Dan Freed has a wonderful preprint out on the arXiv this evening, based on a talk he gave at the celebration of MSRI’s 25th anniversary, entitled Remarks on Chern-Simons Theory. It’s mainly about the current state of attempts to better understand the mathematical significance of the Chern-Simons-Witten quantum field theory.
This is a truly remarkable and very simple 3d quantum gauge theory, the significance of which Witten first came to understand back in 1988. He quickly showed that the theory brought together in an unexpected way several quite different but important areas of mathematics and physics (3d topology, moduli spaces of vector bundles, loop group representations, quantum groups, 2d conformal field theory among others). This work was the main reason he was awarded a Fields medal in mathematics in 1990. While Witten and others worked out many important aspects of this story back then, many important puzzles still remain, and it is these that Freed concentrates on.
Perhaps the biggest puzzle is that of how to actually define the theory in a local manner. The standard definition thrown around is that this is just the QFT with Lagrangian given by the Chern-Simons number CS[A] of a connection A, so all one has to do is evaluate the path integral
$$\int [dA] e^{i2\pi k CS(A)}$$
While this is a good starting point for a perturbative expansion at large k, it doesn’t appear to make much sense non-perturbatively. Freed points out that it is known that the theory must depend on additional topological structure on the 3-manifold (e.g. a 2-framing), whereas the path integral looks like it only depends on the orientation. If you try and think about how you would actually calculate such an integral numerically, by discretizing it and taking a limit, it looks like you will end up with something hopelessly dependent on the details of the discretization and the limit. For a much simpler toy example with some of the same problems, consider the path integral on closed curves on a sphere, taking as Lagrangian the enclosed area.
Freed describes in detail the state of attempts to rigorously define the theory without dealing with the path integral, but instead exploiting the fact that it is supposed to be a topological qft, and thus may have an abstract definition in terms of generators and relations. He describes the current situation as follows:
There is a generators-and-relations construction of the 1-2-3 theory via modular tensor categories for many classes of compact Lie groups G. This includes finite groups, tori, and simply connected groups, the latter via quantum groups or operator algebras. There are new generators-and-relations constructions – at this stage still conjectural – of the 0-1-2-3 theory for certain groups, including finite groups and tori. There is an a priori construction of the 0-1-2-3 theory for a finite group. There is an a priori construction of the dimensionally reduced 1-2 theory for all compact Lie groups G
The bottom line is that we only have a local construction of the theory for the case of finite groups, where one can make perfectly good sense of the path integral. For the case of a 3-manifold that is a product of a circle and a Riemann surface, one can define things in terms of a 2d theory, and Freed explains the connections to the Freed-Hopkins-Teleman theorem.
To convince mathematicians that there is something to the path integral, Freed writes down the asymptotic expansion for large k that it leads to, and shows that this gives a highly non-trivial conjecture relating quite different mathematical objects associated with a 3-manifold. He shows strong numerical evidence for this conjecture.
Finally, he ends with some extensive and interesting comments about the relationship between quantum field theory and mathematics, as it has been pursued by both physicists and mathematicians over the past quarter-century, with some speculation about what direction this might take in the future.
I wouldn’t normally have commented here as the level is usually well above me and the discusssion is usually frantically mathematical (or overly personal). But, it seems like nobody else has commented yet so I’ll take this opportunity to ask the simplest thing: what does this mean:
“For a much simpler toy example with some of the same problems, consider the path integral on closed curves on a sphere, taking as Lagrangian the enclosed area.”
Simple as it may be I’ve forgotten loads of my differential geometry and QFT from college, as without use, maths just seems to evaporate, and would plead an indulgence if there are no better replies t deal with. Does the patah integral relate to the enclosed are via stokes?
-james.
Off topic:
Could you please recommend a good survey of conformal field theory for mathematicians? Something where one could find the main ideas explained in geometric terms, without long formulas with indices all around, e.g. Atiyah’s “The geometry and physics of knots” etc.
James,
Yes, via Stokes you can expressed the enclosed area as an integral over the closed curve. This is well-defined modulo the total area of the sphere (you have a choice to make about what is “inside” the curve, what is “outside”, the difference is the area of the sphere)
This is just the action though. To get a path integral, you are supposed to multiply by i, and exponentiate, getting a phase, then, integrate this phase over the infinite dimensional space of all closed paths. It’s this last step which is highly problematic.
Knight,
One place to start is Gawedzki’s “lectures on conformal field theory”, in the IAS volume (Quantum Fields and Strings, a course for mathematicians).
Graeme Segal’s notes on CFT give a different and quite interesting perspective, definitely few indices there…
Peter,
I have to admit that this one is a little over my head as well, although I seem to remember that path integrals are integral (sorry!) to QFT so just to inquire about your comment,
“If you try and think about how you would actually calculate such an integral numerically, by discretizing it and taking a limit, it looks like you will end up with something hopelessly dependent on the details of the discretization and the limit”,
Forgive my lack of understanding if it seems too simple a question, but would it help to use a Taylor series for the exponential and to find an approximation on the discrete terms or does this action through away the important parts of the result?
mike,
The problem is that the space of closed paths is infinite-dimensional, so you’re trying to integrate over an infinite-dimensional space. Normally what you do in such a situation is pick some way of discretizing the problem, so that it becomes finite dimensional, then take the limit as the discretization goes to zero. The problem isn’t that you can’t do this kind of calculation, it’s that the limit (if it exists) is going to depend on the details of how you discretized.
This kind of path integral is quite different than the usual ones. There you can analytically continue in time, and get something with gaussian fall-off, that can be made well-defined. Here if you analytically continue, the phase stays a phase.
As far as I can tell, the situation here, like Chern-Simons, is that some additional structure is needed that will allow a well-defined formulation of the integral.
Peter,
I am not sure how you would make the integral converge, but an alternate guess is that perhaps you can try representing the discrete function as a complex function, like a z-transform. Then, instead of taking the limit as the discreteness goes to zero, you can just look at the causality of the function which produces a region of convergence for solutions that are stable (ROC < 1). This is sort of a “landscape” solution that graphically may show where the bounds of the solutions lie in complex space. If there is convergence on the path integral, I would guess it should be bounded when represented as a complex function, but maybe that’s a big guess.
Peter,
Many thanks for your reply.
I said I had forgotton most of the maths I once knew – which is true and I regret it – but I did remember the (generalised) Stoke’s theorem – honest! It’s one of those unifying beautiful results that you get taught and time does not erode. Thanks all the same for taking time to explain things, which you did very clearly.
My real concern was about the path integrals, and I was hoping your “toy” example would help me out. I still remember a patchy smattering of measure theory, but, how would you try to get a measure on the space of all – I guess simple closed curves (?) – on a sphere? What branch of maths helps you out here? (probably “measure theory” Doh!), but, from what I’ve read, construction of suitable measures for the job is a real problem.
Do you think that, in the long run, these path integrals will become rigorously defined, and, like Newton/Leibnitz calculus, Dirac deltas, differentials /”one-forms”, etc, become important branches of maths in the future?
-James.
James,
My point here is that the problem of making sense of this kind of path integral is ill-posed. I don’t think you can do it without adding some structure and changing the problem. This is true both for Chern-Simons and for the toy model. You can change the toy model problem into a QM problem where the answer is clear and given by representation theory. There you find you need to add a kinetic term and fermionic terms to the path integral. I suspect something similar needs to be done for CS, but don’t know how to do this.
The abelian Chern Simons theory for the electromagnetic field provides lots of useful insights. For example the spin-density of the electromagnetic field, which can be expressed as S = D x A + HV. The combination of this pseudo vector together with the axial Dirac current forms a preserved quantity.
The concept of an EM spin density has yet to trickle down from QFT to the more general EM audience. I devoted a chapter in my coming QFT book to work out two elementary examples:
The EM spin density of an electron with charge and magnetic moment as well as the spin density of arbitrary polarized radiation from an atomic spin 1 transition. Both examples result in what you would expect from an EM spin density.
http://www.physics-quest.org/Book_Chapter_EM2_ChernSimonsSpin.pdf
Regards, Hans
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