Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

Understand

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).