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February 29, 2008

Charges and Twisted n-Bundles, I

Posted by Urs Schreiber

I want to talk about

\;\;\bullet twisted nn-bundles with connection over dd-dimensional base space

coupled to

\;\;\bullet (electrically) charged nn-particles ((n1n-1)-branes);

how they are

\;\;\bullet sections with covariant derivatives of (n+1)(n+1)-bundles with connection

which can be interpreted as

\;\;\bullet obstructions to lifts through extensions of nn-groups

or equivalently

\;\;\bullet magnetic charges

of

\;\;\bullet magnetically charged (dn1d-n-1)-particles ((dn2d-n-2)-branes).

A crucial new ingredient compared to my former (I,II) discussion of sections of nn-bundles is the method from groupoidification: think of an nn-representation of an nn-group not as an nn-functor, but in terms of the corresponding action nn-groupoid, as described more recently in L L_\infty-associated bundles and sections.

Much of what I say is, in the language of generalized differential cohomology, in the great

D. S. Freed
Dirac Charge Quantization and Generalized Differential Cohomology
(arXiv)

only that what I describe in the language of \infty-parallel transport/L L_\infty-connections (pdf, arXiv, blog, brief slides, detailed slides) is supposed to be a nonabelian generalization of the purely abelian treatment which allows to not just say things like

The Green-Schwarz mechanism says that the Kalb-Ramond 2-bundle coupling to the electric string is twisted by magnetic 5-brane charge.

(discussion around equation (3.5) in Freed’s article)

But also things like

The Green-Schwarz mechanism says that over the 10-dimensional boundary of 11-dimensional supergravity base space the supergravity Chern-Simons 3-bundle obstructing the lift of an SO(10,1)×E 8SO(10,1) \times E_8-bundle to a String(SO(10,1))×E 8)String(SO(10,1))\times E_8)-2-bundle trivializes by admitting a global 2-section: the twisted Kalb-Ramond 2-bundle.

(previewed in section 3 of L L_\infty-connections)

where the underlying SO(10,1)×E 8SO(10,1) \times E_8-bundle as well as its String-2-bundle lift with connections represent nonabelian differential cohomology.

LaTeXified notes on what I will talk about are beginning to evolve as

Sections and covariant derivatives of L L_\infty-algebra connections
(pdf, blog, Bruce Bartlett’s recollection)

and

Twisted L L_\infty-connections
(pdf).

The quick way to see it.

If you have absorbed section 8 then everything more or less follows from the following simple observation (if not jump to the next paragraph right now!):


The more expository approach

It will have to be seen how far I can get right now. But it is fun to start with this

Simple reconsiderartion of ordinary electromagnetism aka abelian 1-gauge theory,

as found, with the emphasis we need here, in section 2 of Freed’s article as well as in most every other review of D-brane physics (aka higher generalized electromagnetism aka abelian nn-gauge theory) like, er, let’s see… p. 111 here:

Clifford Johnson
D-Brane Primer
(arXiv, book homepage).

So the phenomenon of interest here is a generalization to

- a) higher dimensions (meaning higher dimensions of target space);

- b) higher categorical dimensions (meaning higher dimensions of parameter space);

- c) nonabelian structures

of the following high-school facts (well, maybe most highschool don’t teach them this way, which is a pity, because it could easily be taught this way)

So an electromagnetic field on a space XX is a closed 2-form

FΩ closed 2(X). F \in \Omega^2_{closed}(X) \,.

If there are electrons zipping around on XX, then the fact that they are charged under the electromagnetic field means that whenever they trace out closed curves γ\gamma in XX, we can consistently assign a phase

tra(γ)U(1) \mathrm{tra}(\gamma) \in U(1)

to that curve, an element in U(1)U(1), with the property that whenever

γ=Σ \gamma = \partial \Sigma

is the boundary of a disk Σ\Sigma, we have tra(γ)=tra(Σ)=exp(2πi ΣF). \mathrm{tra}(\gamma) = \mathrm{tra}(\partial \Sigma) = \exp(2\pi i \int_\Sigma F) \,.

This implies that FF is actually integral, meaning that its integral over any closed surface in XX is an integer:

Σ= ΣF. \partial \Sigma = \emptyset \;\; \Rightarrow \;\; \int_\Sigma F \in \mathbb{Z} \,.

From this one concludes that FF is in fact the curvature of a line bundle PXP \to X with connection on XX.

[An important special case is X=S 2X = S^2 and PP the Hopf bundle S 3S 2S^3 \to S^2. As Jim Stasheff emphasises, it is the most remarkable fact about the history of science that P. A. M. Dirac found this line bundle from the physics perspective descibed here in the very same year, 1931, that Hopf described his line bundle – and that this was fully appreciated not until decades later (Jim says by Greub and Petry, 1975).]

The electrons zipping around on XX produce themselves an electromagnetic field. It turns out that this affects the electromagnetic line bundle we are talking about by the constraint that

dF=j E, d \star F = j_E \,,

where

:Ω n+1(X)Ω dn1(X) \star : \Omega^{n+1}(X) \to \Omega^{d-n-1}(X)

is the Hodge star operator induced from a choice of (pseudo-)Riemannian metric on XX; and

j EΩ closed 3(X) j_E \in \Omega^3_{closed}(X)

is a 3-form which is Poincaré dual to the curves that our electrons trace out.

Electrons are electrically charged. What happens when magnetically charged particles also zip around on XX (like the two ends of an ideal current-carrying coil (not to mention just yet the two ends of a string))?

It turns out that in that case there is another 3-form, j Bj_B, such that now

dF=0 d F = 0

is replaced by

dF=j B. d F = j_B \,.

Now, that should be a little shocking. A moment ago we had luckily identified two major concepts in physics and math – electromagnetic fields and line bundles with connection. That identification breaks down as be allow nonvanishing j Bj_B! The curvature 2-form of a line bundle with connection has no chance but to be closed.

So what’s going on? What is the mathematical structure which is to dF=j Bd F = j_B like ordinary line bundles with connection are to FΩ closed 2(X)F \in \Omega^2_{closed}(X)?

The answer is: twisted bundles.

And strikingly: twisted bundles are really 2-bundles. There is 2-categorical physics around already in 1931, secretly.

And there is an easy way to understand this heuristically: I just mentioned that we get magnetic charges from ideal (infinitely thin) current carrying coils on XX – which look like strings. 2-Particles. As they move, they trace out a 2-dimensional volume, not just a curve. So they naturally couple to a 2-bundle!

Which 2-bundle could that possibly be? Easy: the line 2-bundle whose curvature 3-form is just the current 3-form

j BΩ closed 3(X). j_B \in \Omega^3_{closed}(X) \,.

Which incidentally implies that

a) - the current 3-form has to be integral, too

j BΩ closed,integral 3(X) j_B \in \Omega^3_{closed, integral}(X)

which incidentally means that “charge is quantized”: it comes in integr multiples of a fixed unit;

b) since it satisfies dF=j Ed F = j_E, it must come from a 2-bundle/1-gerbe whose class in H 3(X,)H^3(X,\mathbb{Z}) is “pure torsion” (vanishes when multiplied with some finite integer).

The math language for electromagnetic fields in the presence of magnetic charge is hence

The magnetic charge is a line 2-bundle which is trivialized by the electromagnetic field, which is a “twisted 1-bundle” or “gerbe module”.

I’ll say more about what this really means later. For the moment, I just want to collect the items in a dictionary relating math and physics here.

Suppose XX is dd-dimensional. And suppose we are talking about electrically charged nn-particles (known as (n1n-1)-branes). Then

An

electrically charged nn-particle

on XX, known as a

electrically charged (n1)(n-1)-brane

couples to an

electromagnetic nn-bundle with connection

on XX, with

curvature (n+1)(n+1)-form FΩ n+1(X)F \in \Omega^{n+1}(X)

also known as a

(n1)(n-1)-gerbe with connection and curving

which is “twisted” by a

magnetic charge (n+1)(n+1)-bundle

with curvature (n+1)(n+1)-form

j BΩ closed n+2(X)j_B \in \Omega^{n+2}_{closed}(X) known as the magnetic current.

And we get the analogous story with electric and magnetic exchanged everywhere. In fact, it is a coincidence that in the four dimensions which we are familiar with, electrically charged particles correspond also to magnetically charged particles. In other dimensions that fails.

For dd dimensional XX, we have that:

if an electrically charged nn-particle couples to a field with

curvature (n+1)(n+1)-form FΩ n+1(X)F \in \Omega^{n+1}(X)

then the Hodge dual form

FΩ dn1(X) \star F \in \Omega^{d-n-1}(X)

may happen to be the curvature (dn1)(d-n-1)-form of

a (dn2)(d-n-2)-bundle with connection which couples to

(dn2)(d-n-2)-particles.

These latter we then call “magnetically charged”.

Posted at February 29, 2008 2:21 PM UTC

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8 Comments & 6 Trackbacks

Read the post Charges and Twisted n-Bundles, II
Weblog: The n-Category Café
Excerpt: Rephrasing Freed's action functional for differential cohomology in terms of L-oo connections in a simple toy example.
Tracked: March 4, 2008 5:00 PM
Read the post Chern-Simons Actions for (Super)-Gravities
Weblog: The n-Category Café
Excerpt: On Chern-Simons actions for (super-)gravity.
Tracked: March 13, 2008 11:16 AM
Read the post What has happened so far
Weblog: The n-Category Café
Excerpt: A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
Tracked: March 27, 2008 4:43 PM
Read the post Charges and Twisted Bundles, III: Anomalies
Weblog: The n-Category Café
Excerpt: On the anomalies arising in (higher) gauge theories in the presence of electric and magnetic charges.
Tracked: April 25, 2008 10:16 PM
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 7:31 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 3:23 PM

Re: Charges and Twisted n-Bundles, I

Maybe this is why there are no other comments on this blog entry yet: the comprehensible stuff starts only after a page of terrifyingly technical references, an offhand remark about how a supergravity Chern-Simons 3-bundle can obstruct the lift of an SO(10,1)×E 8SO(10,1) \times E_8-bundle to a String (SO(10,1))×E 8)(SO(10,1)) \times E_8)-2-bundle, and a huge commutative diagram full of undefined symbols.

But it’s not really that hard once Urs gets going!

So, I urge readers to give this article another try, skipping everything until you see this section header:

The more expository approach

I also urge Urs to rewrite this entry, simply by removing most of the scary stuff at the top.

Posted by: John Baez on April 2, 2009 4:37 PM | Permalink | Reply to this

Re: Charges and Twisted n-Bundles, I

Urs is right. High school students could understand this if they had the right tools. After all, we are probably as smart as we are ever going to be by the time we reach the age of 12 or so – we just don’t know much yet. A book or a set of class notes that describes this to the motivated but uninitiated by first building the minimal technical framework from the ground up could change the whole approach to teaching physics. Maybe this article can become a first step in that direction.

Posted by: Charlie C on April 2, 2009 5:38 PM | Permalink | Reply to this

strategy

Thanks for tagging parts of this entry as readable.

I believe back then my intended strategy had been to:

1st) mention some terms I want to talk about;

2nd) indicate that there is good motivation for understanding those terms by mentioning interesting stuff mentioned previously that can be understood with them;

3rd) give an exposition of these terms.

I also urge Urs to rewrite this entry, simply by removing most of the scary stuff at the top.

Yeah, part 2.

You know, what I really want to do eventually is not to rewrite any blog entries, but to turn them into entries on the nnLab. There it would be split into bite-sized pieces titled “basic idea and exposition” and “scary but actually not so scary details” to be reached by mouse-click or not, as desired.

Currently there’s lots of nnLab-activity on fundamental foundations of fundamental math going on there. Eventually I want to see more energy put into the Physics compartment of the nnLab.

There is no good comprehensive text out there about which modern abstract math concepts formalize correctly which modern physics concepts, so that a coherent picture emerges. The nnLab would be a great place to do so.

All we need to do is

- finish the foundations for Categories and Sheaves

- lift that to Higher Topos Theory

- set up in there differential cohomology

- and then use that machinery to discuss the physics, such as the stuff discussed here and in the anomalous anomaly discussion.

Posted by: Urs Schreiber on April 2, 2009 5:49 PM | Permalink | Reply to this

Re: strategy

Urs wrote:

You know, what I really want to do eventually is not to rewrite any blog entries, but to turn them into entries on the nLab. There it would be split into bite-sized pieces titled “basic idea and exposition” and “scary but actually not so scary details” to be reached by mouse-click or not, as desired.

That sounds great! Please make sure that in every entry, every section, and every subsection, the most gentle and friendly stuff is at ‘the top’ — the first thing people see.

I’m looking forward to ‘retiring’ and writing only expository papers and books. It’s fun and it’s desperately needed. Just turning the material already on the nn-Café and nnLab into well-organized, easily readable prose would already be a huge and wonderful project. To do the same for all of This Week’s Finds, I’d need an afterlife.

Posted by: John Baez on April 2, 2009 6:36 PM | Permalink | Reply to this

Re: strategy

Yes!!!

Posted by: Charlie C on April 2, 2009 8:09 PM | Permalink | Reply to this

Re: strategy

John wrote:

Please make sure that in every entry, every section, and every subsection, the most gentle and friendly stuff is at the top - the first thing people see.

This is also excellent advice for any article or book, for all of us but especially for any of you without tenure.

My (totally non-mathematical) wife was appalled on seeing a paper which began something like:

Let f be a hemi-demi flurgelflipper.

Posted by: jim stasheff on April 3, 2009 1:49 AM | Permalink | Reply to this

Re: Charges and Twisted n-Bundles, I

Neat. I think I missed this one because I get all my updates via RSS and will only get notified when someone makes a comment. No comment = No RSS notification.

This all seems to fit beautifully on diamond lattices. Lattice electromagnetic theory was the motivation (for me anyway) for all our work on discrete differential geometry.

Posted by: Eric on April 2, 2009 6:10 PM | Permalink | Reply to this

Re: Charges and Twisted n-Bundles, I

Although I almost don’t dare to say it now, but if this entry is being re-read now I should maybe point out that the, according to what I am being told, unintelligable remarks before and including that big diagram above have meanwhile evolved into the notes presented at the most recent unreadable entry

Twisted differential String- and Fivebrane structures

Posted by: Urs Schreiber on April 2, 2009 8:43 PM | Permalink | Reply to this

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