Spectral Geometry

Abstract

The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action. The idea is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein–Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one. The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators on Hilbert spaces, the notion of noncommutative residue, Dixmier trace, pseudodifferential operators etc. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus. Its non-compact generalization, namely the Moyal plane, is also investigated. Update: 2017, December 15 1 ar X iv :1 71 2. 05 94 5v 1 [ m at hph ] 1 6 D ec 2 01 7 Motivations: Let us first expose few motivations from physics to study noncommutative geometry which is by essence a spectral geometry. Of course, precise mathematical definitions and results will be given in the other sections. The notion of spectrum is quite important in physics, for instance in classical mechanics, the Fourier spectrum is essential to understand vibrations or the light spectrum in electromagnetism. The notion of spectral theory is also important in functional analysis, where the spectral theorem tells us that any selfadjoint operator A can be seen as an integral over its spectral measure A = ∫ a∈Sp(a) a dPa if Sp(A) is the spectrum of A. This is of course essential in the axiomatic formulation of quantum mechanics, especially in the Heisenberg picture where the tools are the observables namely are selfadjoint operators. But this notion is also useful in geometry. In special relativity, we consider fields ψ(~x) for ~x ∈ R4 and the electric and magnetic fields E, B ∈ Function(M = R4,R3). Einstein introduced in 1915 the gravitational field and the equation of motion of matter. But a problem appeared: what are the physical meaning of coordinates x and equations of fields? Assume the general covariance of field equation. If gμν(x) or the tetradfield eμ(x) is a solution (where I is a local inertial reference frame), then, for any diffeomorphism φ of M which is active or passive (i.e. change of coordinates), e′I ν (x) = ∂x μ ∂φ(x)ν e I μ(x) is also a solution. As a consequence, when relativity became general, the points disappeared and it remained only fields on fields in the sense that there is no fields on a given space-time. But how to practice geometry without space, given usually by a manifold M? In this later case, the spectral approach, namely the control of eigenvalues of the scalar (or spinorial) Laplacian return important informations on M and one can even address the question if they are sufficient: can one hear the shape of M? There are two natural points of view on the notion of space: one is based on points (of a manifold), this is the traditional geometrical one. The other is based on algebra and this is the spectral one. So the idea is to use algebra of the dual spectral quantities. This is of course more in the spirit of quantum mechanics but it remains to know what is a quantum geometry with bosons satisfying the Klein-Gordon equation ( +m2)ψ(~x) = sb(~x) and fermions satisfying (i∂/ − m)ψ(~x) = sf (~x) for sources sb, sf . Here ∂/ can be seen as a square root of and the Dirac operator will play a key role in noncommutative geometry. In some sense, quantum forces and general relativity drive us to a spectral approach of physics, especially of space-time. Noncommutative geometry, mainly pioneered by A. Connes (see [25, 31]), is based on a spectral triple (A,H,D) where the ∗-algebra A generalizes smooth functions on space-time M (or the coordinates) with pointwise product, H generalizes the Hilbert space of above quoted spinors ψ and D is a selfadjoint operator on H which generalizes ∂/ via a connection on a vector bundle over M . The algebra A also acts, via a representation of ∗-algebra, on H. Noncommutative geometry treats space-time as quantum physics does for the phasespace since it gives a uncertainty principle: under a certain scale, phase-space points are indistinguishable. Below the scale Λ−1, a certain renormalization is necessary. Given a geometry, the notion of action plays an essential role in physics, for instance, the Einstein– Hilbert action in gravity or the Yang–Mills–Higgs action in particle physics. So here, given the data (A,H,D), the appropriate notion of action was introduced by Chamseddine and Connes [11] and defined as S(D,Λ, f) := Tr ( f(D/Λ) )