added pointer to

- Alexander M. Polyakov:
*Fermi-Bose transmutation induced by gauge fields*, Modern Physics Letters A**03**03 (1988) 325-328 [doi:10.1142/S0217732388000398]

as the origin of Witten’s argument for the Wilson loop observables

]]>If somebody is interested there is this seminar in couple of hours taking place:

https://researchseminars.org/talk/nhetc/62

**A Chern-Simons Theory for the North Atlantic Ocean**

David Tong (University of Cambridge)

Tue Jan 24 (starts in 3 hours) Livestream access available

]]>Abstract: In some ways the ocean acts like a

topological insulator. There are chiral edge modes, localised at the coast, that go clockwise in the Northern hemisphere and anti-clockwise in the Southern hemisphere. I’ll describe these properties and explain how they can be understood in terms of something more familiar to high energy physicists. I’ll show that the equations that govern the long-time dynamics of the ocean can be recast as a Maxwell-Chern-Simons theory.

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)

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)

]]>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)

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

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

]]>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.

]]>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?

]]>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.

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

]]>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.

started to add something in a new section

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

]]>Added more references under *Chern-Simons theory – References – Perturbative quantization*

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.

]]>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.

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 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 subsection that list references on geometric quantization of CS 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?

]]>OK, 3d quantum gravity and group field 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.

]]>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.

]]>