General NO-GO Result

The previously treated examples ofF W-algebras all point towards the conclusion that, under the assumptions I have been working with so far, it is not possible to find unitary representations of flat space higher-spin algebras that contain non-trivial higher-spin states. In this section, I will argue, on dimensional grounds, that this conclusion is generic for nonlinear flat space higher-spin algebras that can be obtained via ˙Inönü–Wigner contractions of AdS higher-spinW-algebras.

Suppose one starts from two copies of an inherently nonlinearW-algebra. These algebras contain higher-spin generators that will be denoted byWnandW¯n, whose commutation relations can be schematically written as

[W_n, W_m] =. . .+f(c) :AB:_n+m +. . .+ω(c)

s−1

Y

j=−(s−1)

(n+j)δ_n+m,0,

[ ¯W_n,W¯_m] =. . .+f(¯c) : ¯AB¯:_n+m +. . .+ω(¯c)

s−1

Y

j=−(s−1)

(n+j)δ_n+m,0. (11.27a) These commutators contain nonlinear terms, which are given by infinite sums of products of other generators of the algebra that are denoted byAn,BnandA¯n,B¯n, respectively³. The two copies of theW-algebra are characterized by central charges c,¯c. These central charges appear in the nonlinear terms and in the central charge terms in (11.27) via functionsf(c)andω(c). I will keep these functions arbitrary for the sake of generality, apart from the following restriction onf(c)

c→∞lim f(c) = 0, (11.28) which means that the nonlinear terms are subleading contributions in the semi-classical regime of large central charges. The ellipses in (11.27) denote additional linear and other nonlinear terms which can possibly appear on the right-hand side

3Of course for arbitraryW-algebras the nonlinear terms can contain an arbitrary large number of products of an arbitrary number of generators. For the sake of simplicity and without loss of generality, however, I will only use bilinear terms in order to make my argument clear in what follows.

of the commutator whose exact form is not important for this argument.

Starting from these two copies one can obtain a flat space higher-spin algebra via a nonrelativistic contraction, involving a contraction parameter of dimen-sion [length]⁻¹. In analogy to the considerations in Chapter10I define the new generators of the contractedF W-algebra as

U_n:=W_n+ ¯W_n, V_n:=−W_n−W¯_n, (11.29a) C_n:=A_n+ ¯A_n, D_n:=−A_n−A¯_n, (11.29b) E_n:=B_n+ ¯B_n, F_n:=−B_n−B¯_n. (11.29c) This means that the central charges c_L and c_M in terms of c and ¯c are given by (10.25). From these definitions, one can already infer that the generatorsUn,Cn, E_n, as well as the central chargec_Lare dimensionless. Similarly, the generatorsV_n, D_n,F_nand the central chargec_M have dimension [length]⁻¹.

In order to obtain unitarity under the assumptions I am making, i.e. there is a well defined vacuum which is invariant under the wedge algebra of theF W algebra, one has to take the limitcM →0for the reasons explained in the beginning of this chapter. The main point of the following NO-GO theorem is that one cannot take thec_M →0limit in such a way that non-trivial central terms remain on the r.h.s of the higher-spin commutators, which renders all higher-spin states to be null states that can be modded out from the resulting module.

For the commutator[Un, Vm], this is immediate on dimensional grounds. This com-mutator has dimensions of [length]⁻¹, and in order to have the same dimension, the central term appearing on the right hand side necessarily has to be proportional to cM, implying that the[Un, Vm]commutator will be center-less in the limitcM →0.

The commutator[V_n, V_m]is zero upon contraction, so the only non-trivial commuta-tor remaining to be examined is[Un, Um].

I will first look at the structure of the nonlinear terms of this commutator. Performing the contraction, one obtains

→0lim[U_n, U_m] =. . .+ lim

→0

1

4² (f(c) +f(¯c))(² :CE:_n+m+ :DF:_n+m) +(f(¯c)−f(c))

(:CF:_n+m + :DE:_n+m)+. . . . (11.30) For a generic functionf(c), so that (11.28) holds, one finds that

f(c) +f(¯c)∼ O(²) and f(c)−f(¯c)∼ O(). (11.31) The contraction→0can thus consistently be performed. The nonlinear terms that generically survive the contraction are the ones of the form:DF:_n+m,:CF:_n+m and :DE :_n+m, which have dimensions [length]⁻², [length]⁻¹ and [length]⁻¹, respectively. Since the commutator[Un, Um]is dimensionless, these nonlinear terms

have to appear with prefactors that depend oncM, in order to compensate for their dimension. The structure of the[U_n, U_m]commutator is thus schematically given by

[Un, Um] =. . .+O(_c¹2 M

) :DF:_n+m +O(_c¹

M) (:CF:_n+m + :DE:_n+m) + ˜ω(c_L)

s−1

Y

j=−(s−1)

(n+j)δn+m,0, (11.32)

where the central term has to be a function ofc_Lon dimensional grounds.

In order to take the limitc_M →0, one thus has to rescale⁴U_n→U˜_n:=c_MU_n. The central terms in the[ ˜Un,U˜m]commutator, however, do not survive this contraction, showing that all higher spin generators of the flat space algebra lead to null states when acting on the vacuum. Although I have given the argument for the case in which the nonlinear terms are quadratic, it can easily be extended to the case where nonlinear terms of higher order appear. I have thus shown that unitary represen-tations that contain non-trivial higher spin states are not possible for inherently nonlinear flat space higher spin algebras, at least not under the assumptions I have been working with.

Since every NO-GO theorem is only as good as the assumptions it is built upon I want to emphasize that there are at least three possible loopholes that could circumvent this theorem.

I have chosen a highest-weight state that satisfies the conditions (11.1). It could also very well be that a highest-weight representation is simply not the correct representation to consider. One possible candidate for a suitable unitary representation ofF W-algebras could be so-called “rest-frame” states which were proposed in [IX] to explain unitarity of the one-loop partition function in flat space.

Assuming the highest-weight conditions (11.1) hold, then another possible loophole to the NO-GO theorem could be to use a different prescription when determining the norm of states as in comparison to the one I used in this analysis.

If both the highest-weight conditions (11.1) and the prescription to determine the norm of states are the same as the one I used previously, then the most obvious loophole remaining would be the nonlinearity of the higher-spin algebra. If the higher-spin algebra under consideration is an inherently linear one, then the whole argument in Subsection11.3.3would not be valid anymore.

4This rescaling applies for bilinear terms. For higher order terms one would have to use higher powers ofcM as well in order to be able to take the propercM →0limit.

In Section11.4I will show how such a loophole can be used to circumvent the NO-GO theorem presented in this subsection using the linearF W_∞algebra.