Transformation of algebraic Gromov-Witten invariants of three-folds under flops and small extremal transitions, with an appendix from the stringy and the symplectic viewpoint

Abstract

We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its applications. This gives an algebro-geometric account of Witten's wall-crossing formula for correlation functions of the descendant nonlinear sigma model in adjacent geometric phases of a gauge linear sigma model and of the symplectic approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem. A terse account from the stringy and the symplectic viewpoint is given in the appendix to complement and compare to the discussion in the main text.

38+2 pages, 5 figures

References in corpus (1)

• A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups

Cited by in corpus (4)

• Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpoint
• Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves
• Symplectic virtual localization of Gromov-Witten invariants
• D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck