Double ramification cycles and integrable systems

2. Meaning of higher $r$-coefficients in $\mathrm{DR}_g^{d,r}(A)$

Asked by Jonathan Wise: Find a meaning of the $r$-coefficients of $\mathrm{DR}_g^{d,r}(A)$ (a polynomial in $r$) for $g=1$, possibly after changing its definition.

Conjecture (Ranganathan, Wise) In genus $1$, the degree $1$ coefficient vanishes because the reduced DR cycle is equal to the reduced orbifold DR cycle.

For the objects appearing in the conjecture:

• Reduced DR-cycle: a variant of the usual DR-cycle obtained from a compactification of the space of relative maps to $\mathbb{P}^1$, where we disallow contracted genus $1$ components, but allow genus $1$ singularities
• Orbifold DR cycle: obtained from maps to $\mathbb{P}^1$ with $B \mathbb{Z}_r$ at $0,\infty$
• Reduced orbifold DR cycle: Take the DR cycle, but disallow contracted $g=1$ components, allow elliptic singularities

The conjecture is part of a more general story, comparing relative and orbifold invariants of $(X,D)$, where $D$ is a smooth divisor on $X$. Here, one either places a relative condition, or an $r$-th root structure on $D$. It turns out that the genus $g=0$ invariants agree, but by work of Fenglong You, higher genus invariants do not agree. Yet, the $r=0$ terms of the orbifold side and the relative side agree.