## Catherine Meusburger: 3D gravity

We give an overview of Lorentzian and Euclidean gravity in 3 dimensions and show that structures from noncommutative geometry arise naturally there with a clear physical interpretation.

We start by summarising the classification of maximal globally hyperbolic 3d Lorentzian vacuum spacetimes, which leads to a description in terms of group homomomorphisms from the fundamental group of the spacetime into the isometry group of 3d de Sitter, Minkowski or anti de Sitter space. This relates the phase space of 3d gravity to a moduli space of flat connections on an oriented surface and to Teichmüller space.

In the second lecture we explain how moduli spaces of flat connections can be viewed as Poisson-graph gauge theories with Poisson Lie-symmetries as gauge transformations. We discuss applications to 3d gravity and show that this description naturally leads to non-commutative spacetime coordinates. We also comment on the relation of the phase space of 3d gravity to Teichmüller space and show that its symplectic structure can be described in terms of generalised shear coordinates.

In the third lecture we discuss Hamiltonian quantisation formalisms for moduli spaces of flat connections on sufaces. We show that their quantisation can be viewed as a construction of a Hopf algebra gauge theory, where Hopf algebras take the role of gauge symmetries and module algebras over Hopf algebras the role of functions of flat gauge fields. We show how gauge invariance and curvature can be implemented in this description and that this gives rise to a topological invariant.

Background reading and further reading:

- basics of (2+1)-gravity: [5] chapters 2.1, 2.5, 2.6, 4.1 - 4.7, [6] chapters 2.1-2.3,
- basic hyperbolic geometry and Fuchsian groups: [8] chapters 1-3,
- classification of globally hyperbolic (2+1)-spacetimes: [3], summary in [10],
- symplectic structure of moduli spaces of flat connections: [7,1,2],
- Poisson-Lie groups and related structures: [9] chapter 8 or [4] chapters 1-2,
- symplectic structure in 3d gravity and relation to Teichmüller theory [11],
- Hopf algebras and related structures: [9], chapters 1, 2, 6.1 and 7,
- quantisation: [13,12] and references therein.

References:

[1] A. Alekseev, A. Malkin, Symplectic structures associated to Lie-Poisson groups, Communications in Mathematical Physics 162.1 (1994) 147-173.

[2] A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued, moment maps, J. Differential Geometry 48.3 (1998) 445-495.

[3] R. Benedetti, F. Bonsante, (2+1)-Einstein spacetimes of finite type, Handbook in Teichmuller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House (2007), arXiv:0704.2152.

[4] V. Chari, A. Pressley, A guide to quantum groups, Cambridge University press (1995).

[5] S. Carlip, Quantum gravity in 2+1 dimensions. Vol. 50. Cambridge University Press (2003).

[6] S. Carlip, Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe, Living Rev. Relativity 8 (2005), http://www.livingreviews.org/lrr-2005-1.

[7] V. Fock, A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix (1998), arXiv:math/9802054.

[8] S. Katok, Fuchsian groups, University of Chicago press (1992).

[9] S. Majid, Foundations of quantum group theory, Cambridge University press (2000).

[10] C. Meusburger, Global Lorentzian geometry from lightlike geodesics: What does an observer in (2+1)-gravity see?, in Chern-Simons Gauge Theory: 20 years after, AMS/IP Studies in Advanced Mathematics (2011), Vol. 50, 261-276, arXiv:1001.1842.

[11] C. Meusburger, C. Scarinci, Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes, J. Differential Geometry 103 (2016) 425-474, arXiv:1402.2575.

[12] C. Meusburger, Kitaev lattice models as a Hopf algebra gauge theory, arXiv:1607.01144.

[13] V. Schomerus, Poisson structure and quantization of Chern-Simons theory, in: Landsman, Nicholas P., Markus Pflaum, and Martin Schlichenmaier (eds): Quantization of singular symplectic quotients. Vol. 198. Birkhäuser (2001).