## Vertex algebras and the formal loop space.(English)Zbl 1106.17038

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 $$\mathcal {L} (X)$$ for any scheme $$X$$ of finite type over a field is introduced. This is an ind-scheme containing $$\mathcal L^0 (X),$$ the scheme of formal germs of curves on $$X$$. The authors prove that both $$\mathcal L (X)$$ and $$\mathcal L^0 (X)$$ 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 $$\mathcal L\text (X)$$ and $$\mathcal L^0\text (X)$$ 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 $$\tilde{\mathcal L}(X)$$ 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 $$\mathcal D$$-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 $$\mathcal CDR_{ X}$$. 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 $$\mathcal CDR(\omega_X)$$ is a sheaf of vertex algebras on $$X$$ and that for any right $$\mathcal D_{ X}$$-module $$\mathcal M$$ the de Rham complex $$\mathcal CDR(M)$$ is a sheaf of $$\mathcal CDR(\omega_X)$$-modules.

### MSC:

 17B69 Vertex operators; vertex operator algebras and related structures 14F40 de Rham cohomology and algebraic geometry

### Keywords:

Ind-pro-schemes; chiral de Rham complex
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### References:

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