Computation of Superpotentials for D-Branes
Paul S. Aspinwall, Sheldon Katz
TL;DR
This work provides a metric-independent, algebraic-geometric framework to compute tree-level superpotentials for B-type D-branes on Calabi–Yau threefolds by exploiting ${A_ Infinity}$-structures in the derived category $D(X)$. By recasting holomorphic Chern–Simons theory in terms of morphism spaces and their higher products, the authors derive a practical method—via Čech cohomology and spectral sequences—to compute open-string superpotentials from Ext data, applicable to single branes and quiver gauge theories alike. They validate the approach with explicit calculations: reproducing the conifold result $W=\mathrm{Tr}(BCAD-ACBD)$ and analyzing a higher-obstruction $\mathbb{P}^1$ case to obtain a general $X^{n+1}$ and $Y^{n+1}$ structure with cross terms, as well as presenting a new $(1,-3)$ example with $W=\mathrm{Tr}(XY^2)$; in all cases the superpotential is fixed up to ${A_\infty}$-isomorphisms (nonlinear field redefinitions) and elementary to compute from the local brane data. The approach thus provides a practical, broadly applicable route to D-brane superpotentials that blends derived-category theory with concrete sheaf-cohomology computations. { All mathematical expressions are presented within $...$ to reflect the underlying topological B-model structure and its algebraic-geometric realization.}
Abstract
We present a general method for the computation of tree-level superpotentials for the world-volume theory of B-type D-branes. This includes quiver gauge theories in the case that the D-brane is marginally stable. The technique involves analyzing the A-infinity structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern-Simons theory. As an example, we give a more rigorous proof of previous results concerning 3-branes on certain singularities including conifolds. We also provide a new example.