Derived deformation theory of crepant curves

TL;DR

This paper fully characterizes the derived deformation theory of certain rational curves in Calabi-Yau 3-folds, linking noncommutative deformations to superpotential algebras and supporting string theory predictions.

Contribution

It provides a comprehensive description of the derived deformation theory of rational curves in CY 3-folds, including higher A-infinity-products and superpotential algebra formulations.

Findings

Derived deformation theory described via superpotential algebras.

Connection established between noncommutative deformations and string theory predictions.

New conjectures for contractibility of rational curves in CY 3-folds proposed.

Abstract

This paper determines the full derived deformation theory of certain smooth rational curves C in Calabi-Yau 3-folds, by determining all higher A_\infty-products in its controlling DG-algebra. This geometric setup includes very general cases where C does not contract, cases where the curve neighbourhood is not rational, all known simple smooth 3-fold flops, and all known divisorial contractions to curves. As a corollary, it is shown that the noncommutative deformation theory of C can be described as a superpotential algebra derived from what we call free necklace polynomials, which are elements in the free algebra obtained via a closed formula from combinatorial gluing data. The description of these polynomials, together with the above results, establishes a suitably interpreted string theory prediction due to Ferrari, Aspinwall-Katz and Curto-Morrison. Perhaps most significantly, the main results give both the language and evidence to finally formulate new contractibility conjectures for rational curves in CY 3-folds, which lift Artin's celebrated results from surfaces.