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 to a Fano manifold . We say that satisfies Gamma conjecture I if equals the Gamma class . When the quantum cohomology of is semisimple, we say that satisfies Gamma conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by for an exceptional collection in the derived category of coherent sheaves . Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.
"Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures." Duke Math. J. 165 (11) 2005 - 2077, 15 August 2016. https://doi.org/10.1215/00127094-3476593