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Higher spin Chern-Simons theory and KdV hierarchies


This dissertation summarizes my research in the Lifshitz higher spin Chern-Simons theory and its relation to the integrable system KdV hierarchy as a Ph.D. candidate at UCLA. In Chapter 1, I briefly review the higher spin gravity theory and introduce the Chern-Simons theory as a realization of the Vasiliev theory in three dimensional spacetime. In Chapter 2, I review the KdV hierarchies. In Chapter 3, I discuss how to construct a solution to the Chern-Simons theory which yields a spacetime that exhibits Lifshitz scaling, I also calculate the boundary charge algebra and show the asymptotic Lifshitz symmetry is realized in terms of it. In Chapter 4, I reveal the relation between the Lifshitz Chern-Simons theory and the KdV hierarchies (in the non-supersymmetric case), a proof of the general correspondence is also given using the Drinfeld-Sokolov formalism. In Chapter 5, I work out the supersymmetric extension of this correspondence in a particular case, with the boundary charge algebra of the supersymmetric Chern-Simons theory and the second Hamiltonian structure of the super KdV identified. In Chapter 6, I discuss on the results of my study and possible directions of future research.

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