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Zbl 1106.17038
Kapranov, Mikhail; Vasserot, Eric
Vertex algebras and the formal loop space. (English)
[J] Publ. Math., Inst. Hautes Étud. Sci. 100, 209-269 (2004). ISSN 0073-8301; ISSN 1618-1913/e

The mathematical approach of string theory can be cast in terms of analysis on the space of free loops, i.e. smooth maps $S^1\rightarrow X$ where $X$ is a given spacetime manifold. One has the folklore principle that constructions involving the space of free loops lead to vertex algebras. One class of such constructions is $\Omega_X^{ch}$, the chiral de Rham complex of an algebraic variety. Heuristically, this complex should be interpreted in terms of $LX$, the space of free loops and its subvariety $L^0X$ consisting of loops extending holomorphically into the unit disk. That is $\Omega_X^{ch}$ can be thought of as the semi-infinite de Rham complex with coefficients in the space of distributions on $LX$ supported on $L^0X$. Mathematically the definition of $\Omega_X^{ch}$ is of a more computational nature and proceeds by constructing the action of the group of diffeomorphisms on the irreducible module over the Heisenberg algebra. The article has two main goals: First, to give a precise mathematical theorem underlying the above folklore principle about vertex algebras. An algebro-geometric version of the free loop space $\cal {L} (X)$ for any scheme $X$ of finite type over a field is introduced. This is an ind-scheme containing $\cal L^0 (X),$ the scheme of formal germs of curves on $X$. The authors prove that both $\cal L (X)$ and $\cal L^0 (X)$ themselves possess a nonlinear version of the vertex algebra structure (which makes it clear that any natural linear construction applied to them should give a vertex algebra in the usual sense). The authors proves that natural global versions of $\cal L\text(X)$ and $\cal L^0\text(X)$ have natural structures of factorization monoids. To give a good definition of the algebraic analog of the full loop space $LX$ there is a problem: The functor which to any commutative ring $R$ associates the set of $R((t))$-points of $X$ is representable by $\tilde{\cal L}(X)$ when $X$ is affine, but this functor do not glue well together in the general case: When $X$ is e.g. projective, there is no difference between $R[[t]]$-points and $R((t))$-points. To overcome this subtlety, the authors consider formal loops which are "infinitesimal in the Laurent direction". Then they glue well together. The second goal of the authors, is to give a direct geometric construction of $\Omega_X^{ch}$ for smooth $X$ in terms of their model for the loop space. This construction explains the fact that $\Omega_X^{ch}$ is a sheaf of vertex algebras. As with the study of formal arcs and motivic integration, this considerations can be viewed as algebro-geometric analogs of the basic constructions of $p$-adic analysis. The construction of the formal loop space in chapter 1 is written in a way that makes it possible to understand. The generalities on ind-schemes, the scheme of germs of arcs and the nil-Laurent series is explained such that they form the foundation for proving the first result: The proof of the existence of the formal loop space. It is also possible to understand the formal loop space as an ind-pro-object. In chapter 2, the localization of the global loop space in a smooth curve $C$ is proved to have a ``functorial" structure of factorization monoid, gluing well together on the affine covering of a e.g. projective scheme $X$. This result is highly nontrivial, and hard to prove. Some easy examples illustrates the result in a nice way. To introduce the announced vertex algebras, the theory of $\cal D$-modules on ind-pro-schemes are given. The properties of these sheaves are proved to behave well, leading to the de Rham complexes on ind-pro-schemes. The final chapter concentrate on identification on the chiral de Rham complex $\cal CDR_{ X}$. It is proved that ``localized" to a (point on a) smooth curve $C$ this complex has the structure of a factorization algebra. This leads to the theorem aying that the de Rham complex $\cal CDR(\omega_X)$ is a sheaf of vertex algebras on $X$ and that for any right $\cal D_{ X}$-module $\cal M$ the de Rham complex $\cal CDR(M)$ is a sheaf of $\cal CDR(\omega_X)$-modules.
[Arvid Siqveland (Kongsberg)]

MSC 2000:: *17B69 Vertex operators
14F40 De Rham cohomology

Keywords: Ind-pro-schemes; chiral de Rham complex

Cited in: Zbl 1219.14026

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