The asymptotic W5 symmetry in three-dimensional Chern-Simons spin-5 gravitation theory and beyond

Abstract

This thesis is devoted to contributing somewhat to the understanding of gravity, higher spin theories, and conformal field theories by exploring asymptotic symmetries in the light of AdS/CFT correspondence. The motivations are two-fold: firstly, understanding the theoretical description of higher spin theories is a highly interesting task, as they might have better quantum properties than Einstein's gravity (whose quantum mechanical behaviour is not understood). Hence, such theories are a promising generalization of gravity (besides supergravity and string theory). Secondly, how supersymmetry is incorporated into this bosonic theory. The dissertation consists of six main parts: In chapter 1, we give a brief introduction serving as an overview of the thesis, outlining the main objectives, scope, and significance of the research conducted. In chapter 2, firstly, we briefly touch upon some technical details that form the basis of the theory addressed in the thesis. The first part of this chapter lays down the foundations for the mathematical framework in the subsequent sections. In this chapter, our primary goal is to thoroughly investigate some important aspects of three-dimensional anti-de Sitter (AdS_3) higher spin gravity with a negative cosmological constant within the framework of sl(5,ℝ) ⊕ sl(5,ℝ) Chern-Simons gauge theory. We explicitly construct the asymptotic symmetry algebra as two copies of the classical W_5 algebra with central charge c. We derive an explicit solution that can be regarded as a spin-5 generalization of the BTZ solution, thanks to a method recently discovered. We show how the W_5 symmetry algebra and the higher spin Ward identities emerge from the bulk equations of motion coupled to spin-5 currents. Additionally, we indicate how to introduce the chemical potentials and holonomy conditions associated with these higher spin charges in AdS_3 higher spin gravity in a manner that preserves the asymptotic symmetry algebra. We give an explicit generalized entropy expression that is very important to establish a relationship among generalized spin-5 black hole thermodynamics, asymptotic symmetries, and W_5 algebras. This higher spin asymptotic W_5 symmetry algebra that we calculated within the framework of the sl(5,ℝ) ⊕ sl(5,ℝ) Lie algebra-valued Chern-Simons gauge theory with a negative cosmological constant contains a finite number of conformal primary spins s: s = 2, 3, 4, 5. Our results offer an important, concrete supporting example, providing valuable insights into the space-time geometries of three-dimensional anti-de Sitter (AdS_3) higher spin gravity, as formulated through the framework of sl(N,ℝ) ⊕ sl(N,ℝ) Chern-Simons gauge theories. In Chapter 3, we carry out the supersymmetric extension of our work. We perform supersymmetric extension of our study including the supersymmetry transformation laws and demonstrate how to recover super algebras from its asymptotic space-time symmetries. We present a candidate for N=(1,1) extended higher spin AdS_3 supergravity with the most general boundary conditions. We eventually show that the asymptotic symmetry algebra consists of two copies of the osp(3|2)_k affine algebra on the affine boundary and two copies of SW(3/2,2) symmetry algebra on the superconformal boundary respectively in the presence of the most general boundary conditions. Furthermore, we impose certain restrictions on gauge fields for the loosest set of boundary conditions and that leads us to the supersymmetric extension of the Brown-Henneaux boundary conditions. Moreover, we derive the chemical potentials associated with the source fields that manifest through the temporal components of the connection. We also check the consistency of our construction in the presence of two different classes of boundary conditions that appeared in the literature. We discuss the conservation of the charges and also show that the Chern-Simons action, which is compatible with our boundary conditions, leads to a finite, well-defined variational principle for these higher-spin fields. In Chapter 4, we present the first example of N=(2,2) formulation for the extended higher spin AdS_3 supergravity with the most general boundary conditions as an extension of the previous chapter. Using a method recently proposed, we construct a consistent class of the most general boundary conditions to extend it. An important consequence of our approach is that, for the loosest set of boundary conditions, it ensures that their asymptotic symmetry algebras consist of two copies of the sl(3|2)_k. Moreover, we enjoin certain restrictions on the gauge fields for the most general boundary conditions, leading to the supersymmetric extensions of the Brown and Henneaux boundary conditions. Based on these results, we finally find out that the asymptotic symmetry algebras are two copies of the super W_3 algebra for N=(2,2) extended higher spin supergravity theory in AdS_3. We investigate whether or not our theory falls under the same metric class. As discussed in the previous section, we examine the conservation of charges, and further illustrate that the Chern-Simons action, in accordance with our boundary conditions, results in a finite and well-defined variational principle for the higher spin fields. In this way, we establish an explicit holographic dictionary between Chern-Simons supergravity on AdS_3 and two-dimensional CFT_2 with superconformal symmetry. Therefore, it can be considered that our method provides a good laboratory for investigating the rich asymptotic structure of both extended gravity and extended supergravity. The final chapter collects our results and provides guidance on interpreting the results of the dissertation in order to present the conclusions of our work effectively.