Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

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 AF to a Fano manifold F. We say that F satisfies Gamma conjecture I if AF equals the Gamma class ΓˆF. 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 ΓˆFCh(Ei) for an exceptional collection {Ei} in the derived category of coherent sheaves Dcohb(F). Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.