论文标题
较高的正式循环空间
Higher dimensional formal loop spaces
论文作者
论文摘要
如果$ m $是一种符号歧管，则光滑循环的空间$ \ mathrm c^{\ infty}（\ mathrm s^1，m）$ senartions quasi-syplectic形式的继承。我们将重点介绍该结果的代数类似物。 2004年，卡普拉诺夫（Kapranov）和瓦塞洛特（Vasserot）介绍并研究了计划$ x $的正式环空间。 我们将它们的结构推广到更高的循环。对于任何方案$ x $ - 不一定平滑 - 我们将$ \ Mathcal l^d（x）$，尺寸$ d $的循环空间。我们证明它具有（派生的）泰特方案的结构 - 即其切线是一个泰特模块：它是无限的维度，但在二元性方面的行为却足够好。我们还定义了气泡空间$ \ MATHCAL B^D（X）$，这是环路空间的变体。我们证明$ \ Mathcal b^d（x）$在$ x $具有一种（从[ptvv]的意义上）赋予自然符号形式。 在整个本文中，我们将使用$（\ infty，1）$类别和符号派生的代数几何的工具。
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.
