Msc defense by Marieke van Beest

Newton-Cartan Gravity and 3D Chern-Simons Theory

In this thesis, Torsional Newton-Cartan geometry is developed by gauging the d+1 dimensional Bargmann algebra. Before it is applied in the non-relativistic setting, the method of gauging a symmetry is reviewed for the Poincaré algebra. Particular attention is paid to the case of Twistless Torsional Newton-Cartan geometry, for which the most general local Bargmann invariant action in 2+1 dimensions is constructed, and recognised as Horava-Lifshitz gravity. It is demonstrated that this action can be related to a Chern-Simons theory, in the same way as its relativistic counterpart. In particular, this work shows that the centrally extended Bargmann algebra and its generalisation with a non-zero cosmological constant, the centrally extended Newton-Hooke algebra, both have non-degenerate bilinear forms, and derives the most general Chern-Simons action on the latter algebra. It is shown that the piece of the Bargmann invariant Chern-Simons action, which is analogous to the Einstein-Hilbert term in the relativistic case, reproduces the simplest (torsionless) U(1) invariant Horava-Lifshitz action. A set of solutions to this action (including a cosmological constant term) is determined, up to transformations generated by the two central extensions.

Furthermore, this thesis identifies a contraction of the 2D conformal algebra that yields a relativistic symmetry algebra on the boundary, which is associated to a different real form of the Newton-Hooke algebra in the bulk. This gives rise to a novel, non-relativistic bulk geometry, known as pseudo-Newton-Cartan geometry. It is demonstrated that the corresponding limit can also be taken directly in the Anti-de Sitter Chern-Simons action, and that the result is indeed the Chern-Simons action on the contracted algebra. Finally, the contraction can also be employed on the phase space of the relativistic theory to reach pseudo-Newton-Cartan solutions. This is applied to the BTZ black hole, and a novel pseudo-Newton-Cartan geometry is identified. The limit solution is analysed in terms of its holonomies, and the renormalised on-shell action is computed. It is demonstrated that the non-trivial topology of the geometry is preserved by the contraction, however, the smoothness of the time cycle implies a trivial result for the temperature. The renormalised on-shell action is finite and depends on the parameters of the geometry. Thus the interpretation of the novel pseudo-Newton-Cartan solution as a black hole-like geometry is seemingly ambiguous.