Extended r-spin theory in all genera and the discrete KdV hierarchy

n a recent work with E. Clader and R. Tessler we observed that the intersection numbers on the moduli space of disks, constructed by R. Pandharipande, J. Solomon and R. Tessler, are equal to the intersection numbers on the moduli space of genus 0 stable curves with a certain class, which we called the extended 2-spin class. In a joint work with P. Rossi, trying to generalize this result to higher genera, we construct a cohomological field theory type system of classes on the moduli spaces of stable curves in all genera and proved that the intersection numbers with these classes are controlled by the discrete KdV hierarchy. We also conjecture that our intersection numbers should correspond to the Hodge integrals of certain type on the moduli space of Riemann surfaces with boundary