Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

By Sergey Galkin, V. Golyshev, H. Iritani. 2014

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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 $\widehat{\Gamma}_{F}$. 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 $\widehat{\Gamma}_{F}\operatorname{Ch}(E_{i})$ for an exceptional collection $\{E_{i}\}$ in the derived category of coherent sheaves $\mathcal{D}^{b}_{\mathrm{coh}}(F)$. Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4cbfd7dba9f9a1e107ab25b2c15630128b901283