585 research outputs found
Current Interactions from the One-Form Sector of Nonlinear Higher-Spin Equations
The form of higher-spin current interactions in the sector of one-forms is
derived from the nonlinear higher-spin equations in . Quadratic
corrections to higher-spin equations are shown to be independent of the phase
of the parameter in the full nonlinear higher-spin
equations. The current deformation resulting from the nonlinear higher-spin
equations is represented in the canonical form with the minimal number of
space-time derivatives. The non-zero spin-dependent coupling constants of the
resulting currents are determined in terms of the higher-spin coupling constant
. Our results confirm the conjecture that (anti-)self-dual
nonlinear higher-spin equations result from the full system at () .Comment: 38 pages, no figures; V2: 39 pages, minor corrections, to be
published versio
Higher-Rank Fields and Currents
invariant field equations in the space with symmetric
matrix coordinates are classified. Analogous results are obtained for
Minkowski-like subspaces of which include usual Minkowski
space as a particular case. The constructed equations are associated with the
tensor products of the Fock (singleton) representation of of any rank
. The infinite set of higher-spin conserved currents multilinear
in rank-one fields in is found. The associated conserved charges
are supported by dimensional differential forms in , that are closed by
virtue of the rank- field equations. The cohomology groups
with all and , which determine
the form of appropriate gauge fields and their field equations, are found both
for and for its Minkowski-like subspace.Comment: 27 pages; V2: Significant extension of the results to computation of
all cohomologies, 43 pages; V3: Discussion of equations in
generalized Minkowski space from the perspective of usual Minkowski space and
reference added, typos corrected, the journal version, 44 page
On Recurrent Reachability for Continuous Linear Dynamical Systems
The continuous evolution of a wide variety of systems, including
continuous-time Markov chains and linear hybrid automata, can be described in
terms of linear differential equations. In this paper we study the decision
problem of whether the solution of a system of linear
differential equations reaches a target
halfspace infinitely often. This recurrent reachability problem can
equivalently be formulated as the following Infinite Zeros Problem: does a
real-valued function satisfying a
given linear differential equation have infinitely many zeros? Our main
decidability result is that if the differential equation has order at most ,
then the Infinite Zeros Problem is decidable. On the other hand, we show that a
decision procedure for the Infinite Zeros Problem at order (and above)
would entail a major breakthrough in Diophantine Approximation, specifically an
algorithm for computing the Lagrange constants of arbitrary real algebraic
numbers to arbitrary precision.Comment: Full version of paper at LICS'1
Manifest Form of the Spin-Local Higher-Spin Vertex
Vasiliev generating system of higher-spin equations allowing to reconstruct
nonlinear vertices of field equations for higher-spin gauge fields contains a
free complex parameter . Solving the generating system order by order one
obtains physical vertices proportional to various powers of and
. Recently and vertices in the zero-form
sector were presented in 2009.02811 in the -dominated form implying their
spin-locality by virtue of -dominance Lemma of 1805.11941. However the
vertex of 2009.02811 had the form of a sum of spin-local terms dependent on the
auxiliary spinor variable in the theory modulo so-called -dominated
terms, providing a sort of existence theorem rather than explicit form of the
vertex. The aim of this paper is to elaborate an approach allowing to
systematically account for the effect of -dominated terms on the final
-independent form of the vertex needed for any practical analysis. Namely,
in this paper we obtain explicit -independent spin-local form for the vertex
for its -ordered part where
and denote gauge one-form and field strength zero-form higher-spin
fields valued in an arbitrary associative algebra in which case the order of
product factors in the vertex matters. The developed formalism is based on the
Generalized Triangle identity derived in the paper and is applicable to all
other orderings of the fields in the vertex.Comment: LaTeX, 33 pages, no figures, V2: clarifications and references added,
matches the published versio
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