Moduli Spaces for D-branes at the Tip of a Cone

TL;DR

The paper links D-brane probes of Calabi–Yau cones to geometry via derived categories by constructing quiver gauge theories from full strong exceptional collections of line bundles on a Fano surface $V$. The completed quiver on $\omega = K_V$ yields a moduli space of representations whose cone $C(V)$ appears as an irreducible component when FI-terms vanish, with the invariant ring generated by based loops corresponding to the anticanonical section ring. A key result shows that, for such collections generating a simple helix, $C(V)$ embeds as the canonical reduced subscheme of an irreducible component of the moduli space, clarifying how the ambient cone geometry emerges from quiver representations. The paper illustrates the construction in the $V=\mathbb{P}^1\times\mathbb{P}^1$ case, identifying $C(V)$ with the $\mathbb{Z}_2$ orbifold of the conifold and validating the general mechanism via an explicit loop-ring computation.

Abstract

For physicists: We show that the quiver gauge theory derived from a Calabi-Yau cone via an exceptional collection of line bundles on the base has the original cone as a component of its classical moduli space. For mathematicians: We use data from the derived category of sheaves on a Fano surface to construct a quiver, and show that its moduli space of representations has a component which is isomorphic to the anticanonical cone over the surface.

Figures (2)

• Figure 1: The completed quiver for ${\mathbb{P}^{1}} \times {\mathbb{P}^{1}}$.
• Figure :

Theorems & Definitions (1)

• Theorem