Higher dimensional formal loop spaces

Hennion, Benjamin

doi:10.24033/asens.2329

Higher dimensional formal loop spaces
[Espaces des lacets formels de dimension supérieure]

Hennion, Benjamin

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 609-663.

Résumé
Abstract

L'espace des lacets lisses $C^{\infty} (S^{1}, M)$ associé à une variété symplectique $M$ se voit doté d'une structure (quasi-)symplectique induite par celle de $M$ . Nous traiterons dans cet article d'un analogue algébrique de cet énoncé. Dans leur article [14], Kapranov et Vasserot ont introduit l'espace des lacets formels associé à un schéma.

Nous généralisons leur construction à des lacets de dimension supérieure. Nous associons à tout schéma $X$ — pas forcément lisse — l'espace $ℒ^{d} (X)$ de ses lacets formels de dimension $d$ . Nous démontrerons que ce dernier admet une structure de schéma (dérivé) de Tate : son espace tangent est de Tate : de dimension infinie mais suffisamment structuré pour se soumettre à la dualité. Nous définirons également l'espace $𝔅^{d} (X)$ des bulles de $X$ , une variante de l'espace des lacets, et nous montrerons que le cas échéant, il hérite de la structure symplectique de $X$ .

If $M$ is a symplectic manifold then the space of smooth loops $C^{\infty} (S^{1}, M)$ inherits of a quasi-symplectic form. We will focus in this article on an algebraic analog of that result. In their article [14], 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 $ℒ^{d} (X)$ , 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 $𝔅^{d} (X)$ , a variation of the loop space. We prove that $𝔅^{d} (X)$ is endowed with a natural symplectic form as soon as $X$ has one (in the sense of [22]).

Throughout this paper, we will use the tools of $(\infty, 1)$ -categories and symplectic derived algebraic geometry.

MR Zbl

DOI : 10.24033/asens.2329

Classification : 18F99, 55U99
Keywords: Formal loops, derived algebraic geometry, shifted symplectic structures.
Mot clés : Lacets formels, géométrie algébrique dérivée, structure symplectique décalée

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     author = {Hennion, Benjamin},
     title = {Higher dimensional formal loop spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {609--663},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
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     year = {2017},
     doi = {10.24033/asens.2329},
     mrnumber = {3665552},
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Hennion, Benjamin. Higher dimensional formal loop spaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 3, pp. 609-663. doi : 10.24033/asens.2329. http://www.numdam.org/articles/10.24033/asens.2329/

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