Vertex algebras and the formal loop space.

*(English)*Zbl 1106.17038The 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.

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.

Reviewer: Arvid Siqveland (Kongsberg)

##### MSC:

17B69 | Vertex operators; vertex operator algebras and related structures |

14F40 | de Rham cohomology and algebraic geometry |

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\textit{M. Kapranov} and \textit{E. Vasserot}, Publ. Math., Inst. Hautes Étud. Sci. 100, 209--269 (2004; Zbl 1106.17038)

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