A model [9].
1
In what follows, we shall first provide an alternative vectorial Type A model
by modifying the sp(2) gauging without affecting the higher spin gauge algebra nor the
perturbative spectrum. We then extend its field content by a dynamical two-form and an
extra dynamical one-form so as to obtain a vectorial generalization of the four-dimensional
spinorial Frobenius-Chern-Simons model (FCS) proposed in [10]. The FCS model contains
the alternative Type A model as a consistent truncation. At the linearized level, the FCS
model contains additional degrees of freedom arising from non-trivial cohomology elements
in the dynamical two-form. In this paper, we shall restrict, however, the perturbative
analysis to the alternative Type A model.
The FCS model is more predictive than its Type A model truncations, as it possesses
an enlarged bi-fundamental gauge group that drastically reduces the number of higher
spin gauge invariants, and facilitates an off-shell formulation using topological field theory
methods leading to an on-shell action with only a finite number of free parameters. More-
over, the FCS model, which is formulated in terms of a Quillen superconnection [38] valued
in a Frobenius algebra, is akin to a topological open string field theory [11–13]. Topo-
logical strings provide a natural framework for coupling massless higher spin particles to
ordinary tensionless strings in anti-de Sitter backgrounds [14–17] in accordance with holog-
raphy [14, 18–20]. Further developments of the FCS model may thus open up new windows
to holography, permitting access to a wide range of physically relevant quantum field the-
ories in four and higher dimensions, including four-dimensional pure Yang-Mills theories.
The resulting framework provides a path-integral quantization of higher spin gravity
using the language of topological quantum field theories on noncommutative Poisson man-
ifolds [25, 26] (see also [10, 27]) which can be used to introduce the notion of star-product
locality of equations of motion, covariant Hamiltonian Lagrangians and other densities
used for constructing observables [25, 26, 30, 31]. Indeed, as stressed by [22], there is a
tension between the quasi-Riemannian notion of spacetime locality and (nonabelian) higher
spin gauge symmetry that obstructs the Fronsdal program [24], i.e. the application of the
Noether procedure so as to obtain a classical action for deformed Fronsdal fields; for related
work on Vasiliev’s formulation to the deformed Fronsdal program, see [21, 23]. Thus, it
makes more sense to think of the deformed Fronsdal theory, viewed as a stand-alone quan-
tum field theory without any reference to higher dimensional noncommutative geometry, as
a quantum effective theory governed by higher spin gauge symmetry and unitarity without
any classical limit directly on spacetime.
In the topological field theory realization of higher spin gravity, the physical states arise
as boundary states of a topological bulk theory created by generalized Chern classes [10]. A
subset of these do not receive any quantum corrections, mainly due to the conservation of
form degrees at bulk vertices. On-shell, they are related to zero-form charges [29, 31] which
indeed serve as generating functionals for holographic correlation functions [30, 32, 33];
for more recent progress, see [34]. Accordingly, the FCS model subjected to appropriate
boundary conditions should contain a Vasiliev branch that is equivalent on-shell to the
quantum effective deformed Fronsdal theory. Of key importance in this approach is the
1
Strictly speaking, the equivalence between the spinorial and vectorial Type A models in four dimensions
remains to be established beyond the linearized level.
– 2 –
