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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.

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