Connection versus metric description for non-AdS solutions in higher-spin theories

Abstract

We consider recently-constructed solutions of three dimensional SL(N, R)×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 higherspin 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) × 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.

Higher spin holography with Galilean symmetry in general dimensions

“…where ν is a free parameter and k is the Klein operator. Define bilinear oscillators T αβ 1 We only focus on integer z since it is recently argued [55] that Schrödinger solution with fractional dynamical exponent z in 3D higher spin theory may not have well-defined metric like description. The reason is that given the frame field E µ defined above, one cannot solve spin-connection ω uniquely from torsion free equation [52] de…”

“…This solution has no non-trivial holonomy, so one can do a large gauge transformation to relate this solution to empty AdS [55].…”

“…where ǫ 0 (y) does not depend on spacetime coordinates and fully determines the global (internal) symmetry. It is concluded in [55] that the symmetry of Schrödinger higher spin solution in 3D Chern-Simons theory is just SL(N, R) × SL(N, R) by applying the gauge function method above. In the current higher dimensional case, one could also conclude that ǫ 0 (y) exhausts the whole Vasiliev higher spin symmetry group.…”

“…where D = 3 and z > 1 is the dynamical scaling. Translation between the Chern-Simons formalism and the metric like formalism are discussed in [47,[51][52][53][54][55]. Further discussion of this Schrödinger geometry in holographic theories can be found in [56].…”

Null warped AdS in higher spin gravity

“…Several papers in the past few years studied this question (and related ones), see e.g. [27][28][29][30][31][32][33], and some were indeed able to find consistent sets of boundary conditions that permit these solutions, including Lobachevsky holography [34], flat space holography [35,36] and Lifshitz holography [37].…”