From $r$-Spin Intersection Numbers to Hodge Integrals

TL;DR

This paper explores the mathematical structure of $r$-spin intersection numbers and Hodge integrals through matrix models, fermionic representations, and symmetry constraints, revealing deep connections between these geometric invariants.

Contribution

It introduces a fermionic representation of the GKMM for $r$-spin numbers and derives a $W_{1+ abla}$ constraint linking $r$-spin intersection numbers to Hodge integrals.

Findings

Representation of GKMM in fermionic form

Linking $r$-spin numbers with Hodge integrals via operators

Complete determination of Hodge partition function by $W_{1+ abla}$ constraint

Abstract

Generalized Kontsevich Matrix Model (GKMM) with a certain given potential is the partition function of $r$-spin intersection numbers. We represent this GKMM in terms of fermions and expand it in terms of the Schur polynomials by boson-fermion correspondence, and link it with a Hurwitz partition function and a Hodge partition by operators in a $\widehat{GL}(\infty)$ group. Then, from a $W_{1+\infty}$ constraint of the partition function of $r$-spin intersection numbers, we get a $W_{1+\infty}$ constraint for the Hodge partition function. The $W_{1+\infty}$ constraint completely determines the Schur polynomials expansion of the Hodge partition function.