Partially-massless higher-spin algebras and their finite-dimensional truncations

Abstract

Abstract:The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS d+1 are studied. The algebras involving PM generators up to depth 2 (ℓ − 1) are defined as the maximal symmetries of free conformal scalar field with 2 ℓ order wave equation in d dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of (A)dS d+1 isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of ℓ − d/2 , which coincides with the annihilator of the one-row ℓ-box Young diagram representation of so d+2 . Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.

New conformal higher spin gravities in 3d

“…Our proposal is to take advantage of a large class of higher spin algebras available from the study of symmetries of (higher derivative) free CFT's of type k Φ(x) = 0 [12][13][14][15] and partially-massless fields in AdS 4 [15][16][17][18] and to construct some new algebras that are finitedimensional [19,20]. This leads to several observations: (i) there exists a new class of nonlinear conformal higher spin theories in three dimensions; (ii) there is a large class of such higher spin algebras available; (iii) there are finite-dimensional algebras among those, which is not the case for the conformal higher spin fields studied in [9,10].…”

“…explicit structure constants [20,58]. It was also found in various contexts [19,20,59] that there are certain finite dimensional algebras that can be viewed as higher spin algebras, at least up to some point. 2 Below we extend this class of finite dimensional algebras.…”

“…Originally, they were found as symmetries of conformally-invariant higher derivative equations k Φ(x) = 0 [12,14,15]. explicit structure constants [20,58]. It was also found in various contexts [19,20,59] that there are certain finite dimensional algebras that can be viewed as higher spin algebras, at least up to some point.…”

“…Next to the simplest example is to take the rank-two symmetric representation, denoted by , which leads to algebra hs( ) [20] with generators t AB CD that are symmetric and traceless in the upper and lower pairs of indices. Its so(d, 2) decomposition reads…”

A Note on Rectangular Partially Massless Fields

“…Such CFTs were studied in [83] for instance, and were proposed to be dual to HS theories [82] whose spectrum consists, on top of the infinite tower of (totally symmetric) higher-spin massless fields, also partially massless (totally symmetric) fields of arbitrary spin (theories which have been studied recently in [84][85][86]), thereby extending the HS holography proposal of Klebanov-Polyakov-Sezgin-Sundell to the non-unitary case. The corresponding HS algebras were studied in [87] for the simplest case = 2 (as the symmetry algebra of the Laplacian square, thereby generalising the previous characterisation of hs (d) 0 as the symmetry algebra of the Laplacian [88]) and for general values of in [89][90][91]. As we already mentionned, the interesting feature of such HS algebras is that their spectrum, i.e., the set of fields of the bulk theory, contains partially massless (totally symmetric) higher-spin fields [82] (introduced originally in [92][93][94][95], and whose free propagation was described in the unfolded formalism in [96]).…”