A tree expansion formula of a homology intersection numbers on the configuration space $\mathcal{M}_{0,n}$

TL;DR

This paper provides an elementary proof of Mizera's tree expansion formula for homology intersection numbers on the moduli space rac{0,n}, connecting combinatorics, real moduli spaces, and string theory.

Contribution

It offers a new, elementary proof of Mizera's formula, utilizing combinatorics of real moduli spaces and associahedron face numbers.

Findings

Elementary proof of Mizera's tree expansion formula.

Connection between combinatorics of moduli spaces and string theory.

Utilization of associahedron face number identities.

Abstract

In \cite{M}, Sebastian Mizera discovered a tree expansion formula of a homology intersection number on the configuration space $\mathcal{M}_{0,n}$. The formula originates in a study of Kawai-Lewellen-Tye relation in string theory. In this paper, we give an elementary proof of the formula. The basic ingredients are the combinatorics of the real moduli space $\overline{\mathcal{M}}_{0,n}(\R)$ and a combinatorial identity related to the face number of the associahedron.