Double ramification cycles and integrable systems

4. Integrable hierarchies from partial CohFTs

Asked by Paolo Rossi:

Using the DR-method, one can assign to any partial CohFT an integrable hierarchy.

Examples:

• the DR cycle itself defines a partial CohFT, and hence an integrable hierarchy, a non-commutative 2d KdV (on a non-commutative torus), of infinite rank
• orbifold GW theory and taking constant part gives extended Toda hierarchy

The following problems are open:

1. Identify the hierarchy defined by the Gromov–Witten theory of $\mathbb{P}^1$ relative to $0,\infty$. Conjecturally, this is the 2d Toda hierarchy.
2. Identify the hierarchy defined by the constant $r$-part ($r$-spin Witten’s class). Conjecturally, this is the KP hierarchy (relation to class of strata of holomorphic differentials)
3. What about the partial CohFT given by cycles of hyperelliptic curves with marked Weierstrass points?

First question: compute some intersection numbers, e.g. with $\lambda_g$.

More general perspective: can anything be said about integrable systems produced by DR-machinery?