  In this work and in the companion paper arXiv:2403.02301, we initiate an
approach to holography based on the AKSZ formalism. As the first step, we
refine Vasiliev's holography proposal in arXiv:1203.5554 by obtaining 4D
higher-spin gravity (HSG) and 3D coloured conformal higher-spin gravity (CCHSG)
-- i.e., coloured conformal matter fields coupled to conformal higher-spin
gauge fields and colour gauge fields -- as two distinct and classically
consistent reductions of a single parent theory. The latter consists, on-shell,
of a flat superconnection valued in a fractional-spin extension of Vasiliev's
higher-spin algebra. The HSG and CCHSG reductions are characterized by dual
structure groups and two-form cohomology elements, and their embedding in a
common parent model provides a rationale for deriving holographic relations
from multi-dimensional AKSZ partition functions on cylinders with dual boundary
conditions, to appear separately. In this work we i) construct the underlying
non-commutative geometry as a metaplectic operator algebra represented in a
Hermitian module of a pair of conformal particles; ii) identify a discrete
modular group, arising from twisted boundary conditions of the first-quantized
system, and connecting different boundary conditions of the second-quantized
system; and iii) identify the holonomies, structure groups and two-form
cohomology elements that characterize the HSG and CCHSG reductions, and equate
the dual second Chern classes.
