Vasiliev's Equations Research Articles

On z-dominance, shift symmetry and spin locality in higher-spin theory

The paper aims at the qualitative criterion of higher-spin locality. Perturbative analysis of the Vasiliev equations gives rise to the so-called z-dominated non-localities which nevertheless disappear from interaction vertices leaving the final result spin-local in all known cases. This has led one to the z-dominance conjecture that suggests universality of the observed cancellations. Here we specify conditions which include observation of the higher-spin shift symmetry and prove validity of this recently proposed conjecture. We also define a class of spin-local and shift-symmetric field redefinitions which is argued to be the admissible one with respect to spin-locality.

Carrollian conformal scalar as flat-space singleton

We show that, in any space-time dimension, the on-shell (electric) conformal Carrollian scalar can be interpreted as the flat-space limit of the singleton representation of the conformal algebra. In fact, a recently proposed higher-spin algebra for Minkowski spacetime amounts to the Poincaré enveloping algebra on the corresponding module. This higher-spin algebra is a contraction of that entering Vasiliev's equations, which can be constructed analogously from the singleton representation of the conformal algebra. We also show that the higher-spin extension of the Poincaré algebra we consider is a subalgebra of all symmetries of the conformal Carrollian scalar, given by a higher-spin version of the (extended) BMS algebra.

Coupling coefficient in three dimensional higher spin holography

We consider linearised Vasiliev's equations around the background AdS field in three dimensions for the scalar-scalar-higher spin field correlation function. Relating this with the higher spin field determined in the metric formulation allows determination of the corresponding coupling coefficient. The result agrees with the analogous computation for the spin three field.

On current contribution to Fronsdal equations

We explore a local form of second-order Vasiliev equations proposed in [arXiv:1706.03718] and obtain an explicit expression for quadratic corrections to bosonic Fronsdal equations, generated by gauge-invariant higher-spin currents. Our analysis is performed for general phase factor, and for the case of parity-invariant theory we find the agreement with expressions for cubic vertices available in the literature. This provides an additional indication that local frame proposed in [arXiv:1706.03718] is the proper one.

Higher spin double field theory: a proposal

We construct a double field theory coupled to the fields present in Vasiliev's equations. Employing the "semi-covariant" differential geometry, we spell a functional in which each term is completely covariant with respect to $\mathbf{O}(4,4)$ T-duality, doubled diffeomorphisms, $\mathbf{Spin}(1,3)$ local Lorentz symmetry and, separately, $\mathbf{HS}(4)$ higher spin gauge symmetry. We identify a minimal set of BPS-like conditions whose solutions automatically satisfy the full Euler-Lagrange equations. As such a solution, we derive a linear dilaton vacuum. With extra algebraic constraints further supplemented, the BPS-like conditions reduce to the bosonic Vasiliev equations.

Higher-spin gravity as a theory on a fixed (anti) de Sitter background

We study Vasiliev's higher-spin gravity in 3+1d. We formulate the theory in the so-called compensator formalism, where the local isometry group SO(4,1) is reduced to the Lorentz group SO(3,1) by a choice of spacelike direction in an internal 4+1d space. We present a consistent extension of Vasiliev's equations that allows this internal direction to become spacetime-dependent. This allows a new point of view on the theory, where spacetime is identified with the de Sitter space of possible internal directions. We thus obtain an interacting theory of higher-spin massless gauge fields on a fixed, maximally symmetric background spacetime. We expect implications for the physical interpretation of higher-spin gravity, for the search for a Lagrangian formulation and/or quantization, as well as for higher-spin holography.

Holographic geometry of the renormalization group and higher spin symmetries

We consider the Wilson-Polchinski exact renormalization group applied to the generating functional of single-trace operators at a free-fixed point in $d=2+1$ dimensions. By exploiting the rich symmetry structure of free field theory, we study the geometric nature of the RG equations and the associated Ward identities. The geometry, as expected, is holographic, with $AdS$ spacetime emerging correspondent with RG fixed points. The field theory construction gives us a particular vector bundle over the $d+1$-dimensional RG mapping space, called a jet bundle, whose structure group arises from the linear orthogonal bi-local transformations of the bare fields in the path integral. The sources for quadratic operators constitute a connection on this bundle and a section of its endomorphism bundle. Recasting the geometry in terms of the corresponding principal bundle, we arrive at a structure remarkably similar to the Vasiliev theory, where the horizontal part of the connection on the principal bundle is Vasiliev's higher spin connection, while the vertical part (the Faddeev-Popov ghost) corresponds to the $S$-field. The Vasiliev equations are then, respectively, the RG equations and the BRST equations, with the RG beta functions encoding bulk interactions. Finally, we remark that a large class of interacting field theories can be studied through integral transforms of our results, and it is natural to organize this in terms of a large $N$ expansion.

String theory and emergent AdS geometry in higher spin field theories

We analyze the Weyl invariance constraints on higher spin vertex operators in open superstring theory describing massless higher spin gauge field excitations in $d$-dimensional space-time. We show that these constraints lead to low-energy equations of motion for higher spin fields in AdS space, with the leading-order $\ensuremath{\beta}$ function for the higher spin fields producing Fronsdal's operator in ${\mathrm{AdS}}_{d+1}$, despite that the higher spin vertex operators are originally defined in flat background. The correspondence between the $\ensuremath{\beta}$ function in string theory and ${\mathrm{AdS}}_{d+1}$ Fronsdal operators in space-time is found to be exact for $d=4$, while for other space-time dimensions, it requires modifications of manifest expressions for the higher spin vertex operators. We argue that the correspondence considered in this paper is the leading order of a more general isomorphism between Vasiliev's equations and equations of motion of extended open string field theory, generalized to include the higher spin operators.

Higher-order singletons, partially massless fields, and their boundary values in the ambient approach

Using ambient space we develop a fully gauge and o(d,2)-covariant approach to boundary values of AdSd+1 gauge fields. It is applied to the study of (partially) massless fields in the bulk and (higher-order) conformal scalars, i.e. singletons, as well as (higher-depth) conformal gauge fields on the boundary. In particular, we identify the corresponding generalized Fradkin–Tseytlin equations as obstructions to the extension of the off-shell boundary value to the bulk, generalizing the usual considerations for the holographic anomalies to the partially massless fields. We also relate the background fields for the higher-order singleton to the boundary values of partially massless fields and prove the appropriate generalization of the Flato–Fronsdal theorem, which is in agreement with the known structure of symmetries for the higher-order wave operator. All these facts support the following generalization of the higher-spin holographic duality: the O(N) model at a multicritical isotropic Lifshitz point should be dual to the theory of partially massless symmetric tensor fields described by the Vasiliev equations based on the higher-order singleton symmetry algebra.

Twistor space observables and quasi-amplitudes in 4D higher-spin gravity

Vasiliev equations facilitate globally defined formulations of higher-spin gravity in various correspondence spaces associated with different phases of the theory. In the four-dimensional case this induces a map from a generally covariant formulation in spacetime with higher-derivative interactions to a formulation in terms of a deformed symplectic structure on a noncommutative doubled twistor space, sending spacetime boundary conditions to various sectors of an associative star-product algebra. We look at observables given by integrals over twistor space defining composite zero-forms in spacetime that do not break any local symmetries and that are closed on shell. They can be evaluated locally in spacetime and interpreted as building blocks for dual amplitudes. To regularize potential twistor-space divergencies arising in their curvature expansion, we propose a closed-contour prescription that respects associativity and hence higher-spin gauge symmetry. As a sample calculation, we examine next-to-leading corrections to quasi-amplitudes for twistor-space plane waves, and find cancellations that we interpret using transgression properties in twistor space.

Brane partons and singleton strings

We examine p-branes in AdS D in two limits where they exhibit partonic behavior: rotating branes with energy concentrated to cusp-like solitons; tensionless branes with energy distributed over singletonic bits on the Dirac hypercone. Evidence for a smooth transition from cusps to bits are found. First, each cusp yields D − 2 normal-coordinate bound states with protected frequencies; and is dual to a short open p-brane becoming massless on the giant-vacuum brane at the end-of-the-universe. Second, the N bit -sector is an sp ( 2 N bit ) -gauged phase-space sigma model giving rise to symmetrized N bit -pletons of the minimal higher-spin algebra ho 0 ( D − 1 , 2 ) ⊃ so ( D − 1 , 2 ) . The continuum limit leads to a 2d chiral sp ( 2 ) -gauged sigma model which is critical in D = 7 ; equivalent à la Bars–Vasiliev to an su ( 2 ) -gauged spinor string; and furthermore dual to WZW model in turn containing a topological so ˆ ( 6 , 2 ) −2 / ( so ˆ ( 6 ) ⊕ so ˆ ( 2 ) ) −2 coset model giving rise to a chiral ring generated by singleton-valued weight-0 spin fields. Moreover, the two-bit truncation leads via a reformulation à la Cattaneo–Felder–Kontsevich to a topological open string on the phase space of the D-dimensional Dirac hypercone giving rise to the Vasiliev equations based on vector oscillators and weak sp ( 2 ) -projection. Geometrically, the bi-locality reflects broken boundary-singleton worldlines, and the intertwiner κ relates T- and R-ordered deformations of the boundary and the bulk of the worldsheet, respectively.