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**Unformatted text preview: **y three classes of models containing local degrees of
freedom:
Yang-Mills theories, i.e. the theory of self-interacting
set of spin-one ﬁelds;
general relativity, i.e. the theory of a self-interacting
spin-two ﬁeld; and
higher-spin gravity, i.e., the theory of a self-interacting
tower of critically massless even-spin ﬁelds.
Looking to their classical perturbation theories, one sees
that higher-spin gravity distinguishes itself in the sense that it
does not admit a strictly massless perturbative formulation on
shell in terms of massless ﬁelds in ﬂat spacetime. Instead it
admits a generally covariant double perturbative expansion in
powers of the following22:
a dimensionless coupling constant g counting the numbers of weak ﬁelds; and
the inverse of a cosmological constant Ã counting the
numbers of pairs of derivatives.
Although higher-spin gravity still lacks a standard off-shell
formulation, its on-shell properties nonetheless suggest a
quantum theory in AdS spacetime in which localized
higher-spin quanta interact in such a fashion that the resulting
low-energy effective description be dominated by higherderivative vertices such that the standard minimal spin-two
couplings show up only as a subleading term. Thus one may
think of higher-spin gravity as an effective ﬂat-space quantum
ﬁeld theory with an exotic cutoff: a ﬁnite infrared cutoff,
showing up as a cosmological constant in the gravitational
perturbation theory, that at the same time plays the role of a
massive parameter in higher-derivative interactions.
We mention again that the reason for this state of affairs
can be explained directly in terms of the (mainly negative)
results for higher-spin gauge theory in ﬂat spacetime: if one
removes Ã, i.e., attempts to formulate a strictly massless
higher-spin gauge theory without any infrared cutoff, then
one falls under the spell of various powerful (albeit restricted)
no-go theorems concerning the couplings between massless
ﬁelds with spin s > 2 and massless ﬁelds with spins s 2 in
ﬂat spacetime.
As mentioned, the perhaps most striking constraint on
gauge theories with vanishing cosmological constant Ã ¼ 0
is the clear-cut clash between the equivalence principle,
which essentially concerns the non-Abelian nature of spintwo gauge symmetries, and Abelian higher-spin gauge symmetry: on the one hand, all massless (as well as massive)
ﬁelds must couple to a massless spin-two ﬁeld via twoderivative vertices with the same universal coupling constant;
on the other hand, such minimal couplings are actually
incompatible with the free gauge transformations for
spin-s > 2 ﬁelds as long as one assumes that these couplings
play the dominant role at low energies.
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One can also deﬁne a Planck length ‘p ¼ g jÃj, but unlike
general relativity, which contains only two derivatives, higher-spin
gravity has no sensible expansion (in its unbroken phase) in powers
of ‘p . In this sense, the per...

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