Near horizon boundary conditions for spin-3 gravity in flat space

Abstract

The holographic principle proposes a solution to one of the most prominent problems of our time - the search for a consistent, quantized theory of gravity. According to this principle a theory of gravity in (d + 1) dimensions is equivalent to a quantum field theory (without gravity) in d dimensions. An important realization of this conjecture is the Anti-de-Sitter/conformal field theory (AdS/CFT) corre- spondence. However, since this correspondence is a strong-weak correspondence, it is hard to explicitly check the holographic principle by calculating observables on the field theory and the gravitational theory side. Conversely, higher-spin theories lead to weak-weak dualites, which can provide useful insights into aspects of the holographic principle [1-3]. Furthermore, since calculations in three dimensions may be done in the Chern-Simons formulation and are technically less challenging than in higher dimensions, it is often useful to restrict onself to three dimensions to clear up conceptional issues and obtain a better unde standing of the holographic principle. In this thesis, we construct a new set of boundary conditions for spin-3 gravity in three-dimensional flat space. This set of boundary conditions is inspired by the recent 'Soft Heisenberg hair'-proposal for Einstein gravity in three-dimensional Anti-de-Sitter space [4], which has subsequently been extended to flat space [5] and higher-spin gravity in AdS space [6]. In chapter 2 we discuss the peculiarities of restricting oneself to three dimensions and review the Chern- Simons formalism and the canonical analysis. In chapter 3 we discuss boundary conditions for gravity in three-dimensional AdS space and give a review of the Brown-Henneaux boundary conditions [7] and the near horizon boundary conditions proposed in [4]. In chapter 4 we motivate the respective near horizon boundary conditions for spin-3 gravity in three- dimensional flat space and compute the canonical boundary charges and the asymptotic symmetry algebra. As in previous, related work [4-6] the boundary conditions ensure regularity of the solutions (1) current independently of the charges. The asymptotic symmetry algebra is again given by a set of u algebras. We find that the vacuum descendants generated by the charges all have the same energy as the vacuum, i.e. they are higher-spin 'soft hair' in the sense of Hawking, Perry and Strominger [8]. Furthe more, we derive the entropy for solutions that are continuously connected to flat space cosmologies and find the same result as in the spin-2 case: the entropy is linear in the spin-2 zero-mode charges and inde- pendent from the spin-3 charges. Using twisted Sugawara-like constructions of the higher-spin currents we show that our simple result for entropy of higher-spin flat space cosmologies coincides precisely with the complicated earlier results expressed in terms of higher-spin zero-mode charges.7