A note on the Einstein–Hilbert action and Dirac operators on $${\mathbb{R}^n}$$

Abstract

We prove an extension to R n , endowed with a suitable metric, of the relation between the Einstein-Hilbert action and the Dirac operator which holds on closed spin manifolds. By means of complex powers, we first define the regularised Wodzicki residue for a class of operators globally defined on R n . The result is then obtained by using the properties of heat kernels and generalised Laplacians.Keywords Wodzicki residue · Einstein-Hilbert action · Dirac operator Mathematics Subject Classification (2000) Primary 58J40; Secondary 58J42 · 47A10 · 47G30 · 47L15 IntroductionIn 1984 Wodzicki [38] introduced a trace on the algebra of classical pseudodifferential operators on a closed manifold M. A similar result was independently obtained by Guillemin [18], in order to give a soft proof of Weyl formula. The Wodzicki residue became then a standard tool in Non-commutative Geometry. In 1988, Connes [10] proved that, for operator of order − dim(M), Dixmier Trace and the Wodzicki residue are equivalent. Moreover, he conjectured that the Wodzicki residue could connect Dirac operators and Einstein-Hilbert actions on M. In 1995, Kastler [21], Kalau and Walze [19] proved this conjecture. Namely, let / D be the classical Atiyah-Singer

Wodzicki residue for operators on manifolds with cylindrical ends

“…For an overview on the subject, see also the monograph of S. Scott [36]. Wodzicki Residue, sometimes called non-commutative trace, gained a growing interest in the years, also in view of the links with non-commutative geometry and Dixmier trace, see, e.g., A. Connes [7], B. Ammann and C. Bär [1], W. Kalau and M. Walze [15], D. Kastler [17], R. Ponge [31], U. Battisti and S. Coriasco [4]. The concept has been extended to different situations: manifolds with boundary by B.V. Fedosov, F. Golse, E. Leichtnam and E. Schrohe [10], conic manifolds by E. Schrohe [34] and J.B. Gil and P.A.…”

“…n S n−1 S n−1 a −n,−n (θ, θ ′ )dθ ′ dθ,Tr c ψ (A) = 1 (2π) n lim τ →∞ M\Cτ S n−1 a −n,· (x, θ ′ )dθ ′ dx − (log τ ) S n−1 S n−1 a −n,−n (θ, θ ′ )dθ ′ dθ − m2+nk S n−1 S n−1 a −n,m2−k (θ, θ ′ )dθ ′ dθ n lim τ →∞ S n−1 |ξ|≤τ a ·,−n (θ, ξ)dξdθ − (log τ ) S n−1 S n−1 a −n,−n (θ, θ ′ )dθ ′ dθ − m1+nm1−j (m 1 − j) S n−1 S n−1 a m1−j,−n (θ, θ ′ )dθ ′ dθ ,(3 4). and the angular term, analogous to (2.n S n−1 S n−1 d dz (a m1z−n−m1,m2z−n−m2 ) z=1 (θ, θ ′ )dθ ′ dθ.…”

Ellipticity in the SG calculus

“…The ζ-function of operators in L m,µ cl has also been thoroughly investigated in [3], both on R d as well as on manifolds with ends. Incidentally, we also remark that the analysis performed in [65] and [3] allowed to extend the concept of Wodzicki residue to the operators belonging to the SG-classes (see also U. Battisti, S. Coriasco [4]).…”

A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary

“…In [Ac], Ackermann gave a note on a new proof of this theorem by the heat kernel expansion way. The Kastler-Kalau-Walze theorem had been generalized to some cases, for example, Dirac operators with torsion [AT], CR manifolds [Po], R n [BC1] (see also [BC2], [Ni]).…”