D-branes and normal functions

Abstract

We explain the B-model origin of extended Picard-Fuchs equations satisfied by the D-brane superpotential on compact Calabi-Yau threefolds. The domainwall tension is identified with a Poincaré normal function -a transversal holomorphic section of the Griffiths intermediate Jacobian -via the Abel-Jacobi map. Within this formalism, we derive the extended Picard-Fuchs equation associated with the mirror of the real quintic.

Real mirror symmetry for one-parameter hypersurfaces

“…For the same reasons as in the quintic case discussed in [2], only the neighborhood of the set of points {p …”

“…The chain integral then satisfies an inhomogeneous version of the Picard-Fuchs differential equation governing closed string mirror symmetry, with an inhomogeneous term that can be computed explicitly from the curve and the Griffiths-Dwork algorithm. This Abel-Jacobi type method developped in [2] is similar in spirit to the computations in local geometries [9,10,11].…”

“…This can be viewed as an application of the homological Calabi-Yau/Landau-Ginzburg correspondence [25,26] together with elementary methods for the computation of Chern classes. The only point on which we will be slightly less rigorous than in [2] is that we will perform our computation as if we were pretending to be working on the hypersurfaces (1.2)…”

“…Chern classes by the algorithm described in [2] for the quintic. This can be viewed as an application of the homological Calabi-Yau/Landau-Ginzburg correspondence [25,26] together with elementary methods for the computation of Chern classes.…”

“…As explained in [2], the appropriate mathematical concept is that of a Hodge theoretic normal function. In the B-model, it can be represented geometrically as an integral of the holomorphic three-form over a three-chain suspended between homologically equivalent holomorphic curves.…”