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\(2+1\)-dimensional gravity as an exactly soluble system. (English) Zbl 1258.83032

The paper under review, which is seminal and inspiring for people working in mathematical physics, contains a lot of discussions involving different fields of mathematics and physics such as quantum field theory, Chern-Simons theory, gauge theory, theory of algebraic groups, Yang-Mills theory, topology and string theory.
The author starts considering the classical phase space in \((2+1)\)-dimensional gravity (space of all solutions of the classical equations up to gauge transformations). Thus, assuming the absence of a cosmological constant, the Ricci tensor vanishes and for instance the space-time \(\mathbb{M}\) is flat, where \(\mathbb{M}=\Sigma\times\mathbb{R}^1\), \(\Sigma\) a Riemann surface of genus \(g\) for space and \(\mathbb{R}^1\) for time. An interesting result in this paper is the explicit equivalence between gravity and gauge theory in three dimensions.
The novice reader can find two examples of flat spaces with initial singularities as a motivation to follow the rest of the paper. In the first one \(\Sigma\) is considered as a compact smooth two-dimensional surface of genus \(g\), while in the second one a \(2+1\) dimensional Minkowski space is considered. However, the paper involves different specialised areas of mathematics and physics which are not trivial to understand.
The paper is divided in four sections as follows. Section 1 contains a motivating introduction with a summary of results and discussions. In Section 2, the canonical formalism of \((2+1)\)-dimensional gravity at the classical level is discussed. Here a detailed connection with the Chern-Simons action is presented. A supersymmetric generalisation is presented as well, and it is shown that the three-dimensional conformal group \(\mathrm{SO}(3,2)\) acts over the Lorentz Chern-Simons three forms as \(\mathrm{ISO}(2,1)\) acts over \((2+1)\)-dimensional general relativity without a cosmological constant. In Section 3, a canonical formalism to do the quantization is constructed. The starting point is the Lagrangian on a three-dimensional manifold \(\mathbb{M}=\Sigma\times \mathbb{R}^1\), and the Poisson brackets on the physical phase space correspond to Poisson brackets restricted to functions that are invariant under the group generated by the constraints, the so-called constraint group. The polarisation is studied by means of two different approaches. The first one is considering the space phase \(\mathcal{M}\) as a cotangent bundle of some manifold \(\mathcal{N}\) where the quantum Hilbert space is the space of square integrable functions on \(\mathcal{N}\). The second approach presented by the author consists in putting a Kähler structure on the manifold \(\mathcal M\), the space phase, and in quantising it as a Kähler manifold. Ending this section, the author studies the relationship between \((2+1)\)-dimensional gravity and renormalisation theory and concludes with the un-renormalisability of quantum gravity in four dimensions. Section 4 presents a lot of interesting considerations which arose from the discussions throughout the paper. These considerations lead him to suggest some open questions and conjectures which have been useful in the advancement of research in this subject.

MSC:

83C45 Quantization of the gravitational field
81Q60 Supersymmetry and quantum mechanics
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
57R56 Topological quantum field theories (aspects of differential topology)
17B81 Applications of Lie (super)algebras to physics, etc.
53C80 Applications of global differential geometry to the sciences
81T13 Yang-Mills and other gauge theories in quantum field theory
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