Algebraic Aspects of Higher Spin Symmetry

Open Access
Govil, Karan
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
May 26, 2015
Committee Members:
  • Murat Gunaydin, Dissertation Advisor
  • Richard Wallace Robinett, Committee Member
  • Martin Bojowald, Committee Member
  • Ping Xu, Special Member
  • Higher spin algebra
  • AdS/CFT correspondence
  • Quasiconformal
  • Minimal representation
Massless conformal scalar fields in four-dimensional and six-dimensional Minkowski space-time correspond to minimal unitary representations of SO(4,2) (isomorphic to SU(2,2)) and SO(6,2) (isomorphic to SO*(8)), respectively. The Fradkin-Vasiliev type higher spin algebras are defined as the symmetry algebras of massless Klein-Gordon equation for a scalar field which implies that the universal enveloping algebras of the minimal representation are just the conformal higher spin algebras in four and six dimensions, or the AdS_5 and AdS_7 higher spin algebras, respectively. In this thesis, using the quasiconformal methods developed by Günaydin, Koepsell, and Nicolai, we formulate the minimal unitary representation for SU(2,2) and SO*(8) in terms of non-linear twistorial oscillators that transform non-linearly under the respective Lorentz groups, SL(2,C) in four dimensions, and SL(2,H) in six dimensions. Using this formulation, we will define the conformal higher spin algebras hs(4,2) and hs(6,2). We will also define a one-parameter family of continuous deformations, hs(4,2;z), and discrete deformations hs(6,2;t) of these higher spin algebras. Here z is the continuous helicity of massless conformal fields in four dimensions and t is the spin of an SU(2) subgroup of the little group of massless particles, SO(4), in six dimensions. Our results imply the existence of a family of (supersymmetric) HS theories in AdS_5 and AdS_7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in four and six dimensions, respectively. Using the quasiconformal methods, we also construct the minimal unitary representations (and its deformations) for the exceptional supergroup D(2,1;\lambda) which is the most general N=4 superconformal group in one dimension. We shall also review Lorentz covariant twistorial oscillator construction for SO(3,2) (isomorphic to Sp(4,R)), SU(2,2), and SO*(8).