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Handy link: Chern-Simons theory
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.
I just went through your entry and added plenty of links .
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.
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.
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?
Bruce,
thanks, really nice.
I am on the train currently, will see if I can still reply before my connection breaks down...
No references here that aren't also recorded on the relevant nLab page, I hope!
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.
I replied in one of the query boxes
created a stub for Reshetikhin-Turaev construction from there
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.
I have added to Chern-Simons theory a little bit in a new section Classical Chern-Simons theory
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”
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.
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.
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.
OK, 3d quantum gravity and group field theory.
Need to merge material and cross-link with Chern-Simons gravity
and group field theory.
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?
I have added to Chern-Simons theory a subsection that list references on geometric quantization of CS theory.
I have added a futher section Chern-Simons theory – Geometric quantization – In higher codimension in further reply to this MO question.
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
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.
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.
Added more references under Chern-Simons theory – References – Perturbative quantization
started to add something in a new section
Still a bit rough. Will continue later, am out of time now.
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.
I have expanded that paragraph a bit more, here, bringing out also the analytic torsion term
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
the classical CS-invariant;
the exponentiated eta invariant;
analytic torsion.
Now each of these has, for hyperbolic manifolds, a zeta-function expression :
complexified Borel regulator;
special value of Selberg zeta;
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.
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?
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?
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.
My brief look also took in this note if it’s of any interest. It seems to be restricted to odd dimensions.
Thanks! That’s a nice collection of material. I have added a pointer to it to the entries.
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)
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