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Connection vs metric description for non-AdS solutions in higher spin theories
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We consider recently-constructed solutions of three dimensional SL(N,R) x SL(N,R) Chern-Simons theories with non-relativistic symmetries. Solutions of the Chern-Simons theories can generically be mapped to solutions of a gravitational theory with a higher-spin gauge symmetry. However, we will show that some of the non-relativistic solutions are not equivalent to metric solutions, as this mapping fails to be invertible. We also show that these Chern-Simons solutions always have a global SL(N,R) x SL(N,R) symmetry. We argue that these results pose a challenge to constructing a duality relating these solutions to field theories with non-relativistic symmetries.
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Notes on su$(1,2)\oplus$u$(1)$ Chern-Simons theory and Torsional Newton-Cartan gravity
The su(1,2)⊕u(1) Chern-Simons theory is torsional Newton-Cartan gravity whose 1/c expansion reproduces the extended z=2 Schrödinger gravity and whose asymptotic symmetry is the W_3^(2)⊕u(1) algebra.
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