On Superpotentials for D-Branes in Gepner Models

TL;DR

The paper develops a conformal-field-theory toolkit to compute open-string amplitudes and boundary operator products for D-branes in Gepner models, focusing on untwisted A-type branes. It builds from the ${\cal N}=2$ minimal-model coset construction and its fusing-matrix data, then implements a simple-current orbifold projection to obtain Gepner-model boundary states and their OPEs, including explicit quintic examples. A key result is the explicit computation of a cubic superpotential term for a massless modulus on the quintic, showing that the modulus is lifted at the Gepner point and that small- and large-volume moduli spaces can be reconciled with superpotential effects. The framework enables systematic analysis of boundary correlators, bound states, and potential extensions to B-type branes and bulk-boundary couplings, advancing the understanding of D-brane moduli and interactions in Calabi-Yau compactifications.

Abstract

A large class of D-branes in Calabi-Yau spaces can be constructed at the Gepner points using the techniques of boundary conformal field theory. In this note we develop methods that allow to compute open string amplitudes for such D-branes. In particular, we present explicit formulas for the products of open string vertex operators of untwisted A-type branes. As an application we show that the boundary theories of the quintic associated with the special Lagrangian submanifolds Im ω_i z_i = 0 where ω_i^5=1 possess no continuous moduli.