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 $AdS_4$. Quadratic corrections to higher-spin equations are shown to be independent of the phase of the parameter $\eta =\exp i\varphi$ 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 $\eta\bar\eta$. Our results confirm the conjecture that (anti-)self-dual nonlinear higher-spin equations result from the full system at ($\eta=0$) $\bar \eta=0$.Comment: 38 pages, no figures; V2: 39 pages, minor corrections, to be published versio

Higher-Rank Fields and Currents

$Sp(2M)$ invariant field equations in the space ${\cal M}_M$ with symmetric matrix coordinates are classified. Analogous results are obtained for Minkowski-like subspaces of ${\cal M}_M$ which include usual $4d$ Minkowski space as a particular case. The constructed equations are associated with the tensor products of the Fock (singleton) representation of $Sp(2M)$ of any rank ${\mathbf{r }}$. The infinite set of higher-spin conserved currents multilinear in rank-one fields in ${\cal M}_M$ is found. The associated conserved charges are supported by $({\mathbf{r }} M-\frac{{\mathbf{r }} ({\mathbf{r }} -1)}{2})-$dimensional differential forms in ${\cal M}_M$, that are closed by virtue of the rank-$2{\mathbf{r }}$ field equations. The cohomology groups $H^p(\sigma^{\mathbf{r }}_-)$ with all $p$ and ${\mathbf{r }}$, which determine the form of appropriate gauge fields and their field equations, are found both for ${\cal M}_M$ and for its Minkowski-like subspace.Comment: 27 pages; V2: Significant extension of the results to computation of all $\sigma_-$ 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 $\boldsymbol{x}(t)$ of a system of linear differential equations $d\boldsymbol{x}/dt=A\boldsymbol{x}$ 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 $f:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ satisfying a given linear differential equation have infinitely many zeros? Our main decidability result is that if the differential equation has order at most $7$, then the Infinite Zeros Problem is decidable. On the other hand, we show that a decision procedure for the Infinite Zeros Problem at order $9$ (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 ΥωCCCηη\Upsilon^{\eta\eta}_{\omega CCC}

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 $\eta$. Solving the generating system order by order one obtains physical vertices proportional to various powers of $\eta$ and $\bar{\eta}$. Recently $\eta^2$ and $\bar{\eta}^2$ vertices in the zero-form sector were presented in 2009.02811 in the $Z$-dominated form implying their spin-locality by virtue of $Z$-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 $Z$ in the theory modulo so-called $Z$-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 $Z$-dominated terms on the final $Z$-independent form of the vertex needed for any practical analysis. Namely, in this paper we obtain explicit $Z$-independent spin-local form for the vertex $\Upsilon^{\eta\eta}_{\omega CCC}$ for its $\omega CCC$-ordered part where $\omega$ and $C$ 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