D-brane Superpotentials: Geometric and Worldsheet Approaches

TL;DR

This paper establishes a precise connection between geometric and worldsheet methods for computing D-brane superpotentials on Calabi-Yau manifolds, demonstrating their equivalence through explicit examples involving matrix factorizations and chain integrals.

Contribution

It explicitly relates geometric chain integral methods to worldsheet disk correlator calculations for D-branes on Calabi-Yau manifolds, especially for 2-cycle branes.

Findings

Derived explicit formulas for disk correlators using matrix factorizations.

Constructed chain integrals whose periods match worldsheet superpotentials.

Computed superpotentials in different open moduli space branches for a specific Calabi-Yau example.

Abstract

From the worldsheet perspective, the superpotential on a D-brane wrapping internal cycles of a Calabi-Yau manifold is given as a generating functional for disk correlation functions. On the other hand, from the geometric point of view, D-brane superpotentials are captured by certain chain integrals. In this work, we explicitly show for branes wrapping internal 2-cycles how these two different approaches are related. More specifically, from the worldsheet point of view, D-branes at the Landau-Ginzburg point have a convenient description in terms of matrix factorizations. We use a formula derived by Kapustin and Li to explicitly evaluate disk correlators for families of D2-branes. On the geometry side, we then construct a three-chain whose period gives rise to the effective superpotential and show that the two expressions coincide. Finally, as an explicit example, we choose a particular compact Calabi-Yau hypersurface and compute the effective D2-brane superpotential in different branches of the open moduli space, in both geometric and worldsheet approaches.