arXiv subjects

O. A. Gelfond

Publications and source records attributed to O. A. Gelfond.

At least 19 recordsLinked to original sources

arXiv2025

Lorentz Covariance of the $4d$ Nonlinear Higher-Spin Equations via BRST

We propose a BRST extension of the higher-spin gauge theory in $AdS_4$ with the BRST operator associated with the local Lorentz symmetry. Our construction supports manifest local Lorentz covariance and is applicable both to any homotopy scheme of the perturbative analysis including the recently proposed differential homotopy and to the variety of further extended higher-spin models in $AdS_3$ and $AdS_4$ with higher differential forms and Coxeter higher-spin algebras.

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arXiv2023

Moderately non-local $η\barη$ vertices in the $AdS_4$ higher-spin gauge theory

A new concept of moderate non-locality in higher-spin gauge theory is introduced. Based on the recently proposed differential homotopy approach, a moderately non-local scheme, that issofter than those resulting from the shifted homotopy approach available in the literature so far, is worked out in the mixed $η\barη$ sector of the Vasiliev higher-spin theory. To calculate moderately non-local vertices $Υ^{η\bar η}(ω, C,C,C)$ for all ordering of the fields $ω$ and $C$ we apply an interpolating homotopy, that respects the moderate non-locality in the perturbative analysis of the higher-spin equations.

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arXiv2021

Manifest Form of the Spin-Local Higher-Spin Vertex $Υ^{ηη}_{ωCCC}$

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 $η$. Solving the generating system order by order one obtains physical vertices proportional to various powers of $η$ and $\barη$. Recently $η^2$ and $\barη^2$ 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 $Υ^{ηη}_{ωCCC}$ for its $ωCCC$-ordered part where $ω$ 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.

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arXiv2021

Spin-Locality of $η^2$ and $\barη^2$ Quartic Higher-Spin Vertices

Higher-spin theory contains a complex coupling parameter $η$. Different higher-spin vertices are associated with different powers of $η$ and its complex conjugate $\bar η$. Using $Z$-dominance Lemma, that controls spin-locality of the higher-spin equations, we show that the third-order contribution to the zero-form $B(Z;Y;K)$ admits a $Z$-dominated form that leads to spin-local vertices in the $η^2$ and $\bar η^2$ sectors of the higher-spin equations. These vertices include, in particular, the $η^2$ and $\bar η^2$ parts of the $ϕ^4$ scalar field vertex.

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arXiv2021

Homotopy Properties and Lower-Order Vertices in Higher-Spin Equations

New homotopy approach to the analysis of nonlinear higher-spin equations is developed. It is shown to directly reproduce the previously obtained local vertices. Simplest cubic (quartic in Lagrangian nomenclature) higher-spin interaction vertices in four dimensional theory are examined from locality perspective by the new approach and shown to be local. The results are obtained in a background independent fashion.

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arXiv2020

Spin-Locality of Higher-Spin Theories and Star-Product Functional Classes

The analysis of spin-locality of higher-spin gauge theory is formulated in terms of star-product functional classes appropriate for the $β\to -\infty$ limiting shifted homotopy proposed recently in arXiv:1909.04876 where all $ω^2 C^2$ higher-spin vertices were shown to be spin-local. For the $β\to -\infty$ limiting shifted contracting homotopy we identify the class of functions ${\mathcal H}^{+0}$, 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 arXiv:1805.11941 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 constituent 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.

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arXiv2020

Limiting Shifted Homotopy in Higher-Spin Theory and Spin-Locality

Higher-spin vertices containing up to quintic interactions at the Lagrangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a $β\to-\infty$--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 HS 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 $\bar η^2$ 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 $\bar y$. Also the $η^2$ and $\bar η^2$ 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 $η\bar η$. 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.

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arXiv2018

Homotopy Operators and Locality Theorems in Higher-Spin Equations

A new class of shifted homotopy operators in higher-spin gauge theory is introduced. A sufficient condition for locality of dynamical equations is formulated and Pfaffian Locality Theorem identifying a subclass of shifted homotopies that decrease the degree of non-locality in higher orders of the perturbative expansion is proven.

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arXiv2018

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 $η=\exp iφ$ 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 $η\barη$. Our results confirm the conjecture that (anti-)self-dual nonlinear higher-spin equations result from the full system at ($η=0$) $\bar η=0$.

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arXiv2016

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 $H^p(σ^{\mathbf{r }}_-)$ 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.

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arXiv2016

Symmetries of higher-spin current interactions in four dimensions

Current interaction of massless fields in four dimensions is shown to break sp (8) symmetry of free massless equations of all spins down to the conformal symmetry su (2,2). This breaking is in agreement with the form of nonlinear higher-spin field equations.

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arXiv2016

Conserved Higher-Spin Charges in $AdS_4$

Gauge invariant conserved conformal currents built from massless fields of all spins in 4d Minkowski space-time and $AdS_4$ are described within the unfolded dynamics approach. The current cohomology associated with non-zero conserved charges is found. The resulting list of charges is shown to match the space of parameters of the conformal higher-spin symmetry algebra in four dimensions.

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arXiv2015

Unfolded Equations for Current Interactions of 4d Massless Fields as a Free System in Mixed Dimensions

Interactions of massless fields of all spins in four dimensions with currents of any spin is shown to result from a solution of the linear problem that describes a gluing between rank-one (massless) system and rank-two (current) system in the unfolded dynamics approach. Since the rank-two system is dual to a free rank-one higher-dimensional system, that effectively describes conformal fields in six space-time dimensions, the constructed system can be interpreted as describing a mixture between linear conformal fields in four and six dimensions. Interpretation of the obtained results in spirit of AdS/CFT correspondence is discussed.

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arXiv2013

Operator algebra of free conformal currents via twistors

Operator algebra of (not necessarily free) higher-spin conformal conserved currents in generalized matrix spaces, that include 3d Minkowski space-time as a particular case, is shown to be determined by an associative algebra $M$ of functions on the twistor space. For free conserved currents, $M$ is the universal enveloping algebra of the higher-spin algebra. Proposed construction greatly simplifies computation and analysis of correlators of conserved currents. Generating function for $n$-point functions of 3d (super)currents of all spins, built from $N$ free constituent massless scalars and spinors, is obtained in a concise form of certain determinant. Our results agree with and extend earlier bulk computations in the HS $AdS_4/CFT_3$ framework. Generating function for $n$-point functions of 4d conformal currents is also presented.

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arXiv2010

Unfolding versus BRST and currents in Sp(2M) invariant higher-spin theory

The correspondence between BRST and unfolded formulations of field equations on group manifolds and homogeneous spaces is described. The previously introduced nonstandard BRST operator, that underlies Sp(2M) invariant higher-spin field equations, is shown to admit a natural oscillator-like realization. The coordinate independent form of conserved currents in the Sp(2M) invariant higher-spin theory is derived from the BRST formulation on Sp(2M) extended by the Heisenberg group.

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arXiv2009

$Sp(8)$ invariant higher spin theory, twistors and geometric BRST formulation of unfolded field equations

We discuss twistor-like interpretation of the $Sp(8)$ invariant formulation of 4d massless fields in ten dimensional Lagrangian Grassmannian $Sp(8)/P$ which is the generalized space-time in this framework. The correspondence space $\mathbf{C}$ is $SpH(8)/PH$ where $SpH(8)$ is the semidirect product of $Sp(8)$ with Heisenberg group $\HG$ and $PH$ is some quasiparabolic subgroup of $SpH(8)$. Spaces of functions on $Sp(8)/P$ and $SpH(8)/PH$ consist of $Q_P $ closed functions on $Sp(8)$ and $Q_{PH} $ closed functions on $SpH(8)$, where $Q_P $ and $Q_{PH}$ are canonical BRST operators of $P$ and $PH$. The space of functions on the generalized twistor space $\mathbf{T}$ identifies with the $SpH(8)$ Fock module. Although $\mathbf{T}$ cannot be realized as a homogeneous space, we find a nonstandard $SpH(8)$ invariant BRST operator $\QQ$ $(\QQ^2 =0)$ that gives rise to an appropriate class of functions via the condition $\QQ f=0$ equivalent to the unfolded higher--spin equations. The proposed construction is manifestly $Sp(8)$ invariant, globally defined and coordinate independent. Its Minkowski analogue gives a version of twistor theory with both types of chiral spinors treated on equal footing. The extensions to the higher rank case with several Heisenberg groups and to the complex case are considered. A relation with Riemann theta functions, that are $\QQ$-closed, is discussed.

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arXiv2009

Higher Spin Fields in Siegel Space, Currents and Theta Functions

Dynamics of four-dimensional massless fields of all spins is formulated in the Siegel space of complex $4\times 4$ symmetric matrices. It is shown that the unfolded equations of free massless fields, that have a form of multidimensional Schrodinger equations, naturally distinguish between positive- and negative-frequency solutions of relativistic field equations, i.e. particles and antiparticles. Multidimensional Riemann theta functions are shown to solve massless field equations in the Siegel space. We establish the correspondence between conserved higher-spin currents in four-dimensional Minkowski space and those in the ten-dimensional matrix space. It is shown that global symmetry parameters of the current in the matrix space should be singular to reproduce a nonzero current in Minkowski space. The $\D$-function integral evolution formulae for 4d massless fields in the Fock-Siegel space are obtained. The generalization of the proposed scheme to higher dimensions and systems of higher ranks is considered.

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arXiv2006

Higher Spin Conformal Currents in Minkowski Space

Using unfolded formulation of free equations for massless fields of all spins we obtain explicit form of higher-spin conformal conserved charges bilinear in 4d massless fields of arbitrary spins.

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