3-Dimensional General Relativity  

In General > s.a. 3D gravity theories, manifolds and quantum gravity; general relativity; positive energy theorem.
* Action: There are different, classically equivalent ones, including a BF one, except for the fact that some (like the first-order one) admit degenerate metrics; In 8G = 1 units, and with F the curvature of the spin connection ,

SEH = dx2 |g|1/2 (R–2) ,   S1st = (eI FJK –   eI eJ eK) IJK .

* Dynamics: For any , the set of solutions (moduli space) is finite-dimensional; If = 0, the field equations imply F = 0, flat space (Rab = 0 implies Rabcd = 0), and the moduli space of flat connections on the spatial manifold M has dimension 12 (g–1), with g the genus of M.
* Chern-Simons form: The phase space is the moduli space of flat G-connections; G is a typically non-compact Lie group which depends on the signature of spacetime and the cosmological constant; For Euclidean signature with = 0, G is the 3D Euclidean group; For Lorentzian signature with > 0, G = SL(2,C); Can be interpreted in terms of Cartan geometry.
@ Chern-Simons form: Bimonte et al IJMPA(98) [deformed Chern-Simons]; Matschull CQG(99); Meusburger & Schroers CQG(05)gq [boundary conditions and symplectic structure].

Solutions and Special Features > s.a. asymptotic flatness; 3D black holes; FRW spacetimes; gauge choice.
@ Solutions: Duncan & Ihrig GRG(76) [vacuum, static, rotationally symmetric]; Hirschmann & Welch PRD(96) [magnetic]; Williams GRG(98) [rotating kinks]; Brill CQG(04)gq/03-in [cosmology, lattice universes]; Wang & Wu gq/05 [massless scalar, self-similar, kink instability]; Barrow et al CQG(06)gq [cosmology]; Brill et al Pra(07)-a0707 [colliding particles with < 0].
@ Related topics: Hortaçsu et al GRG(03) [vacuum and + scalar, thermodynamics].
> Related topics: see Central Charge; boundaries in field theory; lattice field theory; singularities; time; topological defects.

With Matter and/or Cosmological Constant > s.a. gödel.
* Remark: 2+1 gravity coupled to point particles is a non-trivial example of DSR.
* Metric: When = 0, space is a flat 2D manifold with genus g and n punctures, representing point particles; The metric around each puncture (of mass m (0,2) and spin s R) can be written

ds2 = –(dt + s d)2 + (1–m/2)–1 dr2 + r2 d2 .

* Duality: Lorentzian theory with > 0 is dual to the Euclidean theory with a negative cosmological constant.
@ Point particles: Carlip NPB(89); de Sousa NPB(90); Lancaster PRD(90); Kabat & Ortiz PRD(94)ht/93 [braid invariance]; Menotti & Seminara AP(00)ht/99, NPPS(00) [ADM]; Cantini et al CQG(01)ht/00 [Hamiltonian]; Valtancoli IJMPA(00) [N particles + < 0]; Krasnov CQG(01)ht/00; Matschull CQG(01)gq [phase space]; Cantini & Menotti CQG(03)ht/02 [functional approach]; Freidel et al PRD(04)ht/03 [and DSR].
@ Einstein-Maxwell: Nayak GRG(91); Cataldo & Salgado PRD(96) [static]; Grammenos MPLA(05)gq/04 [magnetic solution, AdS background]; Bañados et al PRD(06)ht/05 [with cosmological constant and Chern-Simons term, Gödel-type black holes].
@ With scalar field: Henneaux et al PRD(02)ht [black holes]; Gegenberg et al PRD(03) [scalar field, action]; de Berredo-Peixoto CQG(03) [static]; Daghan et al GRG(05) [+ cosmological constant, static].
@ Other matter: Carlip & Gegenberg PRD(91) [topological matter]; Peldán NPB(93)gq/92 [+ Yang-Mills]; García & Campuzano PRD(03)gq/02 [fluid, static circularly symmetric].
@ With cosmological constant: Fujiwara & Soda PTP(90) [initial-value]; Corichi & Gomberoff CQG(99) [duality]; Krasnov CQG(02)gq/01 [asymptotically AdS, Euclidean continuation]; Miskovic & Olea PLB(06) [ < 0, boundary conditions]; Witten a0706 [dual conformal field theories]; Li et al a0801 [deformed by gravitational Chern-Simons action].

General References > s.a. models for topology change.
@ Early work: in Bergmann in(65) [comment by Wheeler]; & Leutwyler.
@ Reviews: Brown 88; Carlip JKPS(95)gq-ln; Welling ht/95-in [and point particles]; Carlip CQG(05)gq [especially conformal field theory and black holes].
@ General articles: Giddings et al GRG(84); Gott & Alpert GRG(84); Jackiw NPB(85); Ashtekar & Romano PLB(89); Bengtsson PLB(89); de Sousa Gerbert pr(89); Moncrief JMP(89); Hosoya & Nakao CQG(90); Moncrief JMP(90); Carlip CQG(91) [geometry]; Franzosi & Guadagnini CQG(96); Buffenoir & Noui gq/03; Nelson CQG(04) [global constants].
@ Polygon approach: 't Hooft CMP(88), CQG(92), CQG(93), CQG(93); Waelbroeck & Zapata CQG(96)gq; Welling CQG(97)gq/96 [torus]; Hollmann & Williams CQG(99)gq/98; Kádár & Loll CQG(04)gq/03 [higher-genus data]; Kádár CQG(05) [from first-order formalism]; Eldering gq/06-MS.
@ Hamiltonian form: Puzio CQG(94)gq [Gauss map, holonomies]; Mikovic & Manojlovic CQG(98) [on a torus]; Cantini et al CQG(01)ht/00 [and particles], ht/00-MG9; Kenmoku et al gq/00-in; Menotti gq/01-in; Nelson gq/04-in [ADM]; Bonzom & Livine a0801 [Immirzi-like parameter].
@ Witten formulation: Witten NPB(88); Louko & Marolf CQG(94)gq/93 [R T2]; Louko CQG(95)gq [R Klein bottle].
@ Other connection and holonomy formulations: Bezerra CQG(88); Peldán CQG(92); Manojlovic & Mikovic NPB(92); Unruh & Newbury IJMPD(94)gq/93 [holonomies and geometry]; Barbero & Varadarajan NPB(95)gq [homogeneous], CQG(99)gq [degrees of freedom]; Mikovic & Manojlovic CQG(98)gq/97 [T2, Ashtekar variables, reduced phase space].
@ Moduli space: Mess pr(90); Nelson & Picken mp/05-in [T2, < 0, holonomies and quantization].
@ Related topics: Martin NPB(89), Waelbroeck NPB(91) [observables, time]; Nelson & Regge CMP(93) [observables]; Forni et al JMP(00)gq [null surfaces]; Bengtsson & Brännlund JMP(01)gq/00 [chaos & time machines on R × T2]; Niemi PRD(04)ht/03 [from 2D SU(2) Yang-Mills].

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