Analytic torsion for Calabi-Yau threefolds

Abstract

After Bershadsky-Cecotti-Ooguri-Vafa, we introduce an invariant of Calabi-Yau threefolds, which we call the BCOV invariant and which we obtain using analytic torsion. We give an explicit formula for the BCOV invariant as a function on the compactified moduli space, when it is isomorphic to a projective line. As a corollary, we prove the formula for the BCOV invariant of quintic mirror threefolds conjectured by Bershadsky-Cecotti-Ooguri-Vafa. Contents 1. Introduction 2. Calabi-Yau varieties with at most one ordinary double point 3. Quillen metrics 4. The BCOV invariant of Calabi-Yau manifolds 5. The singularity of the Quillen metric on the BCOV bundle 6. The cotangent sheaf of the Kuranishi space 7. Behaviors of the Weil-Petersson metric and the Hodge metric 8. The singularity of the BCOV invariant I -the case of ODP 9. The singularity of the BCOV invariant II -general degenerations 10. The curvature current of the BCOV invariant 11. The BCOV invariant of Calabi-Yau threefolds with h 1,2 = 1 12. The BCOV invariant of quintic mirror threefolds 13. The BCOV invariant of FHSV threefolds 1 ANALYTIC TORSION FOR CALABI-YAU THREEFOLDS 5an arbitrary Calabi-Yau manifold of arbitrary dimension, which we obtain using determinants of cohomologies [28], Quillen metrics [11], [44], and a Bott-Chern class like A(·). Then the BCOV Hermitian line of a Calabi-Yau manifold depends only on the complex structure of the manifold. The Hodge diamond of Calabi-Yau threefolds are so simple that the BCOV Hermitian line reduces to the scalar invariant τ BCOV in the case of threefolds. Hence Eq. (1.1) on P 1 \ D is deduced from the curvature formula for the BCOV Hermitian line bundles. (See Sect. 4).(b) To establish the formula for log τ BCOV near D, we fix a specific holomorphic extension of the BCOV bundle from P 1 \D to P 1 , which we call the Kähler extension. (See Sect. 5.) Since τ BCOV is the ratio of the Quillen metric and the L 2 -metric on the BCOV bundle, it suffices to determine the singularities of the Quillen metric and the L 2 -metric on the extended BCOV bundle. We determine the singularity of the Quillen metric on the extended BCOV bundle with respect to the metric on T X /P 1 induced from a Kähler metric on X . The anomaly formula for Quillen metrics of Bismut-Gillet-Soulé [11] and a formula for the singularity of Quillen metrics [9], [61] play the central role. (See Sect. 5.).(c) By the smoothness of Def(X ψ ) at ψ ∈ D * [26], [45], [54], the behavior of the L 2 -metric on the extended BCOV bundle near D * is determined by the singularity of Ω WP near D * , which was computed by Tian [54]. (See Sects. 6,7,8.) To determine the behavior of the L 2 metric on the extended BCOV bundle at ψ = ∞, one may assume that π : X → P 1 is semi-stable at ψ = ∞ by Mumford [27]. We consider another holomorphic extension of the BCOV bundle, i.e., the canonical extension in Hodge theory [48]. With respect to the canonical extension, the L 2 -metric has at most an algebraic singularity at ψ = ∞ by Schmid [48]. Comparing the two extensions, we sh...

On the singularity of Quillen metrics

“…(b)] when π : X → S is a family of curves and by the author [16] when π is isolated. In [9], Theorem 1.1 shall play a crucial role in the study of analytic torsion of Calabi-Yau threefolds. Let s be a section of O S ( ) defining the reduced divisor .…”

On solutions to Walcher’s extended holomorphic anomaly equation

“…Then, Walcher [19] further proposed the open string analogue of BCOV, the extended holomorphic anomaly equation, which is a PDE for the B-model topological string amplitude F (g,h) for world-sheets with g handles and h boundaries. 2 1 For genus g = 0, the third covariant derivative of F (0) is the Yukawa coupling, and for g = 1, it is recently proved that F (1) is the Quillen's norm function [6]. For genus g ≥ 2, the mathematical definition of F (g) is yet to be known.…”

On the Birational Invariance of the BCOV Torsion of Calabi-Yau Threefolds

“…See [9, p. 177] or Definition 2.3 below for the precise definition. They conjectured the following (see [9,Sect. 4,Conj H. Fang, Z. Lu and K.-I.…”