### Report-Back on BMC

#### Posted by Urs Schreiber

* – guest post by Bruce Bartlett – *

I was born in a large Welsh industrial town at the beginning of the Great War: an ugly, lovely town (or so it was, and is, to me), crawling, sprawling, slummed,unplanned, jerry-villa’d, and smug-suburbed by the side of a long and splendid-curving shore…

Thus described Dylan Thomas his childhood home of Swansea, Wales - the venue of the British Mathematics Colloquium this year :

Inspired by John’s blurb about the higher categories workshop at Fields earlier this year, I thought I’d send Urs a report-back of the (admittedly less glamorous) “BMC” , and mention a few things possibly of interest to $n$-café patrons.

Alain Connes was one of the plenary speakers. He spoke about his recent work with Ali Chamseddine and Matilde Marcolli on getting the Standard Model (together with gravity) out of an elegant noncommutative geometry framework.

In the beginning of the talk, he showed a slide with the lagrangian for the Standard Model - written out in gory detail :

Gulp! It takes up the whole slide - and that’s apparantly just the
lagrangian on a * flat * background!

In Connes’ picture, spacetime is described as $X = M \times F$ where $M$ is an ordinary (`commutative’) space (it’s a manifold with a spin structure), and $F$ is a finite `noncommutative space’. In fact, the algebra representing $F$ is $F = \mathbb{C} \oplus \mathbb{H} \oplus M_3 (\mathbb{C}).$ So its a copy of the complex numbers, the quaternions, and the $3 \times 3$ matrices.

It’s really cool, at least to a newcomer like me. Physicists have often expressed their gut feeling that somehow spacetime has a discrete texture to it at small length scales… Connes’ picture makes that precise in a really simple and elegant way.

Moreover, apparantly one can turn the “noncommutative geometry crank” on the spacetime $X = M \times F$ and out pops the Standard Model - together with ‘neutrino mixing’ (whatever that means) - in all its gory detail! It’s possible that Urs has gone over all this stuff before… but I think I missed it.

[*Yes, we had a series of posts on that: I, II, III, IV – Urs *]

Another quasi-plenary speaker was Tom Leinster, who gave a cool talk on “New perspectives on Euler characteristic.” Some of this stuff has been discussed on this blog before,

but I was fascinated to see how one can calculate the Euler characteristic of fractals like the Julia set associated to a rational complex-valued function:

We also had a nice talk on “Ricci flow and geometrization of three-manifolds” by Huai-Dong Cao who I think was (back in the day) one of Shing-Tung Yau’s students. Indeed Ricci flow has also been discussed on this blog before - and how it relates to the dilaton field in string theory!

But for me, the thing I was * really * looking forward to in
Swansea was an opportunity to see the legendary “Annie’s Place”
restaurant. As I
understand it, it was at this restaurant in 1988, at the
International Congress of Mathematical Physics, that Witten, Segal and
Atiyah hatched their evil plan to combine geometry with quantum field
theory and thus revolutionize mathematics. It was during dinner that
Witten came up with the idea of Chern-Simons theory to explain the Jones
polynomial, and decided
the next day to not give the talk he had planned, but to talk about
this new theory born only the night before!

Sadly Annie’s restaurant is no longer - it’s now a French restaurant called “Didier and Stephanie” :

Tough noogies.

## Re: Report-Back on BMC

Hi Bruce!

Thanks a lot for the nice report.

Yes, the standard model Lagrangian written out in full detail looks intimidating.

It is important to realize that all the terms are of just a handful of different types, with all the remaining difference being a difference in realization of the same underlying pattern.

First there are the gauge-kinetic terms of the form $F \wedge \star F \,,$ where $F$ is the curvature 2-form of a bundle with connection. Terms of this form describe the dynamics of the “gauge fields”, the “force fields” of nature.

All other terms are bilinears in “fermions”, usually denoted $\psi$, which are sections of spinor bundles associated to the bundles with connection from above.

First there the kinetic terms of the fermions. In the component and coordinate-ridden way these are usually displayed in this context, they look like $\bar \psi \gamma^\mu \frac{\partial}{\partial x^\mu} \psi \,.$

Then there are the “minimal coupling” terms which encode the interaction between the fermions and the gauge bosons from before. With $A$ being the connection 1-form (not only is the background metric usually assumed to be flat here, but also all bundles are assumed to be trivial) these read $\bar \psi \gamma^\mu A_\mu \psi \,.$

It is clear that together these two terms really should thought of as $\langle \psi | D \psi \rangle \,,$ where $D$ is the canonical Dirac operator on the spinor bundles with connection associated to our gauge bundles.

This is clear. One thing that Connes added to this was to observe that also the fourth kind of terms, namely the Yukawa couplings, which read $\bar \psi \phi \psi$ for $\phi$ the

Higgs field, can be absorbed in a term simply of the form $\langle \psi | D \psi \rangle$ if only we allow the Dirac operator to be that associated to a noncommutative algebra.That’s the main point of Connes’ approach as far as pure particle physics is concerned.

The other observation he made is that also the gauge kinitic terms $F \wedge \star F$ may be unified with the Einstein-Hilbert term $R$ by extracting both from a heat kernel expansion of somewthing like the exponential of the trace of the squared Dirac operator.

Given these two observations, the remaining task is to carefully identify precisely the right noncommutative geometry (a spectral triple), such that the associated Dirac operator reproduces all the gory details of the standard model by just expanding $\mathrm{tr}(f(D/\Lambda)) + \langle \psi | D \psi\rangle \,.$ That’s what people have done for almost 20 years now.

For all that time, the noncommutative geometry that had been found yielded something awefully close, but not quite coinciding with the standard model.

Interestingly, all that was missing was a tiny modification in one of the gradings that enter the full definition of a real spectral triple. With that modified grading, suddenly everything nicely falls into place.

The resulting spectral triple describes a noncommutative space whose algebra of functions looks like that on $\mathbb{R}^4$ times the noncommutative algebra you mentioned.

The ordinary spectral dimension of the “internal” noncommutative factor is 0, as it should be for a “compactified” space. Curiously, the dimension of this compact space as seen by K-theory is 6 modulo 8.