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April 25, 2008

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 nn-bundles with connection on a given Riemannian manifold XX.

A line nn-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)(n+1), a Deligne cocycle of degree (n+1)(n+1). Whatever model you use, such a gadget \nabla comes with its curvature (n+1)(n+1)-form F . F_\nabla \,. The higher Yang-Mills like function(al) on the space of all such line nn-bundles with connection is the assignment e S:exp( XF F ), 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 XX.

In a more general setup we are looking at higher Yang-Mills theory with electric charges. As recalled in Charges and twisted nn-bundles, II, “electric charge” for an “electric field” given by a line nn-bundle with connection on a dim(X)dim(X)-dimensional base space is itself a line (dn1)(d-n-1)-bundle with connection, called j^ E\hat j_E.

There is a product of differential cocycles which on their curvature (n+1)(n+1)-forms reproduces the ordinary wedge product of differential forms. If the nn-bundle \nabla has locally the connection form AA and if j^ E\hat j_E has curvature (dn)(d-n)-form j ej_e, then the product line dd-bundle with connection j^ E \nabla \cdot \hat j_E has locally the connection dd-form Aj E. A \wedge j_E \,. The action functional of (higher) abelian Yang-Mills theory in the presence of electric charges is e S:(,j^ E)exp( XF F )exp(2πi Xj^ E). 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 dd-form j EAj_E \wedge A in the case that AA is a globally defined connection nn-form on the line nn-bundle \nabla.

So far, e Se^{-S} is clearly an ordinary function (on the space of pairs consisting of one nn-bundle and one dn1d-n-1-bundle on XX.) No anomaly so far.

Before coming to that, we need to quickly see what e Se^{-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 nn-bundle and a line (n+1)(n+1)-bundle on XX. 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 nn-stack unless we do something about it, but that’s not of relevance for the moment) on smooth test domains UU: conf:U{pairs(,j^ E)onU×X}. conf : U \mapsto \{ pairs (\nabla, \hat j_E) on U \times X\} \,. The action functional e Se^{-S} we have now extends to a function e S:conf e^{-S} : conf \to \mathbb{C} from the smooth space confconf to the complex numbers by thinking of X\int_X, which had been push-forward to the point along XptX \to pt with the coresponding fiber integration obtained by push-forward to UU along X×UUX \times U \to U.

Here the complex numbers are equipped with their canonical structure of a smooth space: :U{smoothcomplexvaluedfunctionsonU}. \mathbb{C} : U \mapsto \{smooth complex-valued functions on U \} \,. So e Se^{-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 nn-bundles, II, magnetic charge for a gauge field given by a line nn-bundle with connection \nabla is itself a line (n+1)(n+1)-bundle with connection, j^ B\hat j_B. But now something changes: if j^ B\hat j_B is nontrivial, then \nabla is no longer an ordinary line nn-bundle, but becomes a “twisted” line nn-bundle, a “section” of j^ B\hat j_B.

As a result of that, one can show that the second term in our action functional e S:(j^ E,0j^ B) X×Uexp( XF F )exp(2πi Xj^ E) 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 UU, by the expression anom:=exp(2πi Xj^ Ej^ B):U{line1bundleswithconnectionover 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 UU) is our anomaly. It is the obstruction to interpreting our expression for e Se^{-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:

anom e S conf. \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.

Posted at April 25, 2008 7:33 PM UTC

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8 Comments & 2 Trackbacks

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?

Posted by: Daniel de Franša MTd2 on April 26, 2008 2:08 AM | Permalink | Reply to this

Re: Charges and Twisted Bundles, III: Anomalies

I meant more ( or different ) information by comparing the results of Generalized Differential Cohomology with the one I cited above.

Posted by: Daniel de Franša MTd2 on April 26, 2008 2:12 AM | Permalink | Reply to this

Re: Charges and Twisted Bundles, III: Anomalies

What is the fundamental difference between this aproach and that old one

I think the big insight is that magnetic charge and electric charge are themselves represented by differential cocycles:

the magnetic current 3-form in ordinary electromagnetism is really the curvature 3-form of a gerbe with connection. The electromagnetic field is a “twisted bundle” for that gerbe, a “gerbe module”.

This strikes me as an important insight. You can think of it as a refinement of Dirac’s old argment:

Dirac discussed the electromagnetic field outside of a magnetic monopole. Since at the location of the magnetic monopole the electromagnetic field strength 2-form is not the curvature of a line bundle, as it is outside, one usually discusses this by removing the location of the magnetic monopole from the underlying manifold: Dirac’s old argument is, in modern words, that the magnetic charge is the Chern-class of a line bundle on 3{0}\mathbb{R}^3 - \{0\}, where the point taken out is that where the magnetic monopole would sit.

Now we see that there is a refined description of this, where we can take the location of the monopole into account: we smear it a bit to make it given by a smooth 3-form supported close to the origin of 3\mathbb{R}^3 and interpret that as a gerbe on 3\mathbb{R}^3 with trivial curvature everywhere except at the location of the monopole. Now we can say what the electromagnetic field is around the origin: no longer a line bundle with connection, but a “twisted line bundle” or “gerbe module”.

It’s a simple observation. But I find it deserves more emphasis: ordinary electromagnetism already involves higher gauge theory, 2-bundles, gerbes. Only using that do we get a coherent global picture which can be discussed on all of base space, without the need to remove points from it before starting the discussion.

And then Freed observes that this observation has a profound implication on the ordinary action functional of electromagnetism in the presence of electric charges: the usual exponentiated coupling term of the electric charge to the electromagnetic field, which locally reads exp(2πi XAj E) \exp( 2\pi i \int_X A \wedge j_E ) is now globally no longer a function, in general, but a section of a line bundle. So it is not, in fact, in general well defined as an action functional.

This is an important insight, it seems to me.

Of course in Freed’s discussions it gains in importance as he applies it to the cases where the electromagnetic field itself is replaced by a higher gauge field.

Posted by: Urs Schreiber on April 27, 2008 3:58 AM | Permalink | Reply to this

Re: Charges and Twisted Bundles, III: Anomalies

A little late… and having a little fun so please don’t take this too seriously…

I was never particularly troubled by Dirac’s ideas. In fact, I thought they were very pretty. Rather than “removing points” I thought you could think of it as “identifying” distinct points. I’m stretching my memory (and maybe even my sanity), but if you were to identity two distinct points in space, then I thought one point would appear to be an electric monopole to its neighboring observers and the other point a magnetic monopole to its neighboring observers. In effect, you could say there are no individual point charges, i.e. just vacuum, and what we observe to be charges are merely points identified in space as flux lines flow into the one point and out the other so that the net flux is still zero (kind of the electric version of a wormhole).

Maybe saying it another way, you might be able to argue that the “anomaly” is due to assuming a trivial topology. If you assume trivial topology, then you need nontrivial bundles. If you have a nontrivial topology, then the bundle is trivial. That has a nice sounding duality to it.

(trivial topology, non-trivial bundle)(non-trivial topology, trivial bundle)(\text{trivial topology, non-trivial bundle}) \Leftrightarrow(\text{non-trivial topology, trivial bundle})

Or something like that…

What if the idea of charges and differential cohomology is a head fake? I think that once in a while really brilliant physicists come along with seemingly really brilliant ideas, which distract efforts by taking a lot of smart people down blind alleys. I hope this isn’t one of them. I’m not very convinced yet that this is getting us closer to the “truth”.

I’ll post a link if I can find a reference. Found one!

Charge and the topology of spacetime

PS: Happy holidays! Maybe I’ve had too much eggnog…

Posted by: Eric on December 24, 2008 6:17 PM | Permalink | Reply to this

Self-dual fields

The Dirac quantization condition (more precisely, the coupling you are discussing) isn’t usually considered “an anomaly” except in the case of self-dual fields (the real subject of interest in Freed’s papers cited above). And, even then, it depends on the theory whether the coupling in question is, in fact, anomalous.

This isn’t to say that the formulation of self-dual theories isn’t fraught with complications. It’s just that not all complications are “anomalies.”

Posted by: Jacques Distler on April 26, 2008 7:16 AM | Permalink | PGP Sig | Reply to this

Re: Self-dual fields

the case of self-dual fields

I hope to come to that.

Posted by: Urs Schreiber on April 27, 2008 3:00 AM | Permalink | Reply to this

Re: Charges and Twisted Bundles, III: Anomalies

My question is related to this observation

“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.”

with something so great, I would like to undestand that there something fundamentaly new, that is, a practical example, at least an evidance, that this will really radicaly change the every day world and/or science view, as the other cited aproaces did.

Posted by: Daniel de Franša MTd2 on April 26, 2008 3:29 PM | Permalink | Reply to this

Re: Charges and Twisted Bundles, III: Anomalies

with something so great,

Yeah, I advertized it quite a bit, didn’t I. But it deserves to be, I think. It is not appreciated as widely as it should be.

Via Jim, I just received a message from Dan Freed pointing out also the review

Daniel S. Freed
K-Theory in Quantum Field Theory
arXiv:math-ph/0206031

which starts with reviewing his ideas about extended QFT #, then the theory of abelian gauge fields and then the anomaly cancellation issue.

I would like to undestand that there something fundamentaly new, that is, a practical example, at least an evidance, that this will really radicaly change the every day world and/or science view, as the other cited aproaces did.

It goes a long way towards better understanding the quantization of gauge fields.

For those neither thrilled by the observation that Dirac’s famous ideas on magnetic charge really have a refined higher interpretation, nor motivated by the wealth of more string-theoretic examples that Freed discusses, I noticed that it might help emphasize the relevance of this by putting it this way:

one of the central motivations of loop quantum gravity has been to find the quantization of the phase space of all “gauge fields” – in their case all n=1n=1 connection for a nonabelian gauge group. This is what one would need for a nonperturbative quantization of gauge theory. But the nonabelian case is very hard to understand and the solution found in LQG (either in Christian Fleischhack’s version or in the LOST version, I talked about that here) pertains only to “generalized connections” where the requirement of smoothness is dropped and the notion of nontrivial bundles disappear.

On the other hand, using their differential cohomology, Freed, Moore and Segal solved this problem I, II (for the proper smooth case) in the special case of abelian, but possibly higher nn, gauge fields. That’s a huge advance concerning the understanding of gauge theory, I’d say.

Of course ultimately we’d want to understand this for smooth and nonabelian: we need to understand the quantization of nonabelian differential cohomology.

Posted by: Urs Schreiber on April 27, 2008 3:32 AM | Permalink | Reply to this
Read the post Charges and Twisted Bundles, IV: Anomaly Canellation
Weblog: The n-Category Café
Excerpt: How the fermionic anomaly may cancel against the charge anomaly in higher gauge theory: the Green-Schwarz mechanism
Tracked: April 27, 2008 7:30 PM
Read the post Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles
Weblog: The n-Category Café
Excerpt: On the electric-magnetic dual formulation of higher abelian Yang-Mills theory.
Tracked: May 17, 2008 3:29 PM

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