nLab spectral triple

Idea

Algebraically

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

1. 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 $X$;

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 $\mathcal{H}$ 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.

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, 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

$$\array{ \bullet \\ & \searrow \\ && \bullet & \to \\ & \nearrow \\ \bullet }$$

subject to the obvious associativity condition.

Then a spectral triple $(A,H,D)$ is the data encoding a sufficiently nice smooth functor

$$Z_{(A,H,D)} : R Cob_{1|1}^{Feyn} \to sVect$$

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

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

• $H$ 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

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

(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

• $A = C^\infty(X)$ 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.

Properties

Metric dimension and KO-dimension

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.

Related concepts

• spectral geometry

• hearing the shape of a drum

• KK-theory

• spectral action

• Connes-Lott-Chamseddine model

• 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)

See also

• Alain Connes, Matilde Marcolli, chapter I, section 10 of Noncommutative Geometry, Quantum Fields and Motives

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 (doi:10.1063/1.531241)

and that of spectral action in

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

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

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

See also

• Alan Carey, John Phillips, Fredholm modules and spectral flow J. Canadian Math. Soc. 50 (1998) 673-718. (publisher)

• R. da Rocha, A. A. Tomaz, Hearing the shape of inequivalent spin structures and exotic Dirac operators (arXiv:2003.03619)

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).

Spectral triples over Jordan algebras:

• Latham Boyle, Shane Farnsworth, The standard model, the Pati-Salam model, and “Jordan geometry” (arxiv:1910.11888)

• Fabien Besnard, Shane Farnsworth, Particle models from special Jordan backgrounds and spectral triples $[$arXiv:2206.07039$]$

See also

• Ludwik Dabrowski, Andrzej Sitarz, Multiwisted real spectral triples (arXiv:1911.12873)

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 Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, AMS 2011

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)

Discussion in the context of D-brane matrix models is in

• T. Asakawa, S. Sugimoto, S. Terashima, D-branes, Matrix Theory and K-homology, JHEP 0203 (2002) 034 (arXiv:hep-th/0108085)

On fermionic path integrals via finite-dimensional spectral triples:

• John W. Barrett: Fermion integrals for finite spectral triples [arXiv:2403.18428]

von Neumann spectral triples

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

This goes back to (Carey-Philips 98) and

• Alan L Carey, John Phillips, Fyodor Sukochev, On unbounded $p$-summable Fredholm modules (arXiv:math/9908091)

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

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)

Examples

On the spectral Riemannian geometry of the fuzzy 2-sphere:

• Francesco D'Andrea, Fedele Lizzi, Joseph Várilly, Metric Properties of the Fuzzy Sphere, Lett. Math. Phys.103 (2013), 183-205 (arXiv:1209.0108)