# Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes

## TL;DR

This paper uses half-spin tautological relations to derive a combinatorial identity equivalent to Faber's formula for kappa-classes, offering new proofs for specific cases and deepening understanding of tautological relations.

## Contribution

It introduces a novel combinatorial identity based on half-spin tautological relations and proves several cases, providing new insights into Faber's proportionalities of kappa-classes.

## Key findings

- Derived a combinatorial identity equivalent to Faber's formula
- Proved several cases of the identity, offering new proofs
- Enhanced understanding of tautological relations on moduli spaces

## Abstract

We employ the $1/2$-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on $\mathcal{M}_g$, $g\geq 2$. We then prove several cases of the combinatorial identity, providing a new proof of Faber's formula for those cases.