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 analog of that result. In their article [?], 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—i.e., its tangent is a Tate module : it is infinite dimensional but behaves nicely enough regarding duality. We also define the bubble space $\mathfrak {B} ^d(X)$, a variation of the loop space. We prove that $\mathfrak {B} ^d(X)$ is endowed with a natural symplectic form as soon as $X$ has one (in the sense of [?]). Throughout this paper, we will use the tools of $(\infty ,1)$-categories and symplectic derived algebraic geometry.