Tautological relations and the r-spin Witten conjecture

In [11], A. Givental introduced a group action on the space of Gromov-Witten potentials and proved its transitivity on the semi-simple potentials. In [], Y.-P. Lee showed, modulo certain results announced by C. Teleman, that this action respects the tautological relations in the cohomology ring of the moduli space M _g,n of stable pointed curves. Here we give a simpler proof of this result. In particular, it implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the geometric Gromov-Witten potential coincides with the potential constructed via Givental's group action. As the most important application we show that our results suffice to deduce the statement of a 1991 Witten conjecture relating the r-KdV hierarchy to the intersection theory on the space of r-spin structures on stable curves. We use the fact that Givental's construction is, in this case, compatible with Witten's conjecture, as Givental himself showed in [10].