Reverse geometric engineering of singularities

TL;DR

This paper discusses a method to reconstruct geometric singularities from quiver gauge theories using noncommutative algebra, providing insights into the inverse process of geometric engineering in string theory.

Contribution

It introduces a framework for reverse geometric engineering using noncommutative quiver algebras and demonstrates the invariance of the resulting geometry under Seiberg dualities.

Findings

Reconstruction of singularities from quiver theories via noncommutative algebra.

Identification of the geometry as the center of the algebra, invariant under dualities.

Conditions under which the reconstructed geometry matches the original singularity.

Abstract

One can geometrically engineer supersymmetric field theories theories by placing D-branes at or near singularities. The opposite process is described, where one can reconstruct the singularities from quiver theories. The description is in terms of a noncommutative quiver algebra which is constructed from the quiver diagram and the superpotential. The center of this noncommutative algebra is a commutative algebra, which is the ring of holomorphic functions on a variety V. If certain algebraic conditions are met, then the reverse geometric engineering produces V as the geometry that D-branes probe. It is also argued that the identification of V is invariant under Seiberg dualities.