The spectral Einstein functional for the nonminimal de Rham-Hodge operator
Hongfeng Li, Yong Wang
TL;DR
This work defines and analyzes non-self-adjoint spectral triples built from the nonminimal de Rham-Hodge operator $\widetilde{D}=a_0 d+b_0 \delta$ on even-dimensional closed manifolds, and uses the noncommutative residue to formulate spectral geometric functionals. It derives explicit closed-form expressions for the spectral metric functional $\mathscr{M}_{\widetilde{D}}$ and the spectral Einstein functional $\mathscr{N}_{\widetilde{D}}$, showing that $\mathscr{M}_{\widetilde{D}} = -2^{2m} \frac{2 \pi^{m}}{\Gamma(m)} \int_M (a_0 b_0)^{-m+1} g(u,v)\,dVol_M$ and $\mathscr{N}_{\widetilde{D}} = -2^{2m} \frac{2 \pi^{m}}{\Gamma(m)} \int_M \frac{(a_0 b_0)^{-m+2}}{6} \mathbb{G}(u,v)\,dVol_M$, where $\mathbb{G}(u,v)=\operatorname{Ric}(u,v)-\frac{1}{2} s g(u,v)$. The authors construct and manipulate symbol expansions for $\widetilde{D}_0$ and analyze the associated Clifford-algebra traces to achieve these results, thereby extending the Kastler-Kalau-Walze-type framework to non-self-adjoint, torsion-free contexts. The paper also provides several illustrative non-self-adjoint spectral-triple examples on both commutative and noncommutative spaces. These findings advance the spectral-gravity correspondence by linking nonminimal differential operators to explicit geometric action-like functionals.
Abstract
In this paper, we give the definitions of the non-self-adjoint spectral triple and its spectral Einstein functional. We compute the spectral Einstein functional associated with the nonminimal de Rham-Hodge operator on even-dimensional compact manifolds without boundary. Finally, several examples of the non-self-adjoint spectral triple are listed.