# What do whitehead towers have to do with physics?

First let me say something that I don't completely understand, since I do not know enough physics. If I say anything wrong, someone please tell me:

For the spinning particle, there is a sigma-model, which is a type of quantum field theory, which describes how a spinning particle can propagate. The input data for this sigma model is an oriented pseudo-Riemannian manifold $X$ equipped with a line bundle with connection. The condition for "quantum anomaly cancellation", can be shown to be that the classifying map $X \to BSO(n)$ corresponding to the tangent bundle has a lift through the map $BSpin(n) \to BSO(n)$. Such a lift is called spin-structure.

The sigma-model for the spinning string, starts similarly, but the role of the line bundle is replaced by that of a bundle-gerbe (that is a gerbe with band U(1)). - I believe what is going on here is that a line bundle is the same data as a principal $U(1)$-bundle, and a bundle-gerbe is the same data as a principal bundle for the $2$-group $[U(1)\to 1]$. Anyhow, for this new quantum-field theory, the condition for "quantum anomaly cancellation" is that the classifying map $X \to BSO(n)$ has a lift through $BString(n) \to BSpin(n) \to BSO(n)$. In fact, $String(n)$ does not exist as a Lie group, but it does exist as a (weak) group object in differentiable stacks, which are in particular sheaves (over the category of manifolds) in homotopy 1-types.

Apparently this can be taken even further, and one can talk about a sigma-model for the so-called $5$-brane, and the condition "quantum anomaly cancellation" is that the classifying map $X \to BSO(n)$ has a lift through $BFiveBrane(n) \to BString(n) \to BSpin(n) \to BSO(n)$, and $Fivebrane(n)$ at least exists as a group object in sheaves (over the category of manifolds) in homotopy $5$-types. (Is this a sigma-model with bundle-gerbe replaced by a principal bundle for $U(1)$ promoted to a $3$-group?)

Note: We should really be starting with $O(n)$ since a lift of $X \to BO(n)$ through $BSO(n) \to BO(n)$ is the same as equipping $X$ with an orientation.

Anyway, my question is, what exactly is "quantum anomaly cancellation" (perhaps in layman's terms) and what does it have to do with the whitehead tower of $O(n)$?

Also, is there more after $5$-branes?

-
I don't know the answer to your question but I'm pretty certain "layman's terms" for "quantum anomaly cancellation" is an oxymoron. – Ryan Budney Mar 28 '11 at 0:06
I had this (perhaps perverse) wish that one needs to lift to every more highly-connected covers of the Spin group to define string-/M-theory properly, and lift so far that the classifying space of the structure group is 11-connected (or poss. more). In this case, since spacetime is 10-/11-dimensional resp., it can't see any of the topology of the classifying space. Then everything that remains is geometric, albeit topologically trivial. – David Roberts Mar 28 '11 at 0:18
Photo at panoramio.com/photo/14652965 – Tom Goodwillie Mar 28 '11 at 14:25
@Tom: that's funny! (Alas it's a different Whitehead: the one with the tower in Bury is the inventor of the torpedo!) – José Figueroa-O'Farrill Mar 28 '11 at 15:45
@Tom: Thanks for that. I needed a laugh. – David Carchedi Mar 28 '11 at 16:06

Hi Dave,

I only just saw this question now. Maybe I can still react anyway.

The anomalies that we are talking about here mean the following: the action functional of a given QFT may turn out to be not quite a function on the configuration space, but instead a section of some line bundle with connection over configuration space. Hence for the theory to make sense, that line bundle with connection must be trivialized, and hence first of all must be trivializable. The Chern class of that line bundle is the the global anomaly, the obstruction to there existing a trivialization of the underlying bundle. It's curvature is the local anomaly, a measure for the obstruction for it to trivialize also as a bundle with connection.

That such anomalous contributions to the action functional appear for heterotic-type super-branes ("spinning branes") comes from the fact that for these the fermionic path integral is not a function on the bosonic configurations, but is a section of the Pfaffian line bundle of the given Dirac operator.

So anomaly cancellation is the process where we identify those constraints on the field content which make these anomaly line bundles become trivialized. This, and the standard references on it, is reviewed here:

http://ncatlab.org/nlab/show/quantum+anomaly

That the anomaly of the spinning particle vaishes if the target space has Spin structure is classical. That the anomaly of the heterotic string vanishes if the target has String structure is due to Killingback and Witten, originally, by an argument that was recently made rigorous by Bunke. See the references at

http://ncatlab.org/nlab/show/differential+string+structure

That the cancellation of the anomaly of the super 5-brane is similarly related to higher topological structures, and the introduction of the term "Fivebrane group" and "Fivebrane structure" is originally due to

Hisham Sati, Urs Schreiber, Jim Stasheff, Fivebrane structures Rev.Math.Phys.21:1197-1240 (2009) (arXiv:0805.0564)

That article also includes a review of the whole story of anomaly cancellation in its section 3.

In the successor

Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential string and fivebrane structures Communications in Mathematical Physics (2012) (arXiv:0910.4001)

we develop the full differential geometry corresponding to this. Various related articles with further developments are listed here

-

The short and dirty answer is that to define the action one needs a global section of a particular vector bundle. One only gets a global section if the structure group (of perhaps another, related bundle) lifts to a group $G\langle n\rangle \to G$ in the Whitehead tower. The toy example is that the orientation bundle, which admits a global section (i.e. an orientation form on the base) iff the tangent bundle (out of which the orientation bundle is constructed) has a reduction of its structure group to the special orthogonal group. Likewise, for the next step, one can only define the action for a fermion if one has a Dirac operator (i.e. a square root of the Laplacian), and this exists (I think! I'm a bit rusty here) iff the base manifold is spin. With this example one can see it is like passing to a Riemann surface of the punctured plane in order to define the function $z \mapsto z^{\frac{1}{n}}$: you need to get rid of the topological obstruction.

-
Whereas it is true that in order to be able to write the action for a Dirac fermion (inded, to have Dirac fermions at all!) you need a spin structure, this answer does not address what this has to do with an anomaly. In fact, the anomaly in this case is a global anomaly of a one-dimensional sigma model describing a spinning particle. I believe this is explained in Edward Witten's "Global anomalies in string theory" (as a warm-up example) in his 1985 contribution to the Argonne Symposium on Anomalies, Geometry and Topology, which sadly I do not have with me nor can I find online. – José Figueroa-O'Farrill Mar 28 '11 at 1:10
Thanks for your answer David. This helps me a lot with the big picture. – David Carchedi Mar 28 '11 at 15:16

I finally got a hold of the paper in question: Edward Witten's Global anomalies in string theory, where the cases of the particle and the string are discussed.

The model in question is a supersymmetric quantum-mechanical model (a zero-dimensional quantum field theory, if you wish) known as a supersymmetric nonlinear sigma model. For those in the know, it is the very same one whose Witten index is the index of the Dirac operator. The model is defined on the circle and the target space is a riemannian manifold $(M,g)$. The model consists of bosonic fields $\phi: S^1 \to M$ and fermionic fields $\psi$ which are sections of the "oddified" pullback by $\phi$ of the tangent bundle of $M$, denoted $\Pi\phi^{-1}TM$.

The path integral quantisation of this model requires one to compute the fermion effective action, which is the square root of the one-dimensional Dirac operator twisted by $\phi^{-1}TM$. This determinant defines a line bundle on the loop space of $M$ and the question is whether this bundle has a square root. What Witten finds is that the obstruction to the existence of the square root is the orientability of the loop space of $M$, or equivalently the obstruction to $M$ admitting a spin structure.

In that paper Witten also discusses the case of the string, paying particular attention to the sigma model for the heterotic string. I could not possibly summarise that part of the paper right now, I'm afraid. I hope somebody else on MO can.

As far as the big picture is concerned, quantum theories (and not just sigma models) are vulnerable to anomalies. These appear in a variety of guises. A rough idea, which is not always correct but gets repeated often, is that an anomaly is the quantum violation of a classical symmetry. This is certainly the case for the chiral and conformal anomalies, but it is not the case for the gauge anomaly. The gauge anomaly can be interpreted as the nontriviality of a certain line bundle on the moduli space of connections on a principal fibre bundle.

Anomalies come in two flavours: perturbative and global. Perturbative anomalies are usually given by the index density of some relevant elliptic operator, whereas global anomalies are usually associated to families of elliptic operators and can be understood in terms of spectral flow,...

The anomalies this question refers to are global anomalies for supersymmetric sigma models in dimensions 1 (particle), 2 (string) and 6 (fivebrane).

-
@Jose: Thanks for your answer. It really clarifies a lot of things for me. But part of what you says confuses me. Perhaps you can help clarify one point for me. If I understand things correctly, we have the gauge theory of the classical particle, which is given by a principal $U(1)$-bundle with connection. We may view this as a sigma-model on $X$ whose target space is the stack $BU(1)$. (cont.) – David Carchedi Mar 28 '11 at 14:56
If we then want the QFT corresponding to a single particle, we should consider the embedding of the world line $\Sigma \to X$, as a new sigma model, where we remember the extra structure on $X$ (i.e. its principal $U(1)$-bundle with connection). But, in your answer, it looks the world line $\Sigma$ is replaced with $S^1$. Why would we assume the world line is a circle, or am I just totally confused? – David Carchedi Mar 28 '11 at 14:56
@David: I think we might be talking about different theories. The gauge theory you are describing sounds to me like a particle in a background electromagnetic field (i.e., a connection on a principal $U(1)$ bundle) and a point particle interacting with it, say, electrically; that is, the interaction term is given by the holonomy along the trajectory of the particle. In principle this seems quite different to a sigma-model, but perhaps this is my ignorance of the word "stack". As for the world-line being a circle, Witten does this for convenience. – José Figueroa-O'Farrill Mar 28 '11 at 15:43
@Jose: I guess I am being a little non-standard in calling such a gauge theory a sigma-model. The reason I am doing so is a sigma-model is essentially just given by a map $\Sigma \to X$ where $X$ is the target space, but using the language of stacks, we can regard a gauge theory as a sigma model, since a principal $G$-bundle on $\Sigma$ is the same thing as a map $\Sigma \to BG$. This point of view was explained to me by Urs Schreiber. – David Carchedi Mar 28 '11 at 16:11
@David: Any classical field theory is defined as a map $\Sigma \to \textrm{something}$, but that does not make it a sigma model. The sigma model action is very particular: it always starts with the action for harmonic maps. That is, if $\phi: \Sigma \to M$, then $d\phi \in \Omega^1(\Sigma,\phi^{-1}TM)$ and the action starts with the $L^2$ norm of $d\phi$. Perhaps the YM action can be thought of as a nonlinear version of harmonic maps, but this is not the case for Chern-Simons, for example. Still, what you say is worth thinking about in some detail. More later, perhaps. – José Figueroa-O'Farrill Mar 28 '11 at 18:20