Gamma class, total positivity and mirror symmetry

Mirror symmetry predicts that for any Fano manifold $X$ there should be a Landau-Ginzburg model $(X^{\vee},W)$ such that the quantum $D$-module of $X$ is isomorphic to the Gauss-Manin system of $(X^{\vee},W)$. In addition, the natural lattice structures on the spaces of flat sections of these $D$-modules, one coming from the image of the Chern character of $X$ and one from certain integral relative homology of $X^{\vee}$, should match, after the former is twisted by the Gamma class. These predictions have been verified for toric Fano manifolds.

In this talk, I will discuss the case when $X$ is a flag variety of arbitrary Lie group type, where $(X^{\vee},W)$ is known to be the Rietsch mirror. I will focus on $1=ch([\mathcal{O}_X])$ and explain the result that this element corresponds to the totally positive part of $X^{\vee}$ in the sense of Lusztig. If time permits, I will explain how to apply this result to prove Gamma conjecture I for these varieties.