Spectral invariants of Joyce orbifolds

TL;DR

This paper introduces two new spectral invariants for torsion-free G2-structures on orbifolds, computes them for Joyce orbifolds, and shows they are more sensitive than existing invariants, with connections to Epstein zeta-functions.

Contribution

It defines novel spectral invariants for G2-structures on orbifolds and demonstrates their effectiveness in distinguishing Joyce orbifolds, extending the tools available in G2-geometry.

Findings

Invariants computed explicitly for all Joyce orbifolds.

Invariants outperform the ar-invariant in distinguishing Joyce orbifolds.

Link established between invariants and twisted Epstein ta-functions.

Abstract

This paper introduces two new spectral invariants of torsion-free $\mathrm{G}_2$-structures on closed orbifolds and computes their values on all Joyce orbifolds. These invariants are shown to be more discerning than the $\overline{\nu}$-invariant of Crowley-Goette-Nordstr\"{o}m when applied to Joyce orbifolds, and thus provide candidate tools for distinguishing between Joyce manifolds. The invariants may be viewed as regularisations of the classical Morse indices of the critical points of the Hitchin functionals on closed and coclosed $\mathrm{G}_2$-structures respectively. In the case of Joyce orbifolds, an interesting link with twisted Epstein $\zeta$-functions is also observed.