# Three-Dimensional Gravity

Does anyone have any references that discuss gravity in three-dimensions? I'm trying to make my way through some papers by Witten relating $SL(2,\mathbb{C})$ Chern-Simons theory and gravity in three spacetime dimensions and would like some explanation about the latter of the two topics.

• Please tell us which papers you have in mind precisely so that we don't give you references you already have. There's e.g. this arxiv.org/abs/0706.3359 and this adsabs.harvard.edu/abs/1988NuPhB.311...46W – Marek May 14 '11 at 10:21
• Yes, those are precisely the two papers I'm starting with. I'm wondering if there are any papers/books which lay out GR in 3 spacetime dimensions. I am only familiar with Carroll's book, Wald, and MTW. I have seen Carlip's book, but still waiting for it to be returned to the library. – klw1026 May 14 '11 at 16:02

The t'Hooft references started the modern field, and they are in my opinion the best for first learning the subject, because they are self-contained and they allow a full simulation of the classical dynamics. These are taken from t'Hooft's web page:

The last paper is available online, and gives a complete numerical implementation and numerical results.

Steve Carlip wrote a book, quantum gravity in 2+1 dimension. He also has a review article on Living Review.

It is unclear to me whether you're asking for a resource on three-dimensional gravity(and general Chern-Simons) or refined resources on three-dimensional gravity and SL(2,C) Chern-Simons. An excellent survey connecting three dimensional gravity to (general) Chern-Simons, via the usual methods -- vector bundle parallel transport, Wess-Zumino-Witten action, etc. -- is "3D Gravity, Chern-Simons and Higher Spins: A Miniature Introduction" [Kiran, KS. 14].

If you are unfamiliar w/ how 2+1 gravity is related to general gauge theories -- w/ (first-order) einstein-hilbert action in (2+1)dim proportional to chern simons three-form, etc, -- this text is an excellent introductory read.

For a more specific read on SL(2,C) Chern-Simons and (2+1)dim gravity, I suggest "Three-Dimensional Quantum Gravity, Chern-SImons Theory, and the A-Polynomial" [Gukov, S. 03]. The text focuses especially on hyperbolic three-manifolds and (Alexander / Jones) knot invariants, most of which likely unrelated to the topic of interest [Witten, E. 07], but the result regarding SL(2,C) Chern-Simons partition function and coloured jones is quite exceptional and may be of some use in your readings.

note: for introductory readings on (2+1)dim gravity itself, there is an excellent survey on "exactly solvable" models in "2+1 Dimensional Gravity as an Exactly Solvable System" [Witten, E. 98]. Not to mention, "Developments in Topological Gravity" [Dijkgraaf, R., et all. 18] surveys the topic, where Topological Gravity is elegantly related to matrix models of (2+1)dim gravity via "Weil Peterson Volumes and Intersection Theory on the Moduli Space of Curves" [Mirzakhani, M. 07], etc.

Lastly, as someone quite familiar w/ the text "Three-Dimensional Gravity Revisited" [Witten, E. 07] I recall the topics of extremal holomorphic CFTs (and dual pure AdS3) where of particular importance, specifically where AdS3 has maximally negative Cosmological Constant and the CFT has central charge c=24.

The paper is particularly interested in the following case due to the CFT partition function being proportional to the graded character of the moonshine module. Similar work regarding the monster CFT was completed by "Vertex Operator Algebras and the Monster" [Frenkel, IB., et all. 88] in the so called "Frenkel-Lepowsky-Meurman conjecture" relating the moonshine to vertex operator algebras of central charge c=24 and j-744 graded dimension -- used to study monster CFT via pure AdS3 gravity w/ maximally negative Cosmological Constant.

The difficulty had in this paper, an papers related to it, i.e. "Chiral Gravity in Three Dimensions" [Strominger, A. 08] (refined arguments on the stability of consistent AdS3 w/ suitable partition function via chiral monster CFT), etc., is the language of monstrous moonshine, monster group, and general moonshine -- especially in the case of newer work; i.e. "Notes on the K3 Surface and the Mathieu Group M24" [Eguchi, T., et all. 10], etc. I recommend "Monstrous Moonshine" [Conway, J., et all. 79] and "Vertex Operator Algebras and the Monster" [Frenkel, IB., et all. 88] as general introductions to the topic, from the point of view of a mathematician and physicist respectively.

I hope any of that was of any help!