Contents

Idea

A spectral triple is algebraic data that mimics the geometric data provided by a smooth Riemannian manifold $X$ 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

1. a ${\mathbb{Z}}_2$-graded Hilbert space $\mathscr{H}$, to be thought of as the space of (square integrable) sections of the spinor bundle of $X$;

2. 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.

3. A Fredholm operator $D$ acting on $\mathscr{H}$, to be thought of as the Dirac operator acing on the spinors.

   This item encodes the Riemannian metric and possibly a connection.

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.

As 1-dimensional FQFTs

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 $R{\mathrm{Cob}}_{1\mid 1}^{\mathrm{Feyn}}$ be the cobordism category of Feynman graph?s 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 $(1\mid 1)$-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

• $A=Z_{(A,H,D)}(•{)}_0$ 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

  $$Z_{(A,H,D)}\left(\begin{array}{c}•\\ & \searrow \\ & & •& \to \\ & \nearrow \\ •\end{array}\right)$$Z_{(A,H,D)} \left( \array{ \bullet \\ & \searrow \\ && \bullet & \to \\ & \nearrow \\ \bullet } \right)

• $H$ is some completion of $Z_{(A,H,D)}(•)$ to a super Hilbert space

• and $D\in \mathrm{End}(H)$ is an odd self-adjoint operator on $H$, which gives the value of the functor on the super-interval $(t,\theta )$ by

  $$(t,\theta )\mapsto \exp (-t{D}^2+\theta D).$$(t,\theta) \mapsto \exp( - t D^2 + \theta D ) \,.

(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

• $A=C^{\mathrm{\infty }}(X)$ 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.

Related concepts

• KK-theory

• spectral action

• higher category theory and physics: Spectral standard model and gravity

• (1,1)-dimensional Euclidean field theories and K-theory

• 2-spectral triple

References

General

The standard textbook is

• Alain Connes, Noncommutative Geometry , Academic Press (1994)

The notion of spectral triple and of spectral action was introduced in

• Alain Connes, Gravity coupled with matter and foundation of non-commutative geometry (arXiv:hep-th/9603053)

The characterization of ordinary compact smooth manifolds in terms of spectral triples is in

• Alain Connes, On the spectral characterization of manifolds (arXiv:0810.2088)

Relation to quantum physics

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

• Yan Soibelman, Collapsing Conformal Field Theories, spaces with non-negative Ricci curvature and non-commutative geometry in Mathematical Foundations of Quantum Field and Perturbative String Theory .

A brief indication of some of the central ideas going into this is at

• Urs Schreiber, Spectral triples and graph field theory

A general introduction to and discussion of spectral triples with an eye on quantum mechanics, quantum field theory and string theory is in

• Jürg Fröhlich, Oiver Grandjean, Andreas Recknagel, Supersymmetric quantum theory, non-commutative geometry, and gravitation Lecture Notes Les Houches (1995) (arXiv:hep-th/9706132)

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:

• Daniel Roggenkamp, Katrin Wendland, Limits and Degenerations of Unitary Conformal Field Theories (arXiv:hep-th/0308143)

A summary of this is in

• Daniel Roggenkamp, Katrin Wendland, Decoding the geometry of conformal field theories (arXiv:0803.0657)

Also

• Sebastiano Carpi, Robin Hillier, Yasuyuki Kawahigashi, Roberto Longo, Spectral Triples and the Super-Virasoro Algebra Communications in Mathematical Physics, Volume 295, Number 1 / April 2010 (Springer pdf)

von Neumann spectral triples

One variation uses von Neumann algebras instead of C-star algebras.

• M-T. Benameur, T. Fack, On von Neumann spectral triples (web)

• Alain Connes, Henri Moscovici, Type III and spectral triples (arXiv:math/0609703)

Relation to K-theory

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)