Characteristic cohomology and observables in higher spin gravity

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.

Slightly broken higher spin symmetry: general structure of correlators

“…Together with [20] the results of the present paper strongly indicate the threedimensional bosonization duality: every CFT with a slightly-broken higher spin symmetry (or any other theory with the same type of symmetry) has to have these correlations functions, irrespective of its microscopical realization (whether it is given by Chern-Simons plus bosonic or fermionic matter). For our results to apply to the widest possible class of models, we do not go into specific details.…”

“…The goal of this paper is to construct invariants of the slightly-broken higher spin symmetry, which should compute the correlation functions. The uniqueness of these invariants in the case of Chern-Simons matter theories [20] implies the three-dimensional bosonization duality [18,[21][22][23][24][25] in the large-N limit.…”

“…[18,21,30,31]. Some interesting facts that have been known include: (i) the higher spin algebras of free boson and free fermion are the same in 3d, which is a necessary condition for the bosonization to take place; (ii) the three-point functions are fixed by the deformed higher spin symmetry [18], see [30] for most and [32] for all of them; (iii) in [20], a complete classification of higher spin invariants was obtained. It was shown that the deformed higher spin symmetry has exactly one invariant to serve as an n-point function and this invariant is unobstructed.…”

“…As a result, many applications of Virasoro rely on its representation theory [1], while L ∞ has to do with the higher cohomology of higher spin algebras. Applications require Chevalley-Eilenberg cohomology, which can be reduced to the cyclic and then to the Hochschild cohomology of higher spin algebras as associative algebras [20]. The latter is the first important property of higher spin symmetry -it originates from associative algebras, which eventually governs all of the Lie-type structures, e.g.…”

“…In the case of free 3d scalar and free 3d fermion CFT's, where the underlying algebra is the Weyl algebra A 2 , more detailed answers can be obtained and the matrix extensions can be dropped: it can be shown that the relevant Chevalley-Eilenberg cohomology can be reconstructed from the Hochschild cohomology of the Weyl algebra. This was computed in [20]. In particular, table 5 of [20] implies that, upon restriction to even spins J s , the A ∞ /L ∞ -algebras depend on two parameters, i.e.…”