Closed formula for higher-genus double ramification Euler characteristics

Obtain a closed formula for the orbifold Euler characteristic of the double ramification locus DR_{g,n}(a) in arbitrary genus.

Background

The paper describes an in-principle computation of the orbifold Euler characteristic of DR_{g,n}(a) in all genera by stratifying the locus according to Hurwitz spaces and expressing the result as a weighted sum of Hurwitz numbers.

Although this procedure can be implemented computationally and used for experimental verification, the authors explicitly state that they are unable to obtain a closed formula from it. The unresolved problem is to derive such a formula in arbitrary genus; the method also does not extend to higher rank.

References

This method is significantly less efficient than the genus one recursion employed above, and we are unable to use it to obtain a closed formula.

— Euler characteristics of higher rank double ramification loci in genus one  (2502.12281 - Battistella et al., 17 Feb 2025) in Introduction, subsection “Higher genus: the Hurwitz stratification”