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**Unformatted text preview: **of Weyl tensors and their derivatives that cannot
be replaced by a single effective Born-Infeld interaction with a
ﬁnite coupling; and (iii) constrain the spectrum and cubic
couplings by solving higher-order consistency conditions in
the g expansion (starting at quartic order).
However, without any guiding principle other than Lorentz
and gauge invariance, this is an a priori intractable problem
essentially due to the fact that the whole cubic tail must be
ﬁxed, which may require going to very high orders in the Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . g expansion. Of course, in the simplest scenario, the complete
cubic action could be ﬁxed by quartic consistency, in which
case there would be no interaction ambiguity at the cubic
level. Thus, of all possible hypothetical outcomes the extreme
cases are (i) quartic consistency sufﬁces to completely ﬁx the
cubic action including its Born-Infeld tail; and (ii) quartic
consistency rules out the cubic action altogether in which
case the choice of free theory initiating the Fronsdal program
would have to be revised.
In summary, to make the situation more tractable, one may
resort to some additional guidance besides Lorentz and gauge
invariance, or bias if one wishes to use that word, on what are
suitable notions for ‘‘higher-spin multiplets,’’ for selection of
spectrum of ﬁelds, and ‘‘higher-spin tensor calculus,’’ for
construction of interactions.
How to proceed with this issue becomes most clear in
higher-spin gravity: higher-spin gauge theories based on
higher-spin algebras given by inﬁnite-dimensional extensions
of ordinary ﬁnite-dimensional spacetime isometry algebras.
At this stage it is natural to rethink how unitary representations of the complete higher-spin algebra are mapped directly
to ﬁelds living in inﬁnite-dimensional geometries containing
ordinary spacetime as a submanifold. Indeed one of the key
instruments going into Vasiliev’s formulation of fully nonlinear equations of motion for higher-spin gravities is
unfolded dynamics (Vasiliev, 1988, 1989, 1990, 1994): a
mathematically precise tool for manifestly diffeomorphisminvariant generalized spacetime reconstructions applying to
ﬁnite-dimensional as well as inﬁnite-dimensional cases.
C. Vasiliev’s equations A working deﬁnition of higher-spin algebras developed by
Fradkin, Konstein, and Vasiliev (Fradkin and Vasiliev,
1987a, 1988; Konshtein and Vasiliev, 1989, 1990) that has
proven to be useful is that of Lie subalgebras of associative
algebras obtained from the enveloping algebras of the spacetime isometry algebra by factoring out annihilators of their
‘‘fundamental,’’ or ultrashort, unitary representations (singletons). In this setting, the higher-spin generators are monomials in the spacetime isometry generators, and higher-spin
multiplets arise by tensoring together singletons (Flato and
Fronsdal, 1978; Vasiliev, 2004c;...

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