Constraints for $b$-deformed constellations

Abstract

Hurwitz numbers count branched covers of the sphere and have been of interest in various fields of mathematics. Motivated by the Matching-Jack conjecture of Goulden and Jackson, Chapuy and Do\l\k{e}ga recently introduced a notion of $b$-deformed double weighted Hurwitz numbers. It equips orientable and non-orientable maps and constellations with $b$-weights defined inductively. It is then unclear whether some elementary properties of orientable maps remain true due to the nature of the $b$-weights. We consider here the case of the Virasoro constraints, which express that choosing a corner is equivalent to rooting in terms of generating functions. We prove this property for two families of $b$-deformed Hurwitz numbers, namely 3-constellations and bipartite maps with black vertices of degrees bounded by 3. The proof is built upon functional equations from Chapuy and Do\l\k{e}ga and a lemma which extracts the constraints provided they close in some appropriate sense for the commutator. This requires to calculate the constraint algebra which in those two families do not form a Lie algebra but a generalization of independent interest with structure operators instead of structure constants.