A tree expansion formula of a homology intersection number on the configuration space \(\mathcal{M}_{0,n} \) (Q2042355)

The paper contains a mathematical proof of a homological intersection formula for certain twisted cycles in \(\mathcal{M}_{0,n}\) found originally in the context of open/closed string duality in [\textit{S. Mizera}, J. High Energy Phys. 2017, No. 8, Paper No. 97, 54 p. (2017; Zbl 1381.83126)]. For the proof, after considering the cellular decomposition of \(\overline{\mathcal{M}}_{0,n}(\mathbb{R})\) in terms of the associahedron, see [\textit{M. M. Kapranov}, J. Pure Appl. Algebra 85, No. 2, 119--142 (1993; Zbl 0812.18003)], the author uses a combinatorial identity (Theorem 7.1), a related formula of [\textit{K. Mimachi} and \textit{M. Yoshida}, Commun. Math. Phys. 250, No. 1, 23--45 (2004; Zbl 1069.32015)] and a rule to describe the intersection of two cells in \(\overline{\mathcal{M}}_{0,n}(\mathbb{R})\) in terms of a tree described in [\textit{F. Cachazo} et al., J. High Energy Phys. 2014, No. 7, Paper No. 33, 33 p. (2014; Zbl 1391.81198)]. A remarkable feature of this formula compared with that of Mimachi-Yoshida, is that only a few trees contributes, namely only those with odd valency vertices. The reason being the combinatorial identity (Theorem 7.1).