  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.
