Double ramification cycles and integrable systems

1. A conjecture of Buryak-Rossi

Asked by Paolo Rossi : Starting with a semisimple CohFT (a generalization for non-semisimple CohFT’s can be formulated), there are two ways to associate to it an integrable hierarchy:

• Dubrovin-Zhang associates to this an integrable hierarchy \[w_{t_i} = F_i^{DZ}(w,w_x,\ldots)\]
• Double ramification gives another integrable hierarchy \[\tilde u_{t_i} = F_i^{DR}(\tilde u,\tilde u_x,\ldots)\]

Conjecture (Buryak,Rossi) There exists a normal Miura transformation relating the two hierarchies.

The conjecture has been tested for the trivial, the Hodge, and Witten (up to $r=5$) CohFT’s.

There exist two families of tautological cycles $A,B$ with explicit combinatorial definition; proving equality proves above conjecture. Conjecture is about intersection numbers, equality of cycles would imply this. \[\int_A \Omega_{g,n}(v_1 \otimes \cdots \otimes v_n) = \int_B \Omega_{g,n}(v_1 \otimes \cdots \otimes v_n)\] Maybe $A=B$?

$A=B$ proved in $g=0,1$ and partial equality in $g=2$.

Side question: Does equality of intersection numbers with any CohFT imply equality of cycles? Would follow from question: How much of $H^*(\overline{\mathcal{M}})_{g,n})$ spanned by CohFTs? See [buryakrossi1].