FQFT and cohomology
A spectral triple is algebraic data that mimics the geometric data provided by a smooth Riemannian manifold with spin structure and generalizes it to noncommutative geometry. It is effectively a Fredholm module refined by the specification of a dense subalgebra of the C-star-algebra of bounded operators on that module.
It consists of
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 and Soibelman (linked to at 2-spectral triple) which methinks is the right one.
(but maybe eventually we should have a traditional idea section and move this here to a subsection on further interpretations)
Let 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 -dimensional super-Riemannian manifolds (i.e. super-intervals) and from a single interaction vertex
subject to the obvious associativity condition.
Then a spectral triple is the data encoding a sufficiently nice smooth functor
is some completion of to a super Hilbert space
and is an odd self-adjoint operator on , which gives the value of the functor on the super-interval by
(For technical details that I am glossing over see the field theory link above).
is the space of square integrable spinor sections;
is the Dirac operator
is the space of smooth functions on .
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 entirely by this algebraic data. In particular the Riemannian metric on is encoded in the operator spectrum of , 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.
The standard textbook is
The notion of spectral triple and of spectral action was introduced in
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)
A brief indication of some of the central ideas going into this is at
A summary of this is in
M-T. Benameur, T. Fack, On von Neumann spectral triples (web)
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)