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Unformatted text preview: Dolan, 2006) which introduce the germ of an extended object23 as well as a precursor
In order to construct higher-spin extensions of fourdimensional gravity, the simplest higher-spin algebras of
this type can be realized in terms of elementary noncommutative twistor variables. As a result the full ﬁeld content of a
special class of higher-spin gravity theories, that we refer to
as the minimal bosonic models and their matter-coupled and
supersymmetrized extensions, is packed up into ﬁnite sets of
23 The idea of treating algebras and their representations on a more
equal footing, namely as various left-, right-, or two-sided modules
arising inside the enveloping algebra and its tensor products, is in
the spirit of modern algebra and deformation quantization. Indeed,
further development of these thoughts lead to ﬁrst-quantized systems linking higher-spin gravities to tensionless strings and branes
(Engquist and Sundell, 2006).
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 1001 ‘‘master’’ ﬁelds living on the product of a commutative
spacetime and a noncommutative twistor space.
The feat of Vasiliev was then to realize that these master
ﬁelds can be taken to obey remarkably simple-looking master
equations built using exterior differential calculus on spacetime and twistor space, and star products on twistor space,
reproducing the standard second-order equations in perturbation theory, in about the same way in which Einstein’s
equations arise inside a set of on-shell superspace constraints
via constraints on the torsion and Riemann two-forms. As a
result, Vasiliev’s equations are diffeomorphic invariant, in the
sense of unfolded dynamics, and perturbatively equivalent to
a standard set of on-shell Fronsdal ﬁelds albeit with interactions given by a nonlocal double perturbative expansion
resulting from the star products.
Looking at the twistor-space structure one sees that it
services two purposes. In naive double perturbation theory,
the expansion in the twistor variables combined with star
products simply generates the higher-spin tensor calculus that
one may take to deﬁne the minimal bosonic models after
which one can naively strip off all the twistor variables by
Taylor expansion and make contact with the standard tensorial equations of motion after having eliminated inﬁnite
towers of auxiliary ﬁelds.
A more careful look at these tensorial equations of motion
reveals, however, Born-Infeld tails that are indeed strongly
coupled, i.e., formally divergent for ordinary localized ﬂuctuation ﬁelds and hence inequivalent to the canonical BornInfeld interactions. Focusing on classical solutions in special
sectors (boundary conditions) one then discovers that their
resummation is tantamount to regularizations of star products
that require one to perform the ﬁeld-theoretic calculations
inside the twistor space and not just by looking at Taylor
In other words, Vasiliev’s complete higher-spin...
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