The n-Category Café

Will there be any talks or discussions that are open to a more general public?

It may be a little bit premature to collect links to relevant material before there is a program and abstracts of the talks, but thanks to the internet I already had the opportunity to watch the talks at the AQFT - the first 50 years conference in Göttingen in 2009, including the ones by Klaus Fredenhagen, Stefan Hollands, Gandalf Lechner and Roberto Longo.

Maybe the organizers will group the talks thematically, so that the AQFT talks are scheduled on two or three consecutive days.

There is a tentative schedule of talks now:

Tue 14.6.: Jacobson, Arnlind, Rovelli

Wed 15.6.: Blau, Ashtekar, Lewandowski, Ambjorn

Thu 16.6.: Baez, Beisert, [seminars by other participants]

Fri 17.6.: Bachas, Hollands, de Goursac, Mukhanov

Mon 20.6.: Dixon, Hoppe, Bossard, Lechner

Tue 21.6.: Longo, Elvang, Shaposhnikov, Barrett

Wed 22.6.: Craps, Reuter, Loll, Freidel

Thu 23.6.: Wulkenhaar, Reiterer, [seminars by other participants]

Fri 24.6.: Chamseddine, Nicolai

Both Ambjørn and Loll will be here, and presumably speak about their work on causal dynamical triangulations. Because this approach to quantum gravity discretizes slices up spacetime into surfaces of “constant time”, I’d always been worried that it was quantizing not general relativity but some other theory—one that has a built-in separation between time and space. Lately there’s been some new work delving into this question.

While general relativity with cosmological constant is singled out as the only theory with a diffeomorphism-invariant action built by integrating low-order derivatives of the metric, a splitting of spacetime into space and time lessens the symmetry of the theory and thus allows for more choices. Apparently all these choices are included in the theory called ‘Hořava-Lifshitz gravity’.

A new paper by Sotiriou, Visser and Weinfurtner suggests that causal dynamical triangulations could be a discretization of Hořava-Lifshitz gravity:

We explore the ultraviolet continuum regime of causal dynamical triangulations, as probed by the flow of the spectral dimension. We set up a framework in which one can find continuum theories that can fully reproduce the behaviour of the latter in this regime. In particular, we show that Hořava-Lifshitz gravity can mimic the flow of the spectral dimension in causal dynamical triangulations to high accuracy and over a wide range of scales. This seems to indicate that the two theories lie in the same universality class.

This paper reports on numerical work in 2+1 dimensions, not 3+1. But if causal dynamical triangulations is quantizing Hořava-Lifshitz gravity, that raises some big questions.

On the 14th Ted Jacobson kicked off the conference with a talk on the generalized 2nd law of thermodynamics, in which entropy increases if areas of ‘causal horizons’ are included as part of the definition of entropy. His student Aron Wall has written 3 papers on this subject, and his talk described this work:

• Aron Wall, Ten proofs of the generalized second law.
• Aron Wall, A proof of the generalized second law for rapidly-evolving Rindler horizons.
• Aron Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices.

The first paper critiques previous work. The second proves the generalized 2nd law in a semiclassical framework on a background spacetime that has both boost and null translation symmetries. The argument makes heavy use of the concept of relative entropy. The third paper generalizes the second one to a much wider class of spacetimes, assuming some axioms about the behavior of quantum fields on these spacetimes.

Here are a few very sketchy notes on the conference. You can see talk slides here.

I think I’ll focus on the ideas of Jens Hoppe, since Urs was interested in his work.

First Joakim Arlind, a student of Hoppe, spoke about Poisson algebraic geometry and matrix regularizations. There’s a well-known ‘matrix regularization’ of membrane theory, developed by Hoppe and others, where you replace the algebra of functions on a Riemann surface by $n \times n$ matrices. In a suitable limit as $n \to \infty$, we’re supposed to get the theory we really care about. The algebra of functions on a Riemann surface is a Poisson algebra thanks to its symplectic structure, and in the matrix regularization we replace multiplication of functions by matrix multiplication, Poisson brackets by commutators, and the integral of a function by the trace of a matrix.

I don’t think this plan ever succeeded in achieving what people want most of all, namely quantizing the supersymmetric 2-bane in 11 dimensions. There are some difficult issues of analysis involved in taking the limit, which are nicely summarized in this thesis:

• Douglas Lundholm, Zero-energy states in supersymmetric matrix models, PhD thesis, Stockholm, 2010.

The later parts of this thesis involve the spectra of Schrödinger operators, and any expert on those who wants to become famous should try to solve the open questions raised by membrane theory!

However, the idea has spawned a lot of interesting side-projects, one of which is the idea of generalizing ideas from geometry from manifolds to matrix algebra: a branch of noncommutative geometry. This is what Arlind’s talk was about.

Hoppe himself gave a blackboard talk, and unfortunately my notes only cover some fairly basic stuff. In a lunchtime conversation he did however explain to me how Poisson brackets are naturally replaced by Nambu brackets when we go to higher-dimensional membranes. I never really understood Nambu brackets in classical mechanics, so I was relieved to hear that Hoppe didn’t either. For him, it seems, Nambu brackets are only interesting when we start with a Riemannian manifold equipped with its volume form. For example, if we have a 3-manifold with volume form $\omega$, we can use the Riemannian metric to turn this into a trivector field with components $\omega^{i j k}$ and then define Nambu brackets by

$$\{f, g, h\} = \omega^{i j k} \partial_i f \partial_j g \partial_k h$$

The analogue of ‘matrix regularization’ in this case seems to remain obscure. One obvious guess is to use 3-index objects replacing matrices, but it’s still not obvious (at least to me, and perhaps to him either) how to generalize the matrix commutator in that case. He said he’d looked at Gelfand and Kapranov’s book on hyperdeterminants, without success.

He really liked my talk on Lie super-2-algebras, because he thinks we need to systematically boost up all concepts as we go from particles to strings to higher-dimenmsional membranes.

On a side note, Douglas Lundholm pointed out a certain Fierz identity which only holds in dimension 2, 3, 5 and 9. I think it appears on page 352 of his thesis, but also somewhere else in there. These dimensions are 1 less than the dimensions that superstrings and super-Yang-Mills theory enjoy most (3,4,6,10), I think because he’s doing light-cone quantization and suppressing one dimension. But as usual they’re related to the reals, complexes, quaternions and octonions.

He showed me a way to quickly check that this Fierz identity only holds in these dimensions, simply by summing over certain indices and showing this implies a relation between the dimension of the space of vectors and the dimension of the space of spinors. I’ve been a bit slow about checking that my favorite Fierz identities only hold in certain special dimensions. I didn’t think it could be this easy!

People interested in the surprisingly good behavior of $N=8$ supergravity in 4 dimensions might like the PDF files of talks by Guillaume Bossard and Henriette Elvang. There are some heroic calculations being done here!

Elvang’s talk gives a nice account of how the theory can be shown finite up to… 7 loops, it seems… using only symmetry arguments. Both $SU(8)$ symmetry and the hidden $E_{7(7)}$ symmetry play a role here.