Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

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.

56 pages, 11 figures, v2: references added, some proofs simplified and shortened; Property O introduced; the discussion on Gamma conjecture for toric manifolds/toric complete intersections and the compatibility with quantum Lefschetz has been removed and appears in arXiv:1508.00719, v3: a rigorous definition of mutation of marked reflection systems given in 4.3, v4: minor changes

References in corpus (2)

• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• The $\hat{\Gamma }$-genus and a regularization of an $S^1$-equivariant Euler class

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