Notes on Lattice Topological Field Theory in Three Dimensions, Part I

I have recently been interested in constructing topological field theories for loop quantum gravity, with a specific eye towards the topspin construction. With this in mind I was lead to the classic paper by Dijkgraaf and Witten, “Topological Gauge Theories and Group Cohomology” (1990). I am working my way through it to try and understand – which usually requires taking notes on the paper and working on things I don’t understand. I thought maybe I would post my notes here.

Introduction

For an oriented, smooth 3-manifold M we can specify a principal bundle E=(M,\pi,G) with a gauge group G. An action on this bundle must be constructed from some kind of invariant, to make sure it is somehow independent of the choice of connection A. If E is trivial, the connection is a Lie-algebra valued one-form A\in \Omega^1(M)\otimes \mathfrak{g} and the action is the Chern-Simons functional

S(A)=\frac{k}{8pi ^2}\int_M Tr(A\wedge dA + \frac{2}{3}A\wedge A\wedge A).

The path integral is Z(M)=\int\mathcal{D}A\exp (2\pi i S(A). The integer k is called the level of the theory, and must be an integer. This is because when we take G to be non-abelian, under a gauge transformation the action goes to S(A^g)=S(A)-2\pi k N for some integer N (http://arxiv.org/abs/hep-th/9902115). Thus in order for the amplitude \exp(S(A)) to remain a gauge-invariant, k must at least be an integer.

If E is not a trivial bundle, the action is not a global gauge-invariant. However, since we know curvatures are good gauge invarients (in 4 dimensions), we can use them to define a global Chern-Simons action. So, assume we can extend the bundle E over a smooth four-manifold B. Then we could write

S(A)=\frac{k}{8\pi^2}\int_B Tr(F\wedge F),

where F is the usual curvature of the connection A. If we have a trivial extension of the connection A(x)=\partial_t A dt+A(\vec{x}) then the curvature is only a function of the coordinates \vec{x} on M:

F(x)=dA+A\wedge A=dA(\vec{x})+A(\vec{x})\wedge A(\vec{x}). The question remains how to do this in general, since one cannot always find such an extension of the connection to the boundary (I basically chose B=\mathbb{R}\times M, but we cannot guarantee that E can be extended in the way that I extended A).

New Paper and Presentation on the Fundamental Group of the Spatial Section in LQG

Today I posted a version of the talk I gave at LOOPS 13 to the archive (1308.2934). You can also see the original slides. In principle this will be a longer paper soon, but I have some new ideas after my attendance at LOOPS 13 and I’d like to add them before doing something official.

Basically, classical gravity has nothing to say about the topology of the universe, and one would hope that quantum gravity would. By topology of the universe I mean large-scale structures – general relativity actually can’t tell if the universe is a sphere or a torus, but it can tell us that it appears to be locally flat. There could be any arbitrarily many numbers of “holes” in the universe, as long as the universe itself and the holes were both large enough that things still look flat in our little piece of things.

This means that we have to specify (read: put in by hand) the topology of the universe. Maybe not such a big deal since the model seems very good, but if quantum gravity is really “fundamental” then we don’t really like having to tell it what to do. Since there are a large number of different options for the topology (something like “as many integers as you can count”) specifying one in particular seems a bit strange. It’d be really nice if quantum gravity told us which one is the correct one – and even better if it matched observations!

That’s what my current work is focusing on, which started with an idea from Denicola, Marcolli, and al-Yasry (1005.1057). Without more details, the basic idea is to modify a current approach to quantum gravity (loop quantum gravity) to include topological data. That would allow us to ask questions about the topology, like “where did it come from?” and “what topology do we predict the universe to have if this theory is correct?” At this stage we are just checking to make sure the various constructions of LQG still work, and in this paper I am showing how to actually calculate some topological information. Hopefully this will lead to some calculations which show that LQG a) does include topological information, and b) predicts the topological structure of the universe.

Parallel Talk at the Perimeter Institute

Today I gave a talk entitled “Studying Topology Change with Topspin Networks” during the LOOP13 conference at the Perimeter Institute:

 

http://pirsa.org/displayFlash.php?id=13070085

 

This is the entire session; I start around minute 43. The talk was in the wrong session and late in the conference, and was thus not too well attended. However, the discussion at the end was very nice, and despite me saying a few strange things (“topologically zero fundamental group”? Even worse, I said “homeomorphism” and meant “homomorphism”…that’s a kindergarten math mistake people).

 

In retrospect, do not let anyone tape you giving talks, ever.