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.

We prove that a stable minimal hypersurface of an open ball which is immersed away from a closed (singular) set of finite codimension 2 Hausdorff measure and weakly close to a multiplicity 2 hyperplane must in the interior be the graph over the hyperplane of a 2-valued function satisfying a local C1, estimate. This regularity is optimal under our hypotheses. As a consequence, we also establish compactness of the class of stable minimal hypersur- faces of an open ball which have volume density ratios uniformly bounded by 3−δ for any fixed δ ∈ (0, 1) and interior singular sets of vanishing co-dimension 2 Hausdorff measure.

The Wess-Zumino-Witten term was first introduced in the low energy σ-model which describes pions, the Goldstone bosons for the broken flavor symmetry in quantum chromodynamics. We introduce a new definition of this term in arbitrary gravitational backgrounds. It matches several features of the fundamental gauge theory, including the presence of fermionic states and the anomaly of the flavor symmetry. To achieve this matching, we use a certain generalized differential cohomology theory. We also prove a formula for the determinant line bundle of special families of Dirac operators on 4-manifolds in terms of this cohomology theory. One consequence is that there are no global anomalies in the Standard Model (in arbitrary gravitational backgrounds).

The paper deals with a variational approach of the subRiemannian geometry from the point of view of Hamilton-Jacobi and Hamiltonian formalism. We present a discussion of geodesics from the point of view of both formalisms, and prove that the normal geodesics are locally length-minimizing horizontal curves.

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