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Jamey Bass, Andrey Todorov

The analogue of the Dedekind eta function for CY manifolds I

This is the first of series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi-Yau metrics acting on (0, q) forms on the moduli space of CY manifolds with a fixed polarization.

It is well known that in the case of elliptic curves, the Kronecker limit formula gives an explicit formula for the regularized determinant of the flat metric with fixed volume on the elliptic curves. The following formula holds in this case: where &eegr;(&tgr;) is the Dedekind eta function. It is well known fact that &eegr;(&tgr;)²⁴ is a cusp automorphic form of weight 12 related to the discriminant of the elliptic curve.

Formula (1) implies that there exists a holomorphic section of some power of the line bundle of the classes of cohomologies of (1, 0) forms of the elliptic curves over the moduli space with an L² norm equal to det &Dgr;_{(0, 1)} (&tgr;). This section is &eegr;(&tgr;)²⁴. The purpose of this project is to find the relations between the regularized determinants of CY metric on (0, 1) forms and algebraic discriminants of CY manifolds.

In this paper we will establish the local analogue of the formula (1) for CY manifolds.

Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter

Print ISSN: 0075-4102
Volume: 2006, 10/2006
Pages: 61 - 96

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