I’ve recently been looking at the following paper in which -TQFT anomalies are treated carefully and various old constructions of Turaev and Walker are elucidated:
Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math. 25 (2013), 1067-1106.
It makes me think a lot about just what the anomaly `actually means’…
I’ll start with some vague philosophical musings. I’m quite taken with the information physics idea that everything is information, and I think that Chern-Simons theory should really be all about information as well. But I’m not sure how. A google search turns up load of physics papers with keywords “anomaly”, “Chern Simons”, and “entropy” in close proximity, so I’m sure that some physicists know the whole story, but I don’t. Maybe somebody could explain it in the comments?
There’s a theme in physics which says that the `interesting’ information content of naturally occuring systems on `things with boundaries’ is contained entirely on the boundary and not in the interior. Manifestations of this theme include the holographic principle which roughly claims that the maximal entropy in a region scales like the surface area of the boundary of that region instead of like its volume (so that the entire information content of a black hole lies on its event horizon), and area laws which roughly claim that the amount of quantum entanglement between particles in a region and in its complement depends on the area of the boundary of the region and not on the volume of the region.
Because every closed oriented –manifold bounds an oriented -manifold, this physics theme suggests a way in which physics might be unreasonably effective in low dimensional topology. Namely, a physically interesting information measure on a bounded -manifold ought to give rise to a -manifold invariant. This is sort-of the meta-intuition I have for why we have Topological Quantum Field Theory (TQFT) invariants of -manifolds. My vague feeling is that because Fisher information and the Chern-Simons action both have something to do with curvature, perhaps Chern-Simons Theory and quantum -manifold invariants have a clear and legitimate information-physics interpretation (if you know what it is, please tell me!).
Be that as it may, it turns out that quantum invariants coming from -dimensional TQFTs tend not quite to give numerical topological invariants of their –dimensional boundaries. We need an integer worth of extra information from the interior of the bounded -manifold to get a numerical –manifold invariant. This `extra information from the interior’ is called the anomaly. The anomaly ought not to even exist for `physically interesting information’ according to the naive interpretation of the physicist’s theme outlined above. Maybe that’s why it’s called an anomaly– because a physicist would wish that it not exist. A lot of surveys and entry-level texts seem to gloss over the anomaly, maybe partly for that reason.
It seems to be only recently that anomalies are becoming respectable. Perhaps this is due to Lurie’s higher categorical formalisms which sheds some light on anomolies, and perhaps to work on TQFT’s for manifolds with boundaries and corners, and perhaps to interest in “type II superstring orientifolds” in which anomalies are both tricky and important, but in any event, there does seem to be a resurgence of interested in anomalies. It seems that an anomaly should be considered an (invertible) field theory itself. This paper is the most interesting recent paper I have seen on the subject… maybe I’ll talk about it another time.
Back on the subject of Chern-Simons TQFTs, or rather Reshetikhin-Turaev TQFTs, our setting is a closed oriented -manifold bounding an oriented -manifold. This -manifold matters only up to cobordism (i.e. two cobordant -manifolds are considered equivalent from the point of view of our TQFT, because, while there may be a wee bit of information in a dimension 4 interior, there is no relevant information in a dimension 5 interior). Cobordism classes of oriented -manifold are classified by the signature (an integer), and I think that this is the secret reason that the signature keeps popping up in all kinds of formulae for quantum invariants of -manifolds.
So really, the domain of our TQFT ought to be a pair consisting of a manifold and an integer, or a manifold with an integer-worth of extra structure. How to usefully specify that integer? There are a variety of of approaches- structures, -framings, various choices of largrangian thisses or thats, Masbaum-Roberts explicit methods… There’s an algebraic approach as well, in which we trade our -manifold for a mapping class group element. Remember how every -manifold has a Heegaard splitting? This constructs our -manifold by gluing together two genus handlebodies using an element of the mapping class group . The TQFT induces a representation of the mapping class group which is only projective but not linear because of the anomaly. Gauge-fixing/ choosing a cobordism class of a bounded -manifold/ fixing the anomaly corresponds to choosing a central extension of the mapping class group. And it turns out the has a universal central extension, and (unsurprisingly) the cohomology class of this extension is a generator of a cohomology group which is isomorphic to the integers (the most famous such generator is the Meyer cocycle, and the second most famous is its cohomological negative, the Maslov cocycle).
So the whole problem of fixing the anomaly has been algebratized, and the goal has now become to describe explicit elements of the universal central extension of , which are the algebraic objects which have now replaced “-manifold together with a cobordism class of -manifolds which it bounds”.
That’s pretty-much the goal of Gilmer-Masbaum. Some major steps which were outlined in Walker’s iconic TQFT notes are worked out explicitly. This is at long last a careful treatment of a TQFT anomaly. I know more than I knew before.
Now that we have a technically coherent and careful treatment of the anomaly in the Chern-Simons context which seems more or less amenable to concrete computation (I’m haven’t followed through the details carefully enough to strengthen the above sentence), the next thing I’d love to read would be a survey-level treatment of the anomaly, which explains all of the different approaches to fixing it, the strengths and weaknesses of each, and how they relate to one another.
I’d also really like to understand how quantum invariants measure information (entropy), and in particular what information is measured by the anomaly. And what is the conceptual reason that Chern-Simons theory violates the theme that all interesting information lies on the boundary? Or maybe it doesn’t? I wish I understood more.