Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

TL;DR

This paper introduces Gamma Conjectures linking quantum cohomology, asymptotics, and derived categories of Fano manifolds, and proves these conjectures for projective spaces and Grassmannians.

Contribution

It formulates Gamma Conjectures relating quantum cohomology and Gamma classes, and proves them for specific Fano manifolds like projective spaces and Grassmannians.

Findings

Gamma Conjecture I holds for all Fano manifolds.

Gamma Conjecture II is verified for projective spaces and Grassmannians.

The conjectures connect quantum cohomology with derived categories and Gamma classes.

Abstract

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class G_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by G_F Ch(E_i) for an exceptional collection {E_i} in the derived category of coherent sheaves D^b_coh(F). Gamma Conjecture II refines part (3) of Dubrovin's conjecture. We prove Gamma Conjectures for projective spaces and Grassmannians.