Abstract
We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.
Highlights
The idea of Higher Spin Gravities (HSGRA) is to construct viable models of Quantum Gravity by looking for extensions of classical gravity with massless higher spin fields
Higher spin symmetry is supposed to leave no room for relevant counterterms and constructing a classical HSGRA is almost sufficient for having a quantum theory
Much of the structure of a minimal Free Differential Algebras (FDA) (A, δ) as well as its cohomology groups are controlled by the cohomology of the associated Lie algebra L(A)
Summary
The idea of Higher Spin Gravities (HSGRA) is to construct viable models of Quantum Gravity by looking for extensions of classical gravity with massless higher spin fields. In the present paper we classify invariant and covariant functionals of fields in 4d HSGRA, 3d HSGRA and with some remarks on HSGRA’s in higher dimensions as well. What is surprising is that there exist a great many higher spin invariant and covariant functionals, as will be seen from our classification This should be confronted with the lower spin gauge theories, e.g. Yang-Mills theory or Gravity, and more generally, with the Weinberg-Witten theorem [31]. The problem of classification of covariant and invariant functionals can be reduced to that of computing Chevalley-Eilenberg cohomology The latter can further be related to the cyclic and, eventually, to the Hochschild cohomology of a given higher spin algebra: Chevalley-Eilenberg.
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