Double ramification cycles and integrable systems

8. Double double ramification cycles

Proposed by Jonathan Wise :

On $\mathcal{M}^{\mathrm{ct}}_{g,n}$, or more generally on treelike curves, the formula \[ [\mathrm{DR}_g(A)]\cdot [\mathrm{DR}_g(B)] =[\mathrm{DR}_g(A)]\cdot [\mathrm{DR}_g(A+B)] \] holds. This does not work on $\overline{\mathcal M}_{g,n}$, but it does work on an appropriate blow-up \[ \pi_{A,B}\colon \overline{\mathcal M}^{A,B}_{g,n}\to \overline{\mathcal M}_{g,n} . \] The problem is to compare \[ (\pi_{A,B})_*([\mathrm{DR}]\cdot[\mathrm{DR}]) \qquad \text{with} \qquad ((\pi_{A,B})_*[\mathrm{DR}])\cdot ((\pi_{A,B})_*[\mathrm{DR}]) \ . \]

Proposed by Dmitry Zakharov:

Use subdivisions of cones of $\mathrm{DR}^{\mathrm{trop}}_g(A)\subseteq \mathcal{M}_{g,n}^{\mathrm{trop}}$ to control the blow-ups that appear.