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Fibrés déterminants, déterminants régularisés et mesures sur les espaces de modules des courbes complexes. (Determinant fiber bundles, regularized determinants and measures on moduli spaces of complex curves.). (French) Zbl 0655.32021
Sémin. Bourbaki, 39ème année, Vol. 1986/87, Exp. 676, Astérisque 152/153, 113-149 (1987).
[For the entire collection see Zbl 0627.00006.]
Starting from Polyakov’s remark on the classical equivalence of the Nambu-Goto action and the so-called Polyakov action for bosonic strings (Riemann surfaces) in D-dimensional Euclidean space $${\mathbb{R}}^ D,$$ to quantize the classical theory, one is lead to consider a functional path integral of the form $$\int \exp (-S(g,x)){\mathcal D}g{\mathcal D}x,$$ where S(g,x) is Polyakov’s action, i.e. essentially the energy integral of the embedding map x of the Riemann surface M (supposed to have fixed genus p in what follows) with metric g into $${\mathbb{R}}^ D.$$ $${\mathcal D}g{\mathcal D}x$$ is an infinite dimensional measure on the product space $${\mathcal M}et\times {\mathcal E}$$, where $${\mathcal M}et$$ is the space of metrics on M and $${\mathcal E}$$ is the space of maps from M into $${\mathbb{R}}^ D.$$ S(g,x) is invariant under the group of motions of $${\mathbb{R}}^ D,$$ the group of diffeomorphisms of M and the conformal group of M. The integral over x actually reduces to a Gaußian integral because the action is quadratic in x, and so one is lead to construct a natural measure on $${\mathcal M}et$$, conformally invariant and invariant under the group of diffeomorphisms, in other words a natural measure on $${\mathcal M}_ p$$, the space of moduli of Riemann surfaces of genus p.
As the integral above doesn’t make sense, one considers first a finite dimensional model, giving a formula for a measure on a finite dimensional $${\mathcal C}^{\infty}$$-manifold, satisfying corresponding invariance requirements. Then, transplanting formally this formula to the Fréchet manifold $${\mathcal M}et$$ one finds a measure on $${\mathcal M}et$$ which descends to the Teichmüller space $${\mathcal T}_ p$$ (and actually $${\mathcal M}_ p)$$ if and only if a certain expression involving infinite dimensional determinants of the Laplacian $$\Delta_ g$$ (or equivalently $${\bar \partial}$$ *$${\bar \partial})$$ of M and the positive differential operator $${\bar \partial}$$ $$*_ T{\bar \partial}_ T$$ on the holomorphic tangent bundle T on M, depends only on the image of g in $${\mathcal T}_ p$$. To give a rigorous meaning to these determinants one uses so-called $$\zeta$$-function regularization. The measure is conformally invariant if and only if $$D=26$$ (formulas of conformal anomalies).
The formulas of conformal anomalies can be derived from a local Riemann- Roch-Grothendieck (RRG) theorem for a holomorphic family of compact Riemann surfaces $$\pi$$ : $$X\to S$$ and tensor powers of the relative tangent bundle $$T=T_{X/S}$$ equipped with a $${\mathcal C}^{\infty}$$ hermitian metric. To have a globally defined $${\mathcal C}^{\infty}$$-metric one uses the Quillen metric $$\| \|_ Q$$ on the determinant bundle DET $${\bar \partial}_ T\otimes n$$. The RRG theorem then gives an expression on the level of differential forms for the first Chern class $$c_ 1(DET {\bar \partial}_ T\otimes n$$, $$\| \|_ Q)$$. The critical dimension $$D=26$$ appears as an avatar of the coefficient 1/12 of the Todd genus occurring in the expression for $$c_ 1.$$
A basic result for the rest of the underlying article is a theorem of Mumford which says that for a family of connected Riemann surfaces as above, one has an isomorphism: $$M_{\pi}: DET {\bar \partial}_{T_{X/S}}\cong (DET {\bar \partial}_{{\mathcal O}})^{\otimes 13}$$, compatible with base change and such that if $$\pi$$ is an algebraic morphism of quasi-projective complex varieties, then $$M_{\pi}$$ is algebraic. A result of Beilinson and Drinfeld, using Mumford’s theorem, implies that the Polyakov measure on $${\mathcal T}_ p$$ coincides (up to a multiplicative constant) with the one defined by the metric on the canonical line bundle on $${\mathcal T}_ p$$, thus establishing the “algebraic” nature of Polyakov’s measure.
Mumford’s theorem can also be applied to obtain a more or less explicit description of the Polyakov measure in the vecinity of the divisor at infinity $$\Delta =\bar {\mathcal M}_ p-{\mathcal M}_ p$$, where $$\bar {\mathcal M}_ p$$ is the compactification of $${\mathcal M}_ p$$ by stable curves.
Explicit formulas for the Polyakov measure on $${\mathcal T}_ p$$ for $$p=1,2,3$$ are derived. On $${\mathcal T}_ 1$$ Dedekind’s $$\eta$$-function is involved, whereas on $${\mathcal T}_ 2$$ and $${\mathcal T}_ 3$$ $$\vartheta$$- functions appear.
Reviewer: W.W.J.Hulsbergen

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14H10 Families, moduli of curves (algebraic) 30Fxx Riemann surfaces 53A30 Conformal differential geometry (MSC2010) 81Txx Quantum field theory; related classical field theories
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