In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way.
The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures.
It was conceived by Alain Connes who was motivated by the Atiyah-Singer index theorem and sought its extension to 'noncommutative' spaces.
A motivating example of spectral triple is given by the algebra of smooth functions on a compact spin manifold, acting on the Hilbert space of L2-spinors, accompanied by the Dirac operator associated to the spin structure.
From the knowledge of these objects one is able to recover the original manifold as a metric space: the manifold as a topological space is recovered as the spectrum of the algebra, while the (absolute value of) Dirac operator retains the metric.
[1] On the other hand, the phase part of the Dirac operator, in conjunction with the algebra of functions, gives a K-cycle which encodes index-theoretic information.
The local index formula[2] expresses the pairing of the K-group of the manifold with this K-cycle in two ways: the 'analytic/global' side involves the usual trace on the Hilbert space and commutators of functions with the phase operator (which corresponds to the 'index' part of the index theorem), while the 'geometric/local' side involves the Dixmier trace and commutators with the Dirac operator (which corresponds to the 'characteristic class integration' part of the index theorem).
Extensions of the index theorem can be considered in cases, typically when one has an action of a group on the manifold, or when the manifold is endowed with a foliation structure, among others.
In those cases the algebraic system of the 'functions' which expresses the underlying geometric object is no longer commutative, but one may be able to find the space of square integrable spinors (or, sections of a Clifford module) on which the algebra acts, and the corresponding 'Dirac' operator on it satisfying certain boundedness of commutators implied by the pseudo-differential calculus.
An odd spectral triple is a triple (A, H, D) consisting of a Hilbert space H, an algebra A of operators on H (usually closed under taking adjoints) and a densely defined self adjoint operator D satisfying ‖[a, D]‖ < ∞ for any a ∈ A.
[1] Let δ(T) denote the commutator of |D| with an operator T on H. A spectral triple is said to be regular when the elements in A and the operators of the form [a, D] for a in A are in the domain of the iterates δn of δ.
When a spectral triple (A, H, D) is p-summable, one may define its zeta function ζD(s) = Tr(|D|−s); more generally there are zeta functions ζb(s) = Tr(b|D|−s) for each element b in the algebra B generated by δn(A) and δn([a, D]) for positive integers n. They are related to the heat kernel exp(-t|D|) by a Mellin transform.
The collection of the poles of the analytic continuation of ζb for b in B is called the dimension spectrum of (A, H, D).
In the even case it is usually assumed that J is even with respect to the grading on H. Given a spectral triple (A, H, D), one can apply several important operations to it.
for the operator norm, then Connes introduces an extended pseudo-metric on the state space
a connected, compact, spin Riemannian manifold, that the restriction of this pseudo-metric to the pure states, i.e. the characters of the C*-algebra
, whose space is naturally homeomorphic (when endowed with the weak* topology) to
, and D is the closure of the usual Dirac operator acting on a dense subspace of the Hilbert space
Moreover, Connes observed that this distance is bounded if, and only if, there exists a state
This construction is reminiscent of the construction by Kantorovich of a distance on the space of Radon probability measures over a compact metric space, as introduced by Kantorovich during his study of Monge's transportation problem.
Guided by this observation, it is natural to wonder what properties Connes' metric shares with Kantorovich's distance.
In general, the topology induced by Connes' distance may not be Hausdorff, or give a finite diameter to the state space of the
, whereas Kantorovich's metric always induces the weak* topology on the space of Radon probability measures over
Rieffel worked out a necessary and sufficient condition on spectral triples (and more generally, on seminorms which play a role of analogue for Lipschitz seminorms) for Connes' distance to indeed induce the weak* topology on the state space of
These observations are the foundations of the study of noncommutative metric geometry, which deals with the geometry of the space of quantum metric spaces, many of which being constructed using spectral triples whose Connes' metric induces the weak* topology on the underlying state space.
In this context, an analogue of the Gromov-Hausdorff distance has been constructed on the space of metric spectral triples, allowing the discussion of the geometry of this space, and the construction of approximations of spectral triples by "simpler" (more regular, or finite dimensional) spectral triples.
The self adjoint unitary F gives a map of the K-theory of A into integers by taking Fredholm index as follows.
In the even case, each projection e in A decomposes as e0 ⊕ e1 under the grading and e1Fe0 becomes a Fredholm operator from e0H to e1H.
When the spectral triple is finitely summable, one may write the above indexes using the (super) trace, and a product of F, e (resp.
This can be encoded as a (p + 1)-functional on A satisfying some algebraic conditions and give Hochschild / cyclic cohomology cocycles, which describe the above maps from K-theory to the integers.