Geometric engineering on flops of length two

TL;DR

This work extends geometric engineering from the conifold to Laufer's length-two flop by employing noncommutative crepant resolutions (NCCR) and matrix factorizations (MF) to extract quivers, exceptional curves, and Weil divisors. It presents a concrete 4d $ ext{N}=1$ quiver gauge theory for Laufer’s singularity, demonstrates a continuous flop between two resolved phases, and identifies two families of divisors that give rise to abelian gauge factors and charged matter in both IIA and IIB descriptions, with implications for F-theory via Mordell–Weil enhancements. The analysis reveals higher-charge hypers in Type IIA and connects to local F-theory models through extra elliptic sections, enriching the toolkit for non-toric, singular Calabi–Yau geometries. Overall, the paper broadens the scope of geometric engineering, linking quiver representations, divisor physics, and elliptic fibration structure in a non-toric setting.

Abstract

Type IIA on the conifold is a prototype example for engineering QED with one charged hypermultiplet. The geometry admits a flop of length one. In this paper, we study the next generation of geometric engineering on singular geometries, namely flops of length two such as Laufer's example, which we affectionately think of as the $\it{conifold\ 2.0}$. Type IIA on the latter geometry gives QED with higher-charge states. In type IIB, even a single D3-probe gives rise to a nonabelian quiver gauge theory. We study this class of geometries explicitly by leveraging their quiver description, showing how to parametrize the exceptional curve, how to see the flop transition, and how to find the noncompact divisors intersecting the curve. With a view towards F-theory applications, we show how these divisors contribute to the enhancement of the Mordell-Weil group of the local elliptic fibration defined by Laufer's example.

Figures (6)

• Figure 1: Quiver gauge theory of $N$ D3-branes probing the conifold $\mathcal{C}$.
• Figure 2: The conifold quiver. $e_i$ is the idempotent of node $i=R,M$ in the path algebra of the quiver.
• Figure 3: $\vec{d}=(1,1)$ quiver representation of the conifold.
• Figure 4: Quiver for the NCCR of Laufer's example with $n=1$aspinwall-morrison. For $n>1$ one must add an extra loop at $R$.
• Figure 5: Four-dimensional $\mathcal{N}=1$ quiver gauge theory for Laufer's example with $n=1$.