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## 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 A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class Γ_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 Γ_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, toric manifolds, certain toric complete intersections and Grassmannians.