Comments on Central Charge of Topological Sigma Model with Calabi-Yau Target Space

TL;DR

This paper investigates the central charge of Calabi-Yau d-folds in the context of topological sigma models, analyzing D-brane charges, intersection forms, and their relation to boundary states and moduli space deformations.

Contribution

It provides a concrete construction of the central charge Z for Calabi-Yau d-folds, linking D-brane charges to Mukai vectors and connecting topological sigma model results with CFT data.

Findings

Charges are described as Mukai vectors involving Chern characters and A-roof genera.

The central charge Z is determined by a set of integers and labels boundary states.

The moduli parameter t describes deformations in the open string moduli space.

Abstract

We study a central charge Z of a one parameter family of Calabi-Yau d-fold embedded in CP^{d+1}. For a d-fold case, we construct the Z concretely and analyze charge vectors of D-branes and intersection forms of associated cycles. We find the charges are described as some kinds of Mukai vectors. They are represented as products of Chern characters of coherent sheaves restricted on the Calabi-Yau hypersurfaces and square roots of A-roof genera of the d-folds. By combining results of the topological sigma model and the data of the CFT calculations in the Gepner model, we find that the Z is determined and is specified by a set of integers. It labels boundary states in special classes where associated states are represented as tensor products of boundary states for constituent minimal models. The Z has a moduli parameter t that describes a deformation of a moduli space in the open string channel with B-type boundary conditions. Also monodromy matrices and homology cycles are investigated.