasserot introduced and studied the formal loop space of a scheme $X$. We generalize their construction to higher dimensional loops. To any scheme $X$ -- not necessarily smooth -- we associate $\mathcal L^d(X)$, the space of loops of dimension $d$. We prove it has a structure of (derived) Tate scheme -- ie its tangent is a Tate module: it is infinite dimensional but behaves nicely enough regarding duality. We also define the bubble space $\mathcal B^d(X)$, a variation of the loop space. We prove that $\mathcal B^d(X)$ is endowed with a natural symplectic form as soon as $X$ has one (in the sense of [PTVV]). Throughout this paper, we will use the tools of $(\infty,1)$-categories and symplectic derived algebraic geometry.