A
bstract
A new class of higher-spin gauge theories associated with various Coxeter groups is proposed. The emphasize is on the
B
p
-models. The cases of
B
1
and its infinite graded-symmetric product
...sym
(×
B
1
)
∞
correspond to the usual higher-spin theory and its multi-particle extension, respectively. The multi-particle
B
2
-higher-spin theory is conjectured to be associated with String Theory.
B
p
-higher-spin models with
p >
2 are anticipated to be dual to the rank-
p
boundary tensor sigma-models.
B
p
higher-spin models with
p
≥ 2 possess two coupling constants responsible for higher-spin interactions in AdS background and stringy/tensor effects, respectively. The brane-like idempotent extension of the Coxeter higher-spin theory is proposed allowing to unify in the same model the fields supported by space-times of different dimensions. Consistency of the holographic interpretation of the boundary matrix-like model in the
B
2
-higher-spin model is shown to demand
N
≥ 4 SUSY, suggesting duality with the
N
= 4 SYM upon spontaneous breaking of higher-spin symmetries. The proposed models are shown to admit unitary truncations.
A
bstract
A new efficient approach to the analysis of nonlinear higher-spin equations, that treats democratically auxiliary spinor variables
Z
A
and integration homotopy parameters in the non-linear ...vertices of the higher-spin theory, is developed. Being most general, the proposed approach is the same time far simpler than those available so far. In particular, it is free from the necessity to use the Schouten identity. Remarkably, the problem of reconstruction of higher-spin vertices is mapped to certain polyhedra cohomology in terms of homotopy parameters themselves. The new scheme provides a powerful tool for the study of higher-order corrections in higher-spin theory and, in particular, its spin-locality. It is illustrated by the analysis of the lower order vertices, reproducing not only the results obtained previously by the shifted homotopy approach but also projectively-compact vertices with the minimal number of derivatives, that were so far unreachable within that scheme.
A
bstract
Properties of the resolution operator d
loc
∗
in higher-spin equations, that leads to local current interactions at the cubic order and minimally nonlocal higher-order corrections, are ...formulated in terms of the condition on the class of master fields of higher-spin theory that restricts both the dependence on the spinor
Y
,
Z
variables and on the contractions of indices between the constituent fields in bilinear terms. The Green function in the sector of zero-forms is found for the case of constituent fields carrying helicities of opposite signs. It is shown that the local resolution d
loc
∗
differs from the conventional De Rham resolution d
Z
∗
by a non-local shift.
The concepts of compact and projectively-compact spin-local spinor vertices are introduced. Vertices of this type are shown to be space-time spin-local, i.e., their restriction to any finite subset ...of fields is space-time local. The known spinor spin-local cubic vertices with the minimal number of space-time derivatives are verified to be projectively-compact. This has the important consequence that spinor spin-locality of the respective quartic vertices would imply their space-time spin-locality. More generally, it is argued that the proper class of solutions of the non-linear higher-spin equations that leads to the minimally non-local (presumably space-time spin-local) vertices is represented by the projectively-compact vertices. The related aspects of the higher-spin holographic correspondence are briefly discussed.
A
bstract
The analysis of spin-locality of higher-spin gauge theory is formulated in terms of star-product functional classes appropriate for the
β
→ −∞ limiting shifted homotopy proposed recently in ...
1
where all
ω
2
C
2
higher-spin vertices were shown to be spin-local. For the
β
→ −∞ limiting shifted contracting homotopy we identify the class of functions
H
+
0
, that do not contribute to the r.h.s. of HS field equations at a given order. A number of theorems and relations that organize analysis of the higher-spin equations are derived including extension of the Pfaffian Locality Theorem of
2
to the
β
-shifted contracting homotopy and the relation underlying locality of the
ω
2
C
2
sector of higher-spin equations.
Space-time interpretation of spin-locality of theories involving infinite towers of fields is proposed as the property that the theory is space-time local in terms of original con- stituent fields
ϕ
and their local currents
J
(
ϕ
) of all ranks. Spin-locality is argued to be a proper substitute of locality for theories with finite sets of fields for which the two concepts are equivalent.
A
bstract
Properties of the functional classes of star-product elements associated with higher-pin gauge fields and gauge parameters are elaborated. Cohomological interpretation of the nonlinear ...higher-spin equations is given. An algebra
ℋ
, where solutions of the nonlinear higher-spin equations are valued, is found. A conjecture on the classes of star-product functions underlying (non)local maps and gauge transformations in the nonlinear higher-spin theory is proposed.
A
bstract
Higher-spin vertices containing up to quintic interactions at the LagTangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a ...𝛽 →-∞-shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter 𝜂 in the higher-spin equations. All obtained vertices are shown to be spin-local containing a finite number of derivatives in the spinor space for any given set of spins. The vertices proportional to 𝜂
2
and
η
¯
2
are in addition ultra-local, i.e., zero-forms that enter into the vertex in question are free from the dependence on at least one of the spinor variables
y
or
y
¯
.
Also the 𝜂
2
and
η
¯
2
vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on
AdS
4
.
This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to
η
η
¯
.
It is shown that the 𝛽-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the 𝛽-dependent deformed star product.
A
bstract
The form of higher-spin current interactions in AdS
4
is derived from the full nonlinear higher-spin equations in the sector of Weyl 0-forms. The coupling constant in front of spin-one ...currents built from scalars and spinors as well as Yukawa coupling are determined explicitly. Couplings of all other higher-spin current interactions are determined implicitly. All couplings are shown to be independent of the phase parameter of the nonlinear higher-spin theory. The proper holographic dependence of the vertex on the higher-spin phase parameter is shown to result from the boundary conditions on the bulk fields.
A new construction for gauge invariant functionals in the nonlinear higher-spin theory is proposed. Being supported by differential forms closed by virtue of the higher-spin equations, invariant ...functionals are associated with central elements of the higher-spin algebra. In the on-shell AdS4 higher-spin theory we identify a four-form conjectured to represent the generating functional for 3d boundary correlators and a two-form argued to support charges for black hole solutions. Two actions for 3d boundary conformal higher-spin theory are associated with the two parity-invariant higher-spin models in AdS4. The peculiarity of the spinorial formulation of the on-shell AdS3 higher-spin theory, where the invariant functional is supported by a two-form, is conjectured to be related to the holomorphic factorization at the boundary. The nonlinear part of the star-product function F⁎(B(x)) in the higher-spin equations is argued to lead to divergencies in the boundary limit representing singularities at coinciding boundary space–time points of the factors of B(x), which can be regularized by the point splitting. An interpretation of the RG flow in terms of proposed construction is briefly discussed.
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