A year ago, username @Greg Graviton asked in a thread here about the Spin group as covering of the spatial rotations.

A subquestion was: What other groups, even larger than SU(2) are there that (could) act on "physics" and are an extension of SO(3) Is it possible to classify all possibilities, in particular the ones that are not direct products?

My answer to this was: Of course, there are many more groups relevant for Quantum Mechanics and other physical theories (like gauge groups, as you mentioned). But concerning your second question, as far as "rotations only" go, SU(2) is already the universal covering group. These are the simple connected Lie-Groups, which lie on top of these upward chains you're searching for.

Now recently I came across a construction for structures lying above the spin structure, namely string stucture and then fivebrane structure. There are mainly the nLab references, and there is a thread on MathOverflow, relating the Whitehead tower to physics. Without knowing too much homotopy, I can still see the definitions seem geometrical and hence natural to me. The definitions seem to be quite general constructions over the manifold and are the discussed in the context of QFTs.

I make some observations which surprise me a little and I want to ask these specific questions:

  • The title of the thread tries to address this question. Their names kind of suggest it, but do these mathematical objects, the objects above the spin structure, necessarily have to do with string theory? Now this question is problematic in as far as string theory people will say QFT is generally the necessary conclusion when studying QFTs. I want to filter this thought out for a moment and ask: If you want to do generally non-stringly quantum field theory, do you for practical reasons want to spend your time learning about these objects? Do you come across them just by learning about Dirac operators?

  • The presentation for field theories discussing this focus on a map $\Sigma\to X$, were $\Sigma$ gives the field configuration and $X$ is a manifold. I figure I have to view this just as generalized element of the manifold. Now I find it curious that the Spin, String and fivebrane structure seems to be something build over $X$ first, prior to considering physically propagating things in that world. This reverses my thinking of the spin being something associated with the particle, not with the space it lives in. When following this approach, to what extend to I think of the spin (bundle, etc.) to be a property of the world the object lives in, and then what is the object itself, if we say it only gets these feature when it's put into $X$?

  • Why fivebrane? Why not some other number?


1 Answer 1


Concerning the first question, as usual in string theory there are two steps: first there is an ordinary 2-dimensional QFT on the worldsheet, and then second string theory proper is what is given by the S-matrix formed by summing the correlators of the 2d QFT over all genera. The first step however is pure QFT and can be regarded as such. From this perspective, string structures are a condition for anomaly cancellation in 2-dimensional QFT with fermions. Indeed, the condition appears in various 2d QFTs which had been motivated from string theory (arise from the string worldsheet QFT in some limit) but are not string theoretic proper anymore, most famously maybe what has come to be known as the beta-gamma system. And, yes, these anomaly cancellation conditions do arise from considering Dirac operators, namely they are the conditions that the fermionic path integral for the fermions in the 2d theory yields a globally well defined bosonic action functional (see at quantum anomaly -- Anomalous action functional).

A good review of this is in

  • Konrad Waldorf, Geometric string structure and supersymmetric sigma-models (pdf)

This leads us to your last question: in the same fashion the worldvolume theories of the higher dimensional fundamental branes by themselves are just ordinary QFTs and if you want to ignore the string theoretic ambient interpretation, then you can regard them as such. It so happens that the next dimension in which worldvolume fermions induce an interesting anomaly is in dimension 5+1. This has to do topologically with the structure of the Whitehead tower of the orthogonal group, as you note. Now this 6-dimensional QFT with this next higher anomaly happens to arise in string theory as the worldvolume theory of the fundamental 5-brane in dual heterotic string theory. Whence the name "5-brane structure", due to Hisham Sati. This is indicated a bit in our "Twisted differential string and fivebrane structures".

Concerning your second question, finally, you should probably first think about this for spin structures: while the worldline theory of the spinning particle is a 1-dimensonal QFT with fermions on the worldline, after quantization the anomaly that these pick up does depend on the topology of the target space. Intuitively you may think of this as being due to the fact that the qiantized object scans the whole target space, via its wavefunction. Very useful discussion of ordinary spin structures as quantum anomaly cancellation conditions for this 1d fermionic field theory is in

  • Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

This article is designed as a warm-up for understanding the anomalies of the superstring and higher superbranes, so maybe try to get hold of it.


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