Introduction to Non-Commutative Geometry

I gave a course of lectures at the ESI in Vienna on Non-Commutative Geometry in October 2022. The draft lecture notes are uploaded here. A similar series of lectures will be given at the Galileo Galilei Institute June 12-16th 2023.

Chapters: 1. Introduction 2. Fuzzy spaces with symmetry 3. Commutative analogues 4. Dirac operator 5. Real structures 6. Real spectral triples 7. Standard model charges 8. Standard Model masses 9. The fuzzy sphere 10. The fuzzy torus (+4 more chapters to come)

The old October version is here for those that did the coursework.

Please let me know any corrections or improvements via email.

Penrose on palatial twistors

The main purpose of this post is to publish a draft manuscript version of the research paper Chameleon Twistor Theory: a Geometric Programme for Describing the Physical World by Roger Penrose, with his permission. This will eventually appear in a book later this year and I will add the bibliographic information when I have it.

Roger Penrose talking about the paper in 2018

Abstract Original motivations are recalled, for the introduction twistor theory, as a distinctive complex-geometric approach to the basic physics of our world, these being aimed at applying specifically to (3+1)-dimensional space-time, but where space-time itself is regarded as a notion secondary to the twistor geometry and its algebra. Twistors themselves may be initially pictured as light rays—with a twisting aspect to them related to angular momentum. Twistor theory provides an economical conformally invariant description of quantum wave functions for massless particles and fields, best understood in terms of holomorphic sheaf cohomology, subsequently leading to a non-linear description of anti-self-dual (“left-handed”) gravitational (and Yang-Mills) fields. Attempts to remove this anti-self-dual restriction (the googly problem) led to a 40-year blockage to the development of twistor theory as a possible overall approach to fundamental physics. However, in recent years, a more sophisticated approach to this problem has been developed—referred to as palatial twistor theory—whose basic procedures are described here, where a novel generating-function approach to Λ-vacuum Einstein equations is introduced.

 

I will add my own thoughts on this paper here later.

Random non-commutative geometry

Lisa Glaser and I have started to investigate random non-commutative geometries in our latest paper
Monte Carlo simulations of random non-commutative geometries. The geometries are all specified by a Dirac operator that is a finite-dimensional matrix, so in this theory the partition function is a finite-dimensional integral. We investigated the phenomena that occur by calculating this integral on a computer, in much the same way as you would measure data from an experiment in a lab.

The most fascinating finding is that by tuning a parameter in the action we are able to find a phase transition (audible here!), and at this transition the spectrum of the Dirac operator looks a lot like the spectrum one would expect for a typical manifold in the usual sense of differential geometry. This is surprising, because non-commutative geometries generally look nothing like differential geometries.

Eigenvalues for the 10×10 fuzzy sphere

The first plot shows the eigenvalue spectrum for the fuzzy sphere using an algebra of 10×10 matrices. The plot has the eigenvalues of the Dirac operator on the horizontal axis and the density of eigenvalues vertically.

The spectrum for the fuzzy sphere is the same as for an ordinary two-sphere in differential geometry, except that the spectrum is cut off after a finite number of eigenvalues (in this case, 200). These spheres are special in that the eigenvalues are in fact all integers, and have a multiplicity proportional to the eigenvalue, due to the spherical symmetry.

A Dirac operator on an arbitrary metric on a surface would not have this symmetry or the degeneracy, but the approximate V-shape of the plot should be the same, at least when the eigenvalues are large enough.

Distribution of eigenvalues for random NCG near a phase transition

This is exactly what one sees in the second plot, an approximate V-shape with some slightly different detail for small eigenvalues. But the second plot is generated by an ensemble of random non-commutative Dirac operators!

Actually, we nearly didn’t find this. We looked at a range of different random NCG models with a free parameter and accumulated a large number of eigenvalue plots that looked nothing like a manifold. Only when we started to write a section of the paper explaining this did it become clear that a very small number of the plots did look something like a manifold. These plots occur only close to a phase transition, where the properties of the ensemble change sharply when the parameter in the model is changed.

Our findings fit into a general theme for discrete theories, like lattice gauge theory, where continuum-like behaviour occurs at, or near, a phase transition. Thus it seems like a great place to look for quantum gravity in non-commutative models.

Matrix Quantum Gravity

The idea behind the project outlined in my Como lectures is to model a space-time geometry with a (finite-dimensional) matrix. Then quantum geometry (or random geometry) can be modelled by a suitable integral over the space of matrices, which is again a finite-dimensional integral, and thus well-defined.

In general terms, integrating over a space of matrices has been studied a lot, and goes by the name of a “matrix model”. However, in this case the matrix variable is a very specific type of matrix, namely a Dirac operator. The Dirac operator satisfies a number of algebraic conditions, so my new quantum gravity can be thought of as a matrix model for a very specific type of matrices.

The point of using Dirac operators is that a Dirac operator specifies a geometry. In the classical case, the Dirac operator is a differential operator on a manifold and one can extract the metric tensor from it. In the case of a matrix, the number of spinor “modes” that the Dirac operator operates on is finite, so that there is a maximum energy. Physically, if there are matrix quantum geometry models that behave like quantum gravity, this maximum energy should be identified as the Planck scale. So the mysterious Planck-scale cutoff that is needed to make sense of quantum field theory is built in to the model in a fundamental way. Whether there really are viable matrix quantum gravity models is very much an open question, and the point of The Project is to try to answer this.

To define a matrix Dirac operator in a consistent way, the notion of geometry is generalised to “non-commutative geometry”. In this setting, a lot of geometric constructions still make sense. One hopes, for example, that there is a suitable notion of the limiting case of large matrices in which a sequence of matrix Dirac operators converges to one of the usual Dirac differential operators (“taking the Planck scale to infinity”). More loosely, one can understand constructions in non-commutative geometry as the algebraic analogues of familiar constructions in ordinary metric geometry on a manifold. So it is important to have all of the machinery of non-commutative geometry (technically, the Dirac operator is part of a “real spectral triple”).

The basic set-up for matrix geometries is explained in my paper “matrix geometries and fuzzy spaces as spectral triples”. The paper contains the general definitions and a lot of useful data, such as operations on gamma matrices and explicit forms for the Dirac operator. I’m hoping it will be a resource for future efforts at modelling quantum geometry.

Como lectures

I’m giving some lectures at the SIGRAV school in Como on “Non-commutative geometry and quantum gravity”. I’ll put the slides up here as they happen.

Lecture 1: The Planck scale. This lecture reviews the experimental evidence for a fundamental Planck scale. Various theoretical approaches to modelling the Planck scale are mentioned very briefly, mainly concentrating on the predictions of the spectral action suggested by non-commutative geometry.

Lecture 2: Non-commutative geometry: axioms, examples. The axioms of non-commutative geometry are given in this lecture alongside an explanation of how they are satisfied by an ordinary (commutative) manifold. The example given by the standard model internal space is also summarised.

Lecture 3: Matrix geometries and fuzzy spheres. The lecture explains my project to replace the usual space-time manifold with a non-commutative geometry based on finite-dimensional matrices. Examples with spherical symmetry are considered.

Lecture 4: State sum models, using 2d examples. A different topic is introduced in this lecture: 2d topological state sum models defined using a diagrammatic calculus. Some recent progress in defining models that are sensitive to the spin structure of the surface (and are thus fermionic) is explained.

Lecture 5: Non-commutative geometry in 2d state sum models. The final lecture connects the topic of state sum models and non-commutative geometry. It shows how defects in a 2d state sum model can carry a non-commutative geometry.

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Giving the lectures turned out to be enormously useful. I’m writing a paper called “matrix geometries and fuzzy spaces as spectral triples“, which I hope will appear soon is out now. The paper will be much better for having had to explain the ideas in the lectures.