Reverse geometric engineering of singularities

TL;DR

This work develops a purely algebraic framework to reverse-engineer Calabi–Yau singularities from quiver gauge theories. By constructing a noncommutative meson algebra $A$ from a quiver and its superpotential and then taking its center $Z(A)$, the authors identify a commutative 3-fold $V$ whose holomorphic functions encode the geometry probed by D-branes, with the brane moduli space forming $Sym^N(V)$ when $A$ is finitely generated over $Z(A)$. The paper demonstrates this construction through explicit examples: deformed $A_{n-1}$ singularities, a non-toric non-orbifold singularity, and the ${ m C}^3/{ m Z}_3$ orbifold, showing that $V$ is well-defined and invariant under Seiberg dualities. The results highlight a powerful correspondence between quiver data and the underlying geometry and point toward a program to classify 3-dim singularities via noncommutative algebraic methods.

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.

Figures (3)

• Figure 1: Composite meson field in the quiver diagram: $\phi^{ab}_{ik} = \phi^a_{ij}\phi^b_{jl}$
• Figure 2: $A_{n-1}$ quiver diagram
• Figure 3: Two dual versions of the ${\mathbb{C}}^3/{\mathbb{Z}}_3$ quiver diagram, dualized at the marked node