Unitarity in three-dimensional flat space higher spin theories

Abstract

We investigate generic flat-space higher spin theories in three dimensions and find a no-go result, given certain assumptions that we spell out. Namely, it is only possible to have at most two out of the following three properties: unitarity, flat space, non-trivial higher spin states. Interestingly, unitarity provides an (algebra-dependent) upper bound on the central charge, like c = 42 for the Galilean W (2−1−1) 4 algebra. We extend this nogo result to rule out unitary "multi-graviton" theories in flat space. We also provide an example circumventing the no-go result: Vasiliev-type flat space higher spin theory based on hs(1) can be unitary and simultaneously allow for non-trivial higher-spin states in the dual field theory.

BMS modules in three dimensions

“…4.2 will then be devoted to the construction of induced modules along the lines described above for Poincaré and bms 3 , and we will see there that the ordering that emerges in the ultrarelativistic limit may be seen as a normal ordering with respect to rest-frame conditions. Along the way we will compare our results to those of the non-relativistic limit described in [26], and we will see that the two limits lead to different quantum algebras.…”

“…4. In both cases we also comment on the emergence of unitary representations from an ultrarelativistic limit of highest-weight representations of the (extended) local conformal algebra, while stressing that non-relativistic limits naturally lead to non-unitary representations as those considered in [25,26].…”

“…In sect. 3.4 we shall also discuss a different contraction from so(2, 2) to iso(2, 1), to be interpreted as a non-relativistic limit giving rise to representations of the type discussed in [25,26].…”

“…This gives a nonlinear algebra that can be obtained as anİnönü-Wigner contraction of the direct sum of two W 3 algebras. This contraction has been discussed at the semiclassical level in [14,15] and a Galilean limit of the quantum algebra has been considered in [26]. Here we are interested instead in an ultrarelativistic limit of W 3 ⊕ W 3 .…”

BMS Modules in Three Dimensions

“…(2.18) of [26]). They can be interpreted as normal-ordered operators with respect to a highest-weight vacuum defined by conditions of the type (3.22) with ∆ = ξ = 0.…”