Talk:Chern–Simons theory

Invention of Chern-Simons theory incorrectly attributed to Chern and Simons[edit]

The introduction incorrectly attributes the invention of the Chern-Simons QFT to Chern & Simons, who invented the topological invariant used as the action for the QFT. I've tried to fix this, and hopefully haven't slighted any condensed matter researchers. —Preceding unsigned comment added by 129.49.61.155 (talk) 20:21, 5 March 2010 (UTC)[reply]

I think that it is also incorrect to say that the Chern-Simons field theory was introduced by Ed Witten. As far as I know he only realized the connection to knot theory, in 1989. I have found a few older papers by Deser, Jackiw and Templeton which contain the Chern-Simons term (>>Phys. Rev. Lett. 48, 975 (1982)<<, >>Annals of Physics 140, 372 (1982)<<, >>Phys. Rev. D 23, 2291 (1981)<<). The origin of the Chern-Simons term in field theory might be much older! But it might be true that Ed Witten was the first person to consider only a Chern-Simons term (ie without a Yang-Mills term added). Anyone that knows the origin? 4tnemele (talk) 10:06, 19 August 2010 (UTC)[reply]

I found another paper from the same year (and even the same month, January) by Schonfeld (>>Nucl. Phys. B185, 157 (1981)<<) which seems to be independent from the Jackiw et al papers. The Chern-Simons term is also introduced in order to get massive gauge fields in 3D, but he calls the theory for $\xi$ -theory since his term is written as ${\mathcal {L}}_{\xi }={\frac {1}{2}}\xi \epsilon _{\mu \nu \lambda }{\text{Tr}}A^{\mu }[\partial ^{\nu }A^{\lambda }-{\frac {3}{2}}igA^{\nu }A^{\lambda }]$ . The way the paper is written suggests that this is close to the origin of the term. He mentions under "Note added in proof" that while preparing the paper he has become aware of a preprint by Jackiw (probably the 1981 paper I cited above) and a paper by Siegel (>>Nucl. Phys. B156 (1979) 135<<) which also contain this term. The Siegel paper considers a super symmetric version. I'm not sure, but I doubt that this can be traced further back in time. 4tnemele (talk) 10:48, 19 August 2010 (UTC)[reply]

It is sure that the origin of Chern-Simons theory consists not of Witten's but of Chern and Simons' titled "Characteristic Forms and Geometric Invariants" in 1974, which refines the Chern-Weil theory. But Witten has very much contibuted to this item. --Enyokoyama (talk) 13:46, 19 August 2012 (UTC)[reply]

I add　some mathematial explanation into the head of "classical theory," and correct "introduced" -> "developed" at the head of this article. And I wonder this article is tend to only string theoretical side. It should contain the other sides, for example massive gauge theory.--Enyokoyama (talk) 10:44, 29 December 2012 (UTC)[reply]

I've added a little explanation of Chern-Simons gravity theory as an paragraph in the other theory, and Deser, Jackiw and Templeton's paper as an reference.--Enyokoyama (talk) 05:29, 29 April 2013 (UTC)[reply]

Mathematical help wanted[edit]

This page links to the disambiguation page Action. I'm not sure which page it's supposed to link to - I don't think it's Group action, which is the only vaguely mathematical page under Action. Could someone more knowledgeable fix the link to point directly to the appropriate page? Soo 22:42, 24 August 2006 (UTC)[reply]

Can someone verify the claim taht SO(N) is Kauffman. I think this is not precisely correct. Maybe it is true for odd level, but I don't think it is general. —Preceding unsigned comment added by 163.1.74.1 (talk) 07:56, 2 June 2009 (UTC)[reply]

Chern-Simons gravity[edit]

In "Chern-Simons Gravity: from 2+1 to 2n+1 dimensions" in Braz. J. Phys. vol.30 no.2 São Paulo June 2000 by Zanelli, he said that

Chern-Simons (CS) theory has a curious history. It was discovered in the context of anomalies in the 70's and used as a rather exotic toy model for gauge systems in 2+1 dimensions ever since [10] . It was only by the mid 80's that it was realized that ordinary Einstein gravity in 2+1 dimensions is a natural example of a CS system, especially through the work of Witten [11]. As it turns out, CS systems are more conspicuous than it might seem at first sight. General Relativity in 2+1 dimensions (with or without cosmological constant) is a CS system (for ISO(2, 1) or SO(2, 2) groups, respectively); any ordinary mechanical system in Hamiltonian form can be viewed as an abelian CS system in 0+1 dimensions [12]. This way of looking at mechanical systems is not completely absurd and it even sheds some light into ancient problems such as the justification for the old quantization rule of Bohr and Sommerfeld.

where [10] is S. Deser, R. Jackiw and S. Templeton, Ann.Phys.(NY) 140, 372 (1982) and B. Zumino Les Houches Summer School (1983), and [11] is E. Witten, Nucl. Phys. B 311, 46 (1989).

This article by Zanelli is very interesting.--Enyokoyama (talk) 13:08, 4 August 2013 (UTC)[reply]

：[12] J. Saavedra, R. Troncoso, J. Zanelli, Degenerate Dynamical Systems J. Math. Phys. 42, 4383 (2001); http://dx.doi.org/10.1063/1.1389088 (8 pages) arXiv:hep-th/0011231--Enyokoyama (talk) 00:04, 24 August 2013 (UTC)[reply]

And, the relation with 2+1 dimensional gravity is described on the above paper.

The last equation implies that the connection can be written as a function of the vielbein and (40) states that the spacetime must have constant curvature at each point. In view of the fact that (2+1) gravity has no propagating degrees of freedom, constant curvature spacetimes were thought to be rather dull configurations. However, one can be surprised by the fact that solving (40) explicitly for spherically symmetric, static configurations does not necessarily produce a globally AdS spacetime, but an AdS spacetime with identifications as well. This is because the field equations only refer to local properties of spacetime and do not restrict the global topology further. Thus, the spacetime manifold can be cut and pasted, identifying points connected by a finite isometry along a Killing vector -as when one makes a cylinder out of a plane-, one should still have a solution6. In fact, one can produce a black hole in this fashion . What is even more remarkable, is the fact that one can generate a solution by a Lorentz boost that sets the black hole in rotation about its symmetry axis. [17]

(40) is

\epsilon _{abc}(R^{ab}+{\frac {1}{\ell ^{2}}}e^{a}e^{b})=0,

the last equ. is

T^{a}=0.

[17], of course, is BTZ.--Enyokoyama (talk) 14:08, 5 August 2013 (UTC)[reply]

Please be more specific[edit]

When introducing notation like T that symbolizes a function, then kindly state in the article what its domain and codomain are. It is not enough to have a link somewhere within the article to an article about this function (or kind of function).

Observing this guideline is enormously helpful to readers trying to understand difficult material. But it is also tremendously helpful to other editors who then have well-defined notation to work with.