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**Unformatted text preview: **ssue, see, e.g.,
Bengtsson (2008) and references therein]. Hence a key
question6 is whether the Fronsdal program can be completed
at the quartic level, even in the case of the aforementioned
minimal bosonic model? This apparently straightforward
problem will keep a number of interesting surprises in store,
in particular, in view of the aforementioned properties of the
AdS/CFT correspondence (Sezgin and Sundell, 2002b;
Giombi and Yin, 2009, 2010) which have been derived using
a rather different approach, as we return to in Sec. V and
summarize in Sec. VI.
As far as more general interacting quantum ﬁeld theories
with higher-spin ﬁelds are concerned, open string ﬁeld theory
in ﬂat spacetime provides a basic example thereof albeit with
a massless sector restricted to spins less than or equal to 1.
Recently, motivated by the similarities between open string
theory and higher-spin gravities mainly at the level of free
ﬁelds (Francia and Sagnotti, 2003, 2006), Sagnotti and
Taronna (2011) deconstructed its ﬁrst Regge trajectory and
arrived at the germs of the non-Abelian interactions for
massless totally symmetric tensors in ﬂat spacetime
(Boulanger and Leclercq, 2006; Boulanger, Leclercq, and
Sundell, 2008) whose deformations into (A)dS spacetimes
(Boulanger, Leclercq, and Sundell, 2008) lead to the FradkinVasiliev cubic vertices. Moreover, Polyakov (2009) proposed
to extend the open superstring in ﬂat spacetime by sectors of
states with novel world-sheet ghost numbers containing
massless higher-spin particles in interaction. He also managed to show (Polyakov, 2010) that these higher-spin states
6
Here we want to stress that it is only by closing the quartic order
that the cubic Lagrangian, including cubic curvature couplings
known as cubic Born-Infeld terms, will be completely ﬁxed (if it
exists). Because of the double perturbative expansion, the BornInfeld couplings dominate over the minimal couplings in physical
amplitudes (assuming a deformed Fronsdal action with ﬁnite BornInfeld ‘‘tail’’) and hence the quartic-closure problem must be
addressed prior to any attempts to do physics with incomplete cubic
actions. In other words, analyses based solely on current exchange
may receive large corrections due to the exotic usage of the
cosmological constant. Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 989 interact with the closed-string graviton and that these interactions reproduce the aforementioned germs of Boulanger
and Leclercq (2006) and Boulanger, Leclercq, and Sundell
(2008).
As far as actual tensionless limits of strings are concerned,
there is a vast literature which we cannot cover here. Of the
various results that have been obtained, we simply point to
the rather drastic difference between tensionless limits of, on
the one hand, the open string in ﬂat space and, on the other
hand, the closed string in anti–de Sitter spacetime. A precise
version of the former was taken by Bonelli (2003),
Buchbinder et al. (2006), Fotopoul...

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