Spectral invariants of Joyce orbifolds
Laurence H. Mayther
TL;DR
The paper defines two spectral invariants for torsion-free $\mathrm{G}_2$-structures on closed orbifolds by regularising the Morse indices of the Hitchin functionals $\mathcal{H}_3$ and $\mathcal{H}_4$, yielding $\mu_3$ and $\mu_4$ on the moduli space $\mathscr{G}^{TF}_2(\mathrm{M})$. On Joyce orbifolds $\mathrm{M}_\Gamma=\Gamma\backslash\mathbb{T}^7$, these invariants are constant and computable via $\mu_3(\mathrm{M}_\Gamma)=\frac{-1}{|\Gamma|}\sum_{\mathcal{A}=(A,t)\in\Gamma}\operatorname{Tr}^{\mathrm{SU}(3)}_8(A)$ and $\mu_4(\mathrm{M}_\Gamma)=\frac{-1}{|\Gamma|}\sum_{\mathcal{A}=(A,t)\in\Gamma}\operatorname{Tr}^{\mathrm{SU}(3)}_{12}(A)$, where $\operatorname{Tr}^{\mathrm{SU}(3)}_8$ and $\operatorname{Tr}^{\mathrm{SU}(3)}_{12}$ are explicit polynomial expressions in traces of $A$. The invariants are shown to be stronger discriminators than the Crowley-Goette-Nordström $\overline{\nu}$-invariant on Joyce orbifolds and are linked to twisted Epstein $\zeta$-functions, suggesting deep connections to analytic number theory. The work motivates extending computable spectral invariants to Joyce manifolds obtained by resolving singularities and contributes a practical framework for distinguishing torsion-free $\mathrm{G}_2$-geometries.
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 $\overlineν$-invariant of Crowley-Goette-Nordström 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 $ζ$-functions is also observed.