Charges and Twisted Bundles, III: Anomalies
Posted by Urs Schreiber
In quantum physics a phenomenon called “(quantum) anomalies” plays a big role.
There are several different phenomena which go by this name, I think, and in the literature they don’t always tell you which one is which.
But generally, anomalies have to do with “global topological twists” (notably nontrivial fiber bundles) related to the configuration space of a field theory.
These twists are called “quantum” because they tend to become visible and/or relevant only when a classical theory is quantized.
They are called “anomalies”, I’d say, because to a large extent in physics the approach is to pretend that working locally is fine – until one happens to run head-on into global issues. A mathematician might say at this point: “We made a mistake at the beginning in assuming that everything is globally well defined, instead there may be obstructions to doing so”. The physicist says: “My naive approach of working locally is fine, but since it fails to work in this situation, it is the situation which is not normal: it is anomalous.”
A matter of perspective.
In any case, when you see the word “(quantum) anomaly” you should think obstruction to some global trivializability problem.
There is one particular kind of anomaly which arises in gauge theory and in higher gauge theory in the presence of electric and magnetic charges. This one is fully understood technically, to a large extent under control in concrete examples, and is the source of some very beautiful deep connections between physics on the one hand and index theory and differential cohomology on the other.
A good and rather exhaustive description, both as far as physical examples and as far as the mathematical machinery goes, of this phenomenon is given in
D. Freed
Dirac Charge Quantization and Generalized Differential Cohomology
arXiv:hep-th/0011220
This is one of the deepest articles on physics that I know of. The insights described there will rank one day with the central conceptual insights in physics of past centuries, I think. After differential equations in the 19th century and then later differential geometry in the 20th century, this identifies differential cohomology as the mathematical concept at the heart of physics.
The idea is simple: the action functional of gauge theory, in the presence of electric and magnetic charges, is, when you look closely, not really, in general, a function, the way they teach you in school. Rather, it is a multivalued function: a section of a line bundle over configuration space.
But whatever path integral quantization really is, it requires you to integrate the action against a measure. For that to be meaningful, the bundle that it is a section of must be trivializable.
The nontriviality of the bundle on configuration space that the action “functional” is a section of is “the” local anomaly: a measure for the failure of the starting point of the quantization procedure to be well defined.
But in fact more is true: the bundle on configuration space here is not just a bundle, but a bundle with connection: a differential cocycle. In order for everything to be well defined we need this bundle not only to be trivializable and have a flat connection, it also needs to have trivial connection. If not, we say we have a global anomaly.
So this kind of “anomaly” appearing in (higher) gauge theory in the presence of electric and magnetic charges is an obstruction which is measured by a class in differential cohomology.
As far as I know this was first realized in the study of the higher gauge theories that appear as effective target space field theories in string theory, notably in Witten’s discussion of the “5-brane anomaly”. But this is just where it was first realized. Remarkably, as nicely discussed at the beginning of section 2 the phenomenon is entirely visible in the ordinary 1.5 centuries old electromagnetism. And all the more complicated cases follow from this one simply by replacing line bundles with connection everywhere by higher differential cocycles (higher line bundles with connection).
Despite its crucial relevance, there is surprisingly little literature on this – which is however certainly due to the fact that the required differential cohomology theory is not widely familiar, and in fact in the process of being worked out more fully.
A big step in the direction of discussing the general theory of differential cohomology is the article
M.J. Hopkins, I.M. Singer
Quadratic functions in geometry, topology,and M-theory
arXiv:math/0211216.
Various aspects of its application to higher (abelian) quantum gauge theory have been discussed in
Daniel S. Freed, Gregory W. Moore, Graeme Segal
The Uncertainty of Fluxes
arXiv:hep-th/0605198
&
Heisenberg Groups and Noncommutative Fluxes
hep-th/0605200.
which I once tried to summarize a bit here and here.
Abelian higher gauge theory of Yang-Mills type is the study of of a certain function on the space of all line $n$-bundles with connection on a given Riemannian manifold $X$.
A line $n$-bundle with connection is modeled by any of the following concepts: an abelian (n-1) (bundle) gerbe with connection, a Cheeger-Simons differential character of degree $(n+1)$, a Deligne cocycle of degree $(n+1)$. Whatever model you use, such a gadget $\nabla$ comes with its curvature $(n+1)$-form $F_\nabla \,.$ The higher Yang-Mills like function(al) on the space of all such line $n$-bundles with connection is the assignment $e^{-S} : \nabla \mapsto \exp\left( - \int_X F_\nabla \wedge \star F_\nabla \right) \in \mathbb{R} \,,$ where “$\star$” is the Hodge star operator with respect to the chosen Riemannian structure on $X$.
In a more general setup we are looking at higher Yang-Mills theory with electric charges. As recalled in Charges and twisted $n$-bundles, II, “electric charge” for an “electric field” given by a line $n$-bundle with connection on a $dim(X)$-dimensional base space is itself a line $(d-n-1)$-bundle with connection, called $\hat j_E$.
There is a product of differential cocycles which on their curvature $(n+1)$-forms reproduces the ordinary wedge product of differential forms. If the $n$-bundle $\nabla$ has locally the connection form $A$ and if $\hat j_E$ has curvature $(d-n)$-form $j_e$, then the product line $d$-bundle with connection $\nabla \cdot \hat j_E$ has locally the connection $d$-form $A \wedge j_E \,.$ The action functional of (higher) abelian Yang-Mills theory in the presence of electric charges is $e^{-S} : (\nabla, \hat j_E) \mapsto \exp\left( - \int_X F_\nabla \wedge \star F_\nabla \right) \exp\left( 2 \pi i \int_X \hat j_E \cdot \nabla \right) \in \mathbb{C} \,.$
Here the integral in the term on the right denotes push-forward in differential cohomology, which reduces to the ordinary integral over the $d$-form $j_E \wedge A$ in the case that $A$ is a globally defined connection $n$-form on the line $n$-bundle $\nabla$.
So far, $e^{-S}$ is clearly an ordinary function (on the space of pairs consisting of one $n$-bundle and one $d-n-1$-bundle on $X$.) No anomaly so far.
Before coming to that, we need to quickly see what $e^{-S}$ really is: we want it in the end to be a smooth function on the space of all gauge field configurations. So far that is the space of all pairs consisting of a line $n$-bundle and a line $(n+1)$-bundle on $X$. To make that configuration space a smooth space we use the reasoning from Space and Quantity and give it the structure of a sheaf (might be an $n$-stack unless we do something about it, but that’s not of relevance for the moment) on smooth test domains $U$: $conf : U \mapsto \{ pairs (\nabla, \hat j_E) on U \times X\} \,.$ The action functional $e^{-S}$ we have now extends to a function $e^{-S} : conf \to \mathbb{C}$ from the smooth space $conf$ to the complex numbers by thinking of $\int_X$, which had been push-forward to the point along $X \to pt$ with the coresponding fiber integration obtained by push-forward to $U$ along $X \times U \to U$.
Here the complex numbers are equipped with their canonical structure of a smooth space: $\mathbb{C} : U \mapsto \{smooth complex-valued functions on U \} \,.$ So $e^{-S}$ is now a function of “smooth spaces”, which are really sheaves on smooth test domains, but apart from that it is still a function. (Freed does not mention sheaves this way, but that’s what he means to say.)
This changes as we further generalize the theory to also include magnetic charges:
as also recalled in Charges and twisted $n$-bundles, II, magnetic charge for a gauge field given by a line $n$-bundle with connection $\nabla$ is itself a line $(n+1)$-bundle with connection, $\hat j_B$. But now something changes: if $\hat j_B$ is nontrivial, then $\nabla$ is no longer an ordinary line $n$-bundle, but becomes a “twisted” line $n$-bundle, a “section” of $\hat j_B$.
As a result of that, one can show that the second term in our action functional $e^{-S} : (\hat j_E, 0 \stackrel{\nabla}{\to} \hat j_B)_{X \times U} \mapsto \exp\left( - \int_X F_\nabla \wedge \star F_\nabla \right) \exp\left( 2 \pi i \int_X \hat j_E \cdot \nabla \right)$ is no longer a complex function, hence a line 0-bundle – but a twisted line 0-bundle: a section of a line 1-bundle. That line 1-bundle turns out to be given, over each local probe $U$, by the expression $$ anom := \exp\left( 2 \pi i \int_X \hat j_E \cdot \hat j_B \right) : U \mapsto \{ line 1-bundles with connection over $U$ \} \,. $$
This line bundle (over each test domain $U$) is our anomaly. It is the obstruction to interpreting our expression for $e^{-S}$ in the naive fashion as a complex function on configuration space: in general it will now only be a section of the anomaly line bundle over configuration space:
$\array{ anom \\ \downarrow & \uparrow^{e^{-S}} \\ conf } \,.$
I wanted to say more, but it took me longer than I thought to get this far, and now boarding for my plane over the pond will start any minute. I’ll stop here for the moment and try to continue later.
Re: Charges and Twisted Bundles, III: Anomalies
What is the fundamental difference between this aproach and that old one using hodge product and integrating over 2 hemispheres, and indentifying it with a Chern Simon character?
Is there any example that by using this method you get more information? Is there any such example with things already indentified in experiment, like gravity and the standard model?