Derived deformation theory of crepant curves

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 non-commutative deformation theory of $C$ is described via 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 (Adv. Theor. Math. Phys. 7 (2003) 619–665), Aspinwall–Katz (Comm. Math. Phys.. 264 (2006) 227–253) and Curto–Morrison (J. Algebraic Geom. 22 (2013) 599–627). 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 (Amer. J. Math. 84 (1962) 485–496) celebrated results from surfaces.