Spectral invariants of Joyce orbifolds

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.