Induced Spin from the ISO(2, 1) Gauge Theory with the Gravitational Chern-Simons Term

In the context of ISO(2, 1) gauge theory, we consider (2 + 1)-dimensional gravity with the gravitational Chern-Simons term (CST). This formulation allows the ‘exact’ solution for the system coupled to a massive point particle (which is not the case in the conventional Chern-Simons gravity). The solution exhibits locally trivial structure even with the CST, although still shows globally nontrivialness such as the conical space and the helical time structure. Since the solution is exact, we can say the CST induces spin even for noncritical case of σ + αm 6= 0. (To appear in Phys. Lett. B) e-mail : [email protected] e-mail : [email protected] In the hope of unifying elementary forces including gravity, it has been one of the most fascinating things in theoretical physics to understand gravity in the context of gauge theory [1]. There have been many approaches to this end starting from Kibble [2]. However, due to the inhomogeneity and non-compactness of the Poincaré group, it is still never trivial to construct a satisfactory theory that can be viewed on an equal footing with other gauge theory. Recently, Witten showed that at least in (2+1)-dimension, it is possible to write Einstein gravity in a completely analogous way with the usual Chern-Simons gauge theory [3]. Further, motivated by the fact that the local ISO(2, 1) symmetry is, on shell, equivalent to the usual diffeomorphism [4], Grignani et al. constructed Poincaré gauge theory coupled with a massive point particle, making use of ‘Poincaré coordinates’ [5, 6]. The above Poincaré gauge gravity is based on a non-degenerate, invariant quadratic form < Ja, Pb >= ηab, < Ja, Jb >=< Pa, Pb >= 0 on the Lie algebra iso(2, 1), where Pa and Ja are the generators satisfying [Pa, Jb] = ǫab Pc, [Ja, Jb] = ǫab Jc, [Pa, Pb] = 0. (1) Note that the conventional Killing metric of iso(2, 1) is degenerate, so we can’t use it here. Incorporating this quadratic form, the usual Chern-Simons Lagrangian for the gauge connection A = ωJa+e Pa leads to just the the Einstein-Hilbert Lagrangian written in the vielbein notation. LEH = < A∧, (dA+ 2 3 A∧A) > = 2e ∧ (dωc + 1 2 ǫcabω a ∧ ω). (2) This raises a question; what about CST ∼ ω ∧ (dωc + 1 3 ǫcabω a ∧ ω) that was first introduced by Deser et al. [7] to give local dynamics to the standard locally trivial (2+1)-dimensional Einstein gravity (hereafter, the resulting Einstein gravity accompanied by CST is called Chern-Simons gravity or CSG). We expect this CST also to be formulated possibly in ISO(2, 1) gauge theory, just from Chern-Simons Lagrangian, since the term is gauge invariant. Indeed such can be done making use of general quadratic form obtained by taking the Poincaré limit (λ → 0) for linear combination of two types of de Sitter invariant quadratic forms which Witten proposed in ref. [3]. However, the same quadratic form is attainable more systematically, that is, without resource to the above de Sitter invariant quadratic forms. We just require those generators Pa, Jb be the Killing vectors for the supposed quadratic form. The resulting general quadratic form of ISO(2, 1) reads as < Ja, Jb >= α ηab, < Ja, Pb >= ηab, < Pa, Pb >= 0, (3)