(geometry $\leftarrow$ Isbell duality $\to$ algebra)
A spectral triple (Connes-Moscovici 95) is operator algebraic data that mimics the geometric data provided by a smooth Riemannian manifold $X$ with spin structure (Connes 08) and generalizes it to noncommutative geometry. It is effectively a Fredholm module with possibly unbounded Fredholm operator and refined by the specification of a dense subalgebra of the C-star-algebra of bounded operators on that module. As such, spectral triples have close ties to algebraic K-theory and so also to the physics described by these (see also at spectral action).
In a little more detail, a spectral triple consists of
a $\mathbb{Z}_2$-graded Hilbert space $\mathcal{H}$, to be thought of as the space of (square integrable) sections of the spinor bundle of $X$;
An associative algebra $A$ with a dense embedding $A \hookrightarrow B(H)$ into the C-star-algebra of bounded operators on $H$, to be thought of as the algebra of smooth functions on $X$;
These two items encode the topology and smooth structure.
A Fredholm operator $D$ acting on $\mathcal{H}$ and satisfying some conditions, to be thought of as the Dirac operator acting on the spinors.
This item encodes the Riemannian metric and possibly a connection.
(This is, or is a slight variant of, the concept of an unbounded Fredholm modules (e.g. Carey-Philips 98))
Below we discuss how one may think of a spectral triple as being precisely the algebraic data of supersymmetric quantum mechanics defining the worldvolume QFT of the quantum super particle propagating on a Riemannian target space (a sigma-model.) Accordingly this is just the beginning of a pattern. One degree up a 2-spectral triple is algebraic data encoding a Riemannian manifold with string structure.
Here is an unorthodox way to state the idea of spectral triple in terms of FQFT, which is in part just the reformulation of the quantum mechanics motivation that Alain Connes derived his definition from in the modern light of FQFT, but which more concretely follows work by Kontsevich-Soibelman, see (Soibelman 11) and see the references at 2-spectral triple.
(but maybe eventually we should have a traditional idea section and move this here to a subsection on further interpretations)
Let $R Cob_{1|1}^{Feyn}$ be the cobordism category of Feynman graphs for the superparticle with a single type of interaction along the lines of (1,1)-dimensional Euclidean field theories and K-theory. So its morphisms are generated from $(1|1)$-dimensional super-Riemannian manifolds (i.e. super-intervals) and from a single interaction vertex
subject to the obvious associativity condition.
Then a spectral triple $(A,H,D)$ is the data encoding a sufficiently nice smooth functor
to the category of super vector spaces.
Here
$A = Z_{(A,H,D)}(\bullet)_0$ is the even part of the super vector space assigned by the functor to the point, equipped with the structure of a algebra whose product is given by the image of the interaction vertex
$H$ is some completion of $Z_{(A,H,D)}(\bullet)$ to a super Hilbert space
and $D \in End(H)$ is an odd self-adjoint operator on $H$, which gives the value of the functor on the super-interval $(t,\theta)$ by
(For technical details that I am glossing over see the field theory link above).
So this is the quantum mechanics of a superparticle. In the simplest case this comes from a spinor particle propagating on a spin structure Riemannian manifold $X$in which case
$H = L^2(S)$ is the space of square integrable spinor sections;
$D$ is the Dirac operator
$A = C^\infty(X)$ is the space of smooth functions on $X$.
One point of a spectral triple is to take the view of world-line quantum mechanics as basic and characterize the spin Riemannian geometry of $X$ entirely by this algebraic data. In particular the Riemannian metric on $X$ is encoded in the operator spectrum of $D$, which is where the notion “spectral triple” gets its name from.
Then with all the ordinary geoemtry re-encoded algebraically this way, in terms of the 1-dimensional quantum field theory that probes this geometry, one can then use the same formulas to interpret spectral triple geometrically that do not come from an ordinary geometry as in the above example.
For a space described by a spectral triple there are several notions of dimension which all coincide when the space is a classical smooth manifold but which may differ on more general spectral spaces.
Notably there is the metric dimension?, which determines the growth of the eigenvalues of the Dirac operator (Connes95, p. 6).
And then there is the KO-dimension.
The standard textbook is
See also
The terminology of spectral triples was introduced in
Alain Connes, Henri Moscovici, The Local Index Formula in Noncommutative Geometry, Geometry and Funct. Analysis 5 (1995) 174-243.
Alain Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (11), 1995 (pdf)
and that of spectral action in
The characterization of ordinary compact smooth manifolds with spin structure in terms of spectral triples is in
See also
Traditionally spectral triples are discussed without specifying their homomorphisms. Proposals to remedy this such as to obtain a sensible category of spectral triples include the following
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul, A Category of Spectral Triples and Discrete Groups with Length Function, Osaka Journal of Mathematics, 43 n. 2, 327-350 (2006) (arXiv:math/0502583)
Paolo Bertozzini, Roberto Conti, Wicharn Lewkeeratiyutkul, Non-Commutative Geometry, Categories and Quantum Physics, East-West Journal of Mathematics “Contributions in Mathematics and Applications II” Special Volume 2007, 213-259 (2008) (arXiv:0801.2826)
(See also the pointers concerning the relation to KK-theory below).
A discussion of spectral triples as FQFT data encoding a representation of a category of 1-dimensional cobordisms with Riemannian structure and vertices is in section 1.4 of
A brief indication of some of the central ideas going into this is at
A general introduction to and discussion of spectral triples with an eye on quantum mechanics, quantum field theory and string theory is in
In section 7.2 of this an outlokk on how to regard the string's worldvolume CFT as a 2-spectral triple is given.
A detailed derivation for how spectral triples arise as point particle limits of vertex operator algebras for 2d CFTs:
A summary of this is in
Also
One variation uses von Neumann algebras instead of C-star algebras.
This goes back to (Carey-Philips 98) and
See also
M-T. Benameur, T. Fack,, Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras, Advances in Mathematics 199: 29-87, 2006.
Alain Connes, Henri Moscovici, Type III and spectral triples (arXiv:math/0609703)
Relation to K-theory and KK-theory is discussed in
(pdf)
Bram Mesland, Spectral triples and KK-theory: A survey (arXiv:1304.3802)
Alan Carey, John Philips, Adam Rennie, Spectral triples: examples and index theory, in Alan Carey (ed.) Noncommutative geometry and physics, Renormalization, Motives, Index theory (2011)