Spectral triple
Encyclopedia
In noncommutative geometry
and related branches of mathematics
and mathematical physics
, a spectral triple is a set of data which encodes 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. Some authors refer to this notion as unbounded K-cycles
or as unbounded Fredholm modules.
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 an 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 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.
||[a, D]|| < ∞ for any a ∈ A. An even spectral triple is an odd spectral triple with a Z/2Z-grading on H, such that the elements in A are even while D is odd with respect to this grading. One could also say that an even spectral triple is given by a quartet (A, H, D, γ) such that γ is a self adjoint unitary on H satisfying a γ = γ a for any a in A and D γ = - γ D.
A finitely summable spectral triple is a spectral triple (A, H, D) such that a.D for any a in A has a compact resolvent which belongs to the class of Lp+-operators for a fixed p (when A contains the identity operator on H, it is enough to require D-1 in Lp+(H)). When this condition is satisfied, the triple (A, H, D) is said to be p-summable. A spectral triple is said to be θ-summable when e-tD2 is of trace class for any t > 0.
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 think of the 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. The collection of the poles of the analytic continuation of ζb for b in B is called the dimension spectrum of (A, H, D).
A real spectral triple is a spectral triple (A, H, D) accompanied with an anti-linear involution J on H, satisfying[a, JbJ] = 0 for a, b in A. In the even case it is usually assumed that J is even with respect to the grading on H.
When the spectral triple is finitely summable, one may write the above indexes using the (super) trace, and a product of F, e (resp. u) and commutator of F with e (resp. u). 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.
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
and related branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, a spectral triple is a set of data which encodes geometric phenomenon in an analytic way. The definition typically involves a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, an algebra
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
of operators on it and an unbounded self-adjoint
Self-adjoint
In mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
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. Some authors refer to this notion as unbounded K-cycles
K-homology
In mathematics, K-homology is a homology theory on the category of compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space...
or as unbounded Fredholm modules.
Motivation
A motivating example of spectral triple is given by the algebra of 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. 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 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 an 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 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.
Definition
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 satisfyingA finitely summable spectral triple is a spectral triple (A, H, D) such that a.D for any a in A has a compact resolvent which belongs to the class of Lp+-operators for a fixed p (when A contains the identity operator on H, it is enough to require D-1 in Lp+(H)). When this condition is satisfied, the triple (A, H, D) is said to be p-summable. A spectral triple is said to be θ-summable when e-tD2 is of trace class for any t > 0.
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
When a spectral triple (A, H, D) is p-summable, one may think of the zeta functions ζb(s) = Tr(b|D|-s) for each element b in the algebra B generated by δn(A) and δn(
A real spectral triple is a spectral triple (A, H, D) accompanied with an anti-linear involution J on H, satisfying
Important concepts
Given a spectral triple (A, H, D), one can apply several important operations to it. The most fundamental one is the decomposition D = F|D| of D into an self adjoint unitary operator F (the 'phase' of D) and a densely defined positive operator |D| (the 'metric' part).Metric on the pure state space
The positive |D| operator defines a metric on the set of pure states on the norm closure of A.Pairing with K-theory
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. Thus e → Ind e1Fe0 defines an additive mapping of K0(A) to Z. In the odd case the eigenspace decomposition of F gives a grading on H, and each invertible element in A gives a Fredholm operator (F + 1) u (F − 1)/4 from (F − 1)H to (F + 1)H. Thus u → Ind (F + 1) u (F − 1)/4 gives an additive mapping from K1(A) to Z.When the spectral triple is finitely summable, one may write the above indexes using the (super) trace, and a product of F, e (resp. u) and commutator of F with e (resp. u). 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.