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.

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 |