Superpotentials for Quiver Gauge Theories

TL;DR

We address the problem of determining the superpotential for quiver gauge theories arising from marginal D-brane decay at del Pezzo singularities in a Calabi–Yau background. The authors develop a framework based on $A_\infty$ products in the derived category to construct the superpotential, proving that it is linear in the fields $\operatorname{Ext}^2$ with each such field multiplying the corresponding relation polynomial in $\operatorname{Ext}^1$. The analysis reduces from branes on the Calabi–Yau to branes on the del Pezzo surface $S$, and then computes the relevant $A_\infty$ data within the derived category of quiver representations, yielding explicit results for $S=\mathbb{P}^2$ and $S=\mathrm{dP}_1$. The approach provides a general algorithm applicable to other del Pezzo quivers and clarifies how quiver relations encode the moduli of D-branes at Calabi–Yau singularities.

Abstract

We compute superpotentials for quiver gauge theories arising from marginal D-Brane decay on collapsed del Pezzo cycles S in a Calabi-Yau X. This is done using the machinery of A-infinity products in the derived category of coherent sheaves of X, which in turn is related to the derived category of S and quiver path algebras. We confirm that the superpotential is what one might have guessed from analyzing the moduli space, i.e., it is linear in the fields corresponding to the Ext2's of the quiver and that each such Ext2 multiplies a polynomial in Ext1's equal to precisely the relation represented by the Ext2.