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**Unformatted text preview: **s the formation of cusps on spiky
closed strings (Gubser, Klebanov, and Polyakov, 2002;
Kruczenski, 2005) [for generalizations to membranes, see
Sezgin and Sundell (2002b)]. At the cusps, solitonic bound
states arise, carrying the quantum numbers of singletons
(Engquist and Sundell, 2006). In the case of folded long
strings, the resulting two-singleton closed-string states are
massless symmetric tensors with large spin realized by Flato
and Fronsdal (1978). In the extrapolation of this spectrum to
small spins, which is tantamount to taking a tensionless
limit, resides the anti–de Sitter graviton. Engquist and
Sundell (2006) argued that in order for the tensionless limit
to lead to a closed-string ﬁeld theory with nontrivial interactions, it should be combined with sending the cosmological constant to inﬁnity in a discretized model with ﬁxed mass
parameter. This yields ﬁrst-quantized (0 þ 1)-dimensional
models describing multisingleton states. These have continuum limits given by Wess-Zumino-Witten models with
gauged W algebras (rather than Virasoro algebras) that can
be realized in terms of symplectic bosons (Engquist and
Sundell, 2006; Engquist, Sundell, and Tamassia, 2007) and
real fermions.
Engquist and Sundell (2006) furthermore argued that the
coupling of these ﬁrst-quantized models to higher-spin background ﬁelds requires their extension into Poisson sigma
models in one higher dimension containing the original
systems on their boundaries. In particular, in the case of a
single singleton, that represents one string parton or membrane parton, these couplings are mediated via boundary and
bulk vertex operators of a topological open string in the phase
space of a singleton that is a particular example of the
C model of Cattaneo and Felder (2000); the consistency of
this ﬁrst-quantized system with disk topology then requires
Vasiliev’s equations.
The resulting physical picture provides a concrete realization for an extended object that is already present in the FlatoFronsdal formula. This picture also matches well with the
holographic framework: just as the weak-coupling stress
tensor is deformed directly into the strong-coupling stress
tensor on the CFT side, the graviton in higher-spin gravity is
the continuation of that in closed-string theory. Moreover, the
fact that topological C models underlie general associative
algebras directly explains why Vasiliev’s equations are compatible with internal Chan-Paton factors.
One is thus led to contemplate a more profound underlying framework for quantum ﬁeld theory in general, based
on Poisson sigma models and topological summation and
that would naturally incorporate the gauge principle as well
as radiative corrections; in the case of the topological open
string, the additional zero modes arising from cutting holes
in the disk can then provide a ﬁrst-quantized realization of
the massive Goldstone modes of the Girardello-PorratiZaffaroni mechanism (Girardello, Porrati, and Zaffar...

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