Quantcast
Processing math: 100%
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.
W3Counter Web Stats

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public β tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

206 submissions , 164 unreviewed
5,106 questions , 2,251 unanswered
5,412 answers , 23,075 comments
1,470 users with positive rep
822 active unimported users
More ...

  Some basic questions about Chern-Simons theory

+ 6 like - 0 dislike
3924 views

Let the Chern-Simons lagrangian for a group G be, L=kϵμνρTr[AμνAρ+23AμAνAρ]

Then it is claimed that on "infinitesimal" variation of the gauge field ("connection") the lagrangian changes by,

δL=kϵμνρTr[δAμFνρ]

where the "curvature" Fμν is given as μAννAμ+[Aμ,Aν]

Under gauge transformations on A by gG it changes to say A whose μ component is given as g1Aμg+g1μg (This makes sense once a representation of G has been fixed after which A and g are both represented as matrices on the same vector space)

Say the Lagrangian under the above gauge transformations change to L and then one has the relation,

LL=kϵμνρμTr[νgg1Aρ]k3ϵμνρTr[g1μgg1νgg1ρg]

The second term of the above expression is what is proportional to the "winding number density" of the Chern-Simons lagrangian and thats what eventually gets quantized.

I would like to know the following things,

  • Is there a neat coordinate free way of proving the above two variation change equations? Doing this in the above coordinate way is turning out to be quite intractable!

  • Since the Lagrangian is just a complex number one can talk of the "real" and the "imaginary" part of it. But I get the feeling that at times a split of this kind is done at the level of the gauge field itself. Is this true and if yes the how is it defined? (Definitely there is lot of interest in doing analytic continuation of the "level" k)

  • The Euler-Lagrange equations of this action give us only the "flat" connections and in that sense it is a topological theory since only boundary conditions seem to matter. Still all the flat connection configurations are not equivalent but are labelled by homomorphisms from the first fundamental group of the 3-manifold on which the theory is defined to G. How to see this? What is the background theory from which this comes? And why is this called "holonomy"? (I am familiar with "holonomy" as in the context of taking a vector and parallel transporting it around a loop etc)

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Anirbit
asked Jul 14, 2010 in Theoretical Physics by Anirbit (585 points) [ no revision ]
retagged Jun 30, 2014

4 Answers

+ 9 like - 0 dislike

The secret to understanding the Chern-Simons functional over a 3-manifold is to realise that it's a 4-dimensional functional in disguise.

If X is a closed, oriented 4-manifold, and PX a principal SU(n)-bundle, one has the Chern-Weil formula for the second Chern number, c2(P)[X]=18π2XTrF2A,

where FA is the curvature of a connection A in P. If now Y is a closed, oriented 3-manifold, and PYY a principal SU(n)-bundle (necessarily trivial) then we can define the Chern-Simons functional cs:Connections(PY)R/Z

for a connection A by choosing any compact oriented 4-manifold X bounding Y (one always exists), any SU(n)-bundle PXX extending PYY (one again exists), and any connection B in PX extending A (ditto), and setting cs(A)=18π2XTrF2A.
This is well-defined (mod integers) because given another choice (X,PX,B), we can glue together the two manifolds along Y to get a closed 4-manifold (we reverse the orientation of one of them), and express the difference in values of cs(A) as a Chern-Weil integral as above, which is an integer.

To obtain a formula for cs(A), take PX to be a trivial bundle, and take B to be a connection that is trivial except over the collar Y×[1,0]. Over that collar, the bundle is [1,0]×PY, and we can let B the 4-dimensional connection A+t(AA0), where A0 is a reference connection trivialising PY (i.e., flat with trivial holonomy). Then we can explicitly perform the integral defining cs(A) over the [0,1]-factor (try it!) and get a formula for real-valued lift of cs(A) (here we use A0 to trivialise PY), CS(A)=18π2YTr(AdA+23AAA),

which is the formula you gave in indices (up to a factor).

From this formula, it's straightforward to compute the first variation of CS, and to interpret it in terms of FA=dA+AA.

The gauge transformation formula says that if g is an automorphism of PY of degree d then CS(gA)CS(A)=d. This has a conceptual explanation: linearly interpolate from A to gA to obtain a connection over Y×[0,1] whose Chern-Weil integral is CS(gA)CS(A). Use g as a gluing recipe to obtain a bundle over Y×S1 whose second Chern number is d (this defines d, if you will), and apply the Chern-Weil formula.

Reference: S.K. Donaldson, "Floer homology groups in Yang-Mills theory".

Very briefly on the second question: one can work with SL(n,C)-bundles instead. The Lie algebra is the complexfication of that of SU(n), so (after the bundle is trivialised), connections have real and imaginary parts which are 1-forms valued in su(n).

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Tim Perutz
answered Jul 14, 2010 by Tim Perutz (90 points) [ no revision ]
Most voted comments show all comments
And the Chern-Simons paper is jstor.org/stable/1971013

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user José Figueroa-O'Farrill
And the shorter announcement: jstor.org/stable/60680

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user José Figueroa-O'Farrill
Beware that Anirbit, Paul, Jose and I all chose different normalisations for the Chern-Simons functional. The level k is optional; it becomes significant if one wants to consider exp(2πikcs(A)) (in my normalisation).

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Tim Perutz
That's a nice summary! Does the existence of your bundle PX use that SU(n) is simply connected? I remember a paper of Dijkgraaf and Witten where they describe this story for arbitrary groups, and they had to introduce certain additional factors...

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Konrad Waldorf
Konrad, sorry for the slow reply to your comment (I'm travelling...). The simple connectedness of the group has the convenient effect that bundles over an oriented 3-manifold are trivial so one needn't worry. In general, one would want to choose X so that the bundle does extend over it, but I don't honestly know whether this is ever problematic (it's straightforward for SO(3), for instance).

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Tim Perutz
Most recent comments show all comments
+1. A very beautiful answer!

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user José Figueroa-O'Farrill
Jose, thanks - I should make clear that this is the story told in the original paper of Chern and Simons!

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Tim Perutz
+ 7 like - 0 dislike

I'll address your first question, in some computational detail; although I cannot do better than the answers which have already been given in terms of what is conceptually going on.

Let G be a 1-connected Lie group with Lie algebra g. We assume that the group admits a bi-invariant metric or, equivalently, that the Lie algebra admits an ad-invariant inner product, denoted Tr by the traditional abuse of notation in the Physics literature. Let M be an orientable 3-manifold which, for simplicitly, I will take without boundary. Then every principal G-bundle on M is trivial and choosing a trivialisation once and for all we can identify connections with g-valued one-forms on M. Let A be one such connection.

In this language, the Chern-Simons action for A can be written as I[A]=k4πMTr(AdA+23AAA).

There is of course some abuse of notation here. Both A and dA are g-valued, whence the first term in the action is the composition of their wedge product with the ad-invariant inner product on g. This results in a 3-form, which can be integrated on M. The second term is perhaps best understood if one rewrites 23TrAAA=13Tr[A,A]A,
where [A,A] is the composition of wedging and the Lie bracket on g, hence it is again a g-valued 2-form, and the second term in the action is (up to the factor of one-third) again the composition of the wedge product of [A,A] and A with the ad-invariant inner product. (Notice that [A,A] is not zero, because although [,] is antisymmetric, so is between two one-forms.)

Now let us vary I: δI=k4πMTr(δAdA+AdδA+2δAAA).

Using the fact that d is an odd derivation, we have that Tr(AdδA)=dTr(AδA)+Tr(dAδA)
whence, δI=k2πMTr(δA(dA+AA))+k4πMTr(δAA)
The boundary term drops out because we have assumed M=. In summary, the Euler-Lagrange equation is the flatness of the curvature F=dA+AA, which can also be written as F=dA+12[A,A].

Under a gauge transformation g:MG, AAg=gAg1dgg1.

Again there's some abuse of notation in that one is assuming that g is a matrix Lie algebra, but this only notational and it's perfectly possible to write this in a way that makes sense in general; namely, Ag=AdgA+gθ
where θ is the right-invariant Maurer-Cartan form on G.

In order to vary the action is perhaps best to write it in an equivalent way; namely, I[A]=k4πMTr(AF13AAA),

since the curvature transforms covariantly: Fg=gFg1.

The next calculation is a little long and I will not reproduce it here. It is not difficult: just plug in the expressions for Ag and Fg into the action I[Ag] and expand using the fact that Tr is Ad-invariant. One must also use the Maurer-Cartan structure equation dθ=θθ. The result of the calculation is I[Ag]I[A]=k4πMTr(d(g1dgA)+13(dgg1)3)

where the first term vanishes by Stokes's theorem since M has no boundary, leaving I[Ag]I[A]=k12πMTr((dgg1)3)

If G is a compact simple group, then one can rescale the inner product Tr in such a way that 112πMTr((dgg1)3)Z

whence the Chern-Simons path integral is gauge-invariant provided that kZ as well, in units where =1.

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user José Figueroa-O'Farrill
answered Jul 14, 2010 by José Figueroa-O'Farrill (2,315 points) [ no revision ]
Jose: not every G-bundle over a 3-manifold is trivial. It is if G is simply connected

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Paul
Thanks. I guess I'm used to the 3-sphere... I'll edit.

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user José Figueroa-O'Farrill
+ 5 like - 0 dislike

on your 3rd point: The holonomy of a connection, defined as you say by parallel transport, defines a function from the loop space of the manifold (of any dimension) to G (parallel transport of vectors should be generalized to frames for =GL(n), and more generally if you work in a principal G bundle the holonomy lies in G:this is either a defintion of a simle exercise depending on what "connection" means to you. The connection is flat if and only if the holonomy is trivial for all small enough loops. expressing an arbitrary loop as a product of small loops you see this is equivalent to the holonomy being homotopy invariant, hence defining a function (easily seen to be a homomorphism) on the fundamental group. Gauge equivalent flat connections yield conjugate holonomy representations, and in fact Flat/Gauge is identified with a subset of Hom(π1,G)/conj.

For your first 2 points, given any connection A, all others are of the form A+a for an appropriate Lie(G) valued 1-form a. Then the equations F(A+ta)=F(A)+tdAa+t2[a,a]/2, g.(A+a)=A+gAg1+dgg1 (up to signs!) and cs(A)=Tr(F(A)A1/3AAA) lead to answers to your questions. I'm guessing from your notation that you are a physicist: all of this is explained in any mathematical treatment of the Chern-Simons function on connections.

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Paul
answered Jul 14, 2010 by Paul (85 points) [ no revision ]
+ 4 like - 0 dislike

There is a nice geometrical treatment of the Chern-Simons action functional for a general Lie group G - not necessarily simply connected - that relates it to "holonomy". It also clarifies why Chern-Simons theories are classified by H4(BG,Z).

This point of view is described in a paper by Carey et. al. (http://arxiv.org/abs/math/0410013).

The group H4(BG,Z) classifies geometrical objects over G called "multiplicative bundle gerbes". Given a multiplicative bundle gerbe G and a closed oriented 3-manifold M with a principal G-bundle P with connection over it, Carey et. al. describe a beautiful construction of a bundle 2-gerbe CS(G,P) over M, the Chern-Simons 2-gerbe. They prove that this 2-gerbe carries a connection such that its holonomy around M, HolCS(G,P)(M)S1

is the (exponentiated) Chern-Simons action. This works without any assumptions on the manifold M or on the Lie group G.

For a little background, recall that connections on n-gerbes have holonomies around closed oriented (n+1)-manifolds. Ordinary bundles are 0-gerbes. Part of a connection on an n-gerbe is a curving: a (n+1)-form on some covering space. In case of the 2-gerbe CS(G,P), the covering space is the total space of the given bundle P, and the curving 3-form is the Chern-Simons 3-form CSA on P.

This post imported from StackExchange MathOverflow at 2014-06-30 07:41 (UCT), posted by SE-user Konrad Waldorf
answered Jul 15, 2010 by Konrad Waldorf (110 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol in the following word:
pysisOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...