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• CommentRowNumber1.
• CommentAuthorBruce
• CommentTimeOct 12th 2009
I started an entry on Chern-Simons theory, after having been inspired by news that Witten has thought up a new Morse theory approach to the defining the path integral nonperturbatively (see Not Even Wrong), and Urs' encouragement. There is so much material on Chern-Simons theory, and I am only familiar with a small portion of it, that having to do with "extended topological quantum field theory".
• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeOct 12th 2009

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 12th 2009

Hi Bruce!

Great to see you here. I was getting worried where you'd been.

Thanks a lot for starting the entry on Chern-Simons theory. That was badly needed, for instance requested at Dijkgraaf-Witten theory.

You should have a look. As you can see, while you were on vacation from the n-side of life, we weren't being lazy around here. ;-): things like modular form, path integral, extended topological quantum field theory and many others have by now entries here (even though all of them would deserve lots of further improvement).

On the other hand, a bunch of keywords that you mentioned don't exist yet as entries, but would be very desireable: such as knot, knot invariant etc. I put them all in double square brackets, too, such that their grayish non-links will eventually make somebody here create them.

Apart from that I didn't work on your text, as I am occupied with something else. But I think at the point where you say you seem to have forgotton something, essentially what you need to insert in the path integral is the term that denotes the holonomy of the given connection over the knot. I left a remark there, but didn't feel like editing the formula myself.

Apart from that, I'd seriously be interested in learning more about any new progress on making sense of the path integral. But I won't have much time looking into Witten's new stuff right now. I am already working with my available time much like the financial market used to with its assets: I have a negative amount of it left but keep pushing the point where I need to realize that into the future...

But anything regarding formalization of the path integral I am interested in I am thinking that behind all the trees there should be lurking a nice forest of abstract nonsense that noone has seen as yet and which we here should try to identify. Eventually. Unless, of course, Lurie beats us again (more likely, but that shouldn't stop us!)

So when you find out more details of the latest Witten-approach, please Labify it, and I'll have a look.

• CommentRowNumber4.
• CommentAuthorBruce
• CommentTimeOct 28th 2009
Ok I labified some more details of the latest Witten approach. I corrected the knot holonomy section. I've also added a big section on "Geometric quantization and the path integral" detailing some ideas we once spoke about and that I sent in an email.
• CommentRowNumber5.
• CommentAuthorbwebster
• CommentTimeOct 28th 2009

I added a section at the end with my own musings about Chern-Simons theory and Khovanov homology. I'm sure these two have something to do with each other, but I'm not so sure what.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeOct 28th 2009

Ben,

could it be that in the analogy we are looking for if we replace Chern-Simons theory with its toy version, Dijkgraaf-witten theory, then the piece you are looking for is the Yetter model 4d TFT?

Do you know if anyone looked into this?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 28th 2009

Bruce,

thanks, really nice.

I am on the train currently, will see if I can still reply before my connection breaks down...

• CommentRowNumber8.
• CommentAuthorBruce
• CommentTimeOct 28th 2009
Ben, I had been labouring under the impression that you folks who are experts in Khovanov homology already knew the answer to this, so I'm surprised to see you ask that. I remember a talk a few years ago by Sergei Gukov when I got the impression he could explain Khovanov homology in gauge theory terms somehow (for instance, this paper). If I google now, I get a link to Not Even Wrong and also something about the Potts model by Kauffman. Mmm, seems to be still an open question! What is the latest word on this subject?
• CommentRowNumber9.
• CommentAuthorBruce
• CommentTimeOct 28th 2009
I have moved my stuff about the "path integral inside the path integral" to my personal nLab page and left a reference on the main page. I think it makes more sense this way.
• CommentRowNumber10.
• CommentAuthorBruce
• CommentTimeOct 28th 2009
Ben, searching a bit more, it seems that "Surface operators and knot homologies" is closer to the type of thing you are looking for. Basically, it seems that you can either interpret Khovanov homology as a string theory (as in this paper) or as a gauge theory (as in this "Surface operators" paper). Looking at it as a gauge theory is apparantly closest in spirit to the mathematical "n-functor" approach.
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 28th 2009

No references here that aren't also recorded on the relevant nLab page, I hope!

• CommentRowNumber12.
• CommentAuthorbwebster
• CommentTimeOct 29th 2009

I seen Gukov talk about this stuff several times, and I roughly understand the general picture, but I've never found any kind of satisfactory connections to actual Khovanov homology, etc., so as far as I'm concerned, the question is basically unanswered.

• CommentRowNumber13.
• CommentAuthorBruce
• CommentTimeOct 29th 2009
I am interested to know what you mean, because I don't have the expertise to go through Gukov's paper and try and understand what he is saying. What is the element which is missing which causes you to say "never found any kind of satisfactory connections to actual Khovanov homology, etc.". I am kinda glad, because it just means there is still one helluva open problem out there.
• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 29th 2009
• (edited Oct 29th 2009)

I replied in one of the query boxes

created a stub for Reshetikhin-Turaev construction from there

• CommentRowNumber15.
• CommentAuthorbwebster
• CommentTimeOct 30th 2009

What I mean is that Gukov has a nice picture for how to get knot homology out of 4d gauge theory, but I've asked several times what gauge theory gives Khovanov-Rozasnky homology and never gotten a straight answer. There is this paper, which has a bit more substance but is definitely at a ahem physical level of rigor. So there are some ideas out there, but I certainly haven't effectively digested them, and I don't think any of them have been done in a way approaching mathematical rigour.

Also, I've never heard any claims from physicists about groups other than sl(n), which my new work includes, so there's some piece of their picture missing.

• CommentRowNumber16.
• CommentAuthorBruce
• CommentTimeNov 1st 2009
Ok, thanks for this useful info.
• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeMay 3rd 2011

I have added to Chern-Simons theory a little bit in a new section Classical Chern-Simons theory

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMay 3rd 2011

also reorganized the entry a little, making explicit the (stubby) section on the Jones polynomial, which was previously hidden under the headline “Background and history”

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeAug 27th 2011
• (edited Aug 27th 2011)

I have been reorganizing (and slightly expanding) the paragraphs at Chern-Simons theory, trying to make the story become more systematic. But there are still huge gaps and jumps in that entry.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeSep 1st 2011

added to Chern-Simons theory a brief Properties-paragraph on “Chern-Simons theory as 3d quantum gravity”.

Also added references on perturbative quantization of CS theory.

• CommentRowNumber21.
• CommentAuthorzskoda
• CommentTimeSep 1st 2011
• (edited Sep 1st 2011)

Maybe we should have a separate page for 3d quantum gravity. I mean not only Chern-Simons theory, but also connection to spin-foam models and so called group field theory. I do not feel competent to write it, but I am interested in the connection between group field theory approach and noncommutative geometry. There is a case of quantum group Fourier transform which plays role here, see e.g. papers by Shahn Majid, Etera Livine, Florian Girelli and L. Freidel, e.g. those cited at ncFourier (zoranskoda). I wish I understand this, but this aspect I started looking at only few days ago. Some work we have been doing in Zagreb about Lie algebra type noncommutative spaces may be relevant. If I manage I will write a stub for group field theory today and/or tomorrow at least to record some references there as well.

• CommentRowNumber22.
• CommentAuthorzskoda
• CommentTimeSep 1st 2011
• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeSep 2nd 2011

Need to merge material and cross-link with Chern-Simons gravity

I tend to feel a bit hesitant here. But I should have another look at it. Did you? Do you trust all the references that you have included?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeFeb 28th 2012

I have added to Chern-Simons theory a subsection that list references on geometric quantization of CS theory.

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeJul 6th 2012
• (edited Jul 6th 2012)
• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeJul 12th 2012

I have added to Chern-Simons theory a section Geometric quantization - The space of states to go along with the further discussion in the comments below this MO reply

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeJul 12th 2012

I have also tried to brush-up the whole entry Chern-Simons theory a little:

• expanded and polished the Idea-section

• merged the two “Classical CS-theory”-sections as subsections of a single section;

• merged the “Properties”-subsection with the “Further aspects”-subsection;

• cleaned up some things, such as moving the pointer to the conference “20 years CS theory” from a “Further aspects”-subsection to the References;

• finally: emailed Bruce Bartlett and Ben Webster, asking them to do something about their paragraphs here and here.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeAug 13th 2012
• (edited Aug 13th 2012)

Prof. Deser kindly notified me by email that his seminal article with Jackiw and Templeton introduces the Chern-Simons action functional a good bit before 89. So I have added that to the References.

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeDec 27th 2012
• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeFeb 19th 2013
• (edited Feb 19th 2013)

started to add something in a new section

Still a bit rough. Will continue later, am out of time now.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeDec 10th 2014

added a long overdue paragraph under Quantization – Perturbative quantization – Path integral quantization.

So far it (only) says this:

Witten (1989), section 2 indicates the perturbative path integral quantization of Chern-Simons theory and finds that the result is essentially the exponentiated eta invariant (hence the Selberg zeta function) times the contributions of the CS action functional on classical trajectories (Witten 89 (2.17) (2.23)).

For more on this see at eta invariant – Boundaries, determinant line bundles and perturbative Chern-Simons.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeDec 10th 2014

I have expanded that paragraph a bit more, here, bringing out also the analytic torsion term

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeJun 25th 2015
• (edited Jun 25th 2015)

I have been making little additions (such as cross-links and pointers to the literature) to Selberg zeta function, Ruelle zeta function, eta invariant, analytic torsion, Borel regulator.

I am after the following, but not quite there yet in terms of fine print:

the perturbative quantum CS invariant is, as reviewed in the CS-theory entry here, a product of three factors

1. the classical CS-invariant;

2. the exponentiated eta invariant;

3. analytic torsion.

Now each of these has, for hyperbolic manifolds, a zeta-function expression :

1. complexified Borel regulator;

2. special value of Selberg zeta;

3. special value of Ruelle zeta;

up to some fine print such as special values at $s = 0$ versus special values at $s = n$, which may be absorbed into shifts of Dirac operators, which in turn shows up as variations of the other invariants; also there are some powers of $-1$ and of 2 here and there.

Then of course Selberg and Ruelle zeta functions may be variously expressed in terms of each other, which might make one hope that there is one single natural expression producing all three of these factors.

Clearly there wants to be some more unified story here. But enough for tonight.

• CommentRowNumber34.
• CommentAuthorDavid_Corfield
• CommentTimeJun 26th 2015

Do you mean the relationship between Selberg and Ruelle zeta functions that Fried expresses in one situation at the bottom of p. 498 of The zeta functions of Ruelle and Selberg. I, then more generally on p.499?

• CommentRowNumber35.
• CommentAuthorDavid_Corfield
• CommentTimeJun 26th 2015

Do you mean the relationship between Selberg and Ruelle zeta functions that Fried expresses in one situation at the bottom of p. 498 of The zeta functions of Ruelle and Selberg. I, then more generally on p.499?

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeJun 26th 2015

These are in principle the kinds of relations that I mean, yes, but Fried there speaks of the case of surfaces, whereas what is relevant here is the odd-dimensional case. For this case the relations in question are discussed in Bunke-Olbrich 94.

• CommentRowNumber37.
• CommentAuthorDavid_Corfield
• CommentTimeJun 26th 2015

My brief look also took in this note if it’s of any interest. It seems to be restricted to odd dimensions.

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeJun 26th 2015

Thanks! That’s a nice collection of material. I have added a pointer to it to the entries.

• CommentRowNumber39.
• CommentAuthorDavid_Corfield
• CommentTimeNov 21st 2016

My brief foray into the super world turned up these, which I’ve added at Chern-Simons:

• Victor Mikhaylov, Aspects of Supergroup Chern-Simons Theories, (thesis)

• Victor Mikhaylov, Analytic Torsion, 3d Mirror Symmetry, And Supergroup Chern-Simons Theories (arXiv:1505.03130)

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeNov 5th 2018

under “Perturbative quantization” there wasn’t any mentioning of the direct Feynman perturbation series computation by Axelrod-Singer. Have now added a few lines on that [here](https://ncatlab.org/nlab/show/Chern-Simons theory#FeynmanPerturbationSeries)

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeNov 1st 2019

have started as References-subsection with pointers to realization of 3d CS on brane worldvolumes.

On 3d Chern-Simons theories arising from the higher WZ term on super p-brane worldvolumes (notably on D2-branes):

• {#Brodie00} John Brodie, D-branes in Massive IIA and Solitons in Chern-Simons Theory, JHEP 0111:014, 2001 (arXiv:hep-th/0012068)

• Yosuke Imamura, A D2-brane realization of Maxwell-Chern-Simons-Higgs systems, JHEP 0102:035, 2001 (arXiv:hep-th/0012254)

• Mitsutoshi Fujita, Wei Li, Shinsei Ryu, Tadashi Takayanagi, Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy, JHEP 0906:066, 2009 (arXiv:0901.0924)

• Gyungchoon Go, O-Kab Kwon, D. D. Tolla, $\mathcal{N}=3$ Supersymmetric Effective Action of D2-branes in Massive IIA String Theory, Phys. Rev. D 85, 026006, 2012 (arXiv:1110.3902)

Specifically on D8-branes in the context of geometric engineering of 2d QCD (AdS/QCD):

• Ho-Ung Yee, Ismail Zahed, Holographic two dimensional QCD and Chern-Simons term, JHEP 1107:033, 2011 (arXiv:1103.6286)