# Aspects of Spin 3 Bianchi Identities in 2 + 1 Dimensional Conformal Higher-Spin Theories

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The higher spin theories in 2 + 1 dimensions considered in this thesis are of great physical interest since they are an important part in our understanding of both string theory, the AdS/CFT correspondance and M-theory. These conformal higher-spin theories are introduced by obtaining a conformal basis to the spin-2 algebra so(2, 3). This algebra neatly generalizes to the higher spin algebra, giving rise to a theory containing fields of all spins. Looking at the projection of the vacuum equations of motion, F = 0, and the Bianchi identities, DF = 0, onto the conformal basis, the content of these equations is explored. Using the conformal spin-2 basis, the curvature equations from 2+1 dimensional (conformal) general relativity is obtained as a confirmation that the method used is correct. A similar projection onto the conformal spin-3 basis is found and it is shown that assuming only the parabolic part of F = 0 is enough to satisfy the Bianchi identity. Hence, the Bianchi identity allows for coupling the system to matter.

**Nyckelord: **higher-spin theory, conformal gravity, Chern-Simons, gauge theory