Higher-spin Model Research Articles

Strange higher-spin topological systems in 3D

Motivated by the generation of action principles from off-shell dualisation, we present a general class of free, topological theories in three dimensional Minkowski spacetime that exhibit higher-spin gauge invariance. In the spin-two case, we recover a dual reformulation of the triplet system already known, while the higher-spin systems that we obtain seem to be new. They are associated with wild quivers. We study in which situations these exotic (or strange) higher-spin models can be extended to dS3 and AdS3 backgrounds, revealing that the flat limit of such models, when they exist, admits a one-parameter freedom. Interactions are studied in the simplest higher-spin case featuring spin-2 and spin-3 fields. We then give several higher-spin generalizations of these strange systems.

Z_{2} Spin Liquids in the Higher Spin-S Kitaev Honeycomb Model: An Exact Deconfined Z_{2} Gauge Structure in a Nonintegrable Model.

The higher spin Kitaev model prominently features the extensive locally conserved quantities the same as the spin-1/2 Kitaev honeycomb model, although it is not exactly solvable. It remains an open question regarding the physical meaning of these conserved quantities in the higher spin model. In this Letter, by introducing a Majorana parton construction for a general spin-S we uncover that these conserved quantities are exactly the Z_{2} gauge fluxes in the general spin-S model, including the case of spin-1/2. Particularly, we find an even-odd effect that the Z_{2} gauge charges are fermions in the half integer spin model, but are bosons in the integer spin model. We further prove that the fermionic Z_{2} gauge charges are always deconfined; hence, the half integer spin Kitaev model would have nontrivial spin liquid ground states regardless of interaction strengths in the Hamiltonian. The bosonic Z_{2} gauge charges of the integer spin model, on the other hand, could condense, leading to a trivial product state, and this is indeed the case at the anisotropic limit of the model.

Duality-invariant superconformal higher-spin models

We develop a general formalism of duality rotations for bosonic conformal spin-$s$ gauge fields, with $s\geq 2$, in a conformally flat four-dimensional spacetime. In the $s=1$ case this formalism is equivalent to the theory of $\mathsf{U}(1)$ duality-invariant nonlinear electrodynamics developed by Gaillard and Zumino, Gibbons and Rasheed, and generalised by Ivanov and Zupnik. For each integer spin $s\geq 2$ we demonstrate the existence of families of conformal $\mathsf{U}(1)$ duality-invariant models, including a generalisation of the so called ModMax Electrodynamics ($s=1$). Our bosonic results are then extended to the $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetric cases. We also sketch a formalism of duality rotations for conformal gauge fields of Lorentz type $(m/2, n/2)$, for positive integers $m $ and $n$.

Field theories with (2,0) AdS supersymmetry in N=1 AdS superspace

In three dimensions, it is known that field theories possessing extended $(p,q)$ anti-de Sitter (AdS) supersymmetry with ${\cal N}=p+q \geq 3$ can be realised in (2,0) AdS superspace. Here we present a formalism to reduce every field theory with (2,0) AdS supersymmetry to ${\cal N}=1$ AdS superspace. As nontrivial examples, we consider supersymmetric nonlinear sigma models formulated in terms of ${\cal N}=2$ chiral and linear supermultiplets. The $(2,0) \to (1,0)$ AdS reduction technique is then applied to the off-shell massless higher-spin supermultiplets in (2,0) AdS superspace constructed in [1]. As a result, for each superspin value $\hat s$, integer $(\hat s= s)$ or half-integer $(\hat s= s+1/2)$, with $s=1,2,\dots $, we obtain two off-shell formulations for a massless ${\cal N}=1$ superspin-$\hat s$ multiplet in AdS${}_3$. These models prove to be related to each other by a superfield Legendre transformation in the flat superspace limit, but the duality is not lifted to the AdS case. Two out of the four series of ${\cal N}=1$ supersymmetric higher-spin models thus derived are new. The constructed massless ${\cal N}=1$ supersymmetric higher-spin actions in AdS${}_3$ are used to formulate (i) higher-spin supercurrent multiplets in ${\cal N}=1$ AdS superspace; and (ii) new topologically massive higher-spin off-shell supermultiplets. Examples of ${\cal N}=1$ higher-spin supercurrents are given for models of a complex scalar supermultiplet. We also present two new off-shell formulations for a massive ${\cal N}=1$ gravitino supermultiplet in AdS${}_3$.

From Coxeter higher-spin theories to strings and tensor models

A new class of higher-spin gauge theories associated with various Coxeter groups is proposed. The emphasize is on the Bp-models. The cases of B1 and its infinite graded-symmetric product sym (×B1)∞ correspond to the usual higher-spin theory and its multi-particle extension, respectively. The multi-particle B2-higher-spin theory is conjectured to be associated with String Theory. Bp-higher-spin models with p > 2 are anticipated to be dual to the rank-p boundary tensor sigma-models. Bp 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 B2-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.

Metric versus observable operator representation, higher spin models

.We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

Supersymmetric Higher Spin Models in Three Dimensional Spaces

We review the component Lagrangian construction of the supersymmetric higher spin models in three-dimensional (3D) Minkowski and anti de Sitter ( A d S ) spaces. The approach is based on the frame-like gauge-invariant formulation, where massive higher spin fields are realized through a system of massless ones. We develop a supersymmetric generalization of this formulation to the Lagrangian construction of the on-shell N = 1 , 3D higher spin supermultiplets. In 3D Minkowski space, we show that the massive supermultiplets can be constructed from one extended massless supermultiplet by adding the mass terms to the Lagrangian and the corresponding corrections to the supertransformations of the fermionic fields. In 3D A d S space, we construct massive supermultiplets using a formulation of the massive fields in terms of the set of gauge-invariant objects (curvatures) in the process of their consistent supersymmetric deformation.

Quantum mechanics and hidden superconformal symmetry

Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp(1|2) superconformal symmetry. We study the problem of constructing all quantum mechanical models with a hidden osp(1|2) symmetry on a given space of states. This problem stems from interacting higher spin models coupled to gravity. In one dimension, we show that the solution to this problem is the Plyushchay family of quantum mechanical models with hidden superconformal symmetry obtained by viewing the harmonic oscillator as a one dimensional Dirac system, so that Grassmann parity equals wavefunction parity. These models--both oscillator and particle-like--realize all possible unitary irreducible representations of osp(1|2).

Noncommutative Wilson lines in higher-spin theory and correlation functions of conserved currents for free conformal fields

We first prove that, in Vasiliev’s theory, the zero-form charges studied in Sezgin E and Sundell P 2011 (arXiv:1103.2360 [hep-th]) and Colombo N and Sundell P 20 (arXiv:1208.3880 [hep-th]) are twisted open Wilson lines in the noncommutative Z space. This is shown by mapping Vasiliev’s higher-spin model on noncommutative Yang–Mills theory. We then prove that, prior to Bose-symmetrising, the cyclically-symmetric higher-spin invariants given by the leading order of these n-point zero-form charges are equal to corresponding cyclically-invariant building blocks of n-point correlation functions of bilinear operators in free conformal field theories (CFT) in three dimensions. On the higher spin gravity side, our computation reproduces the results of Didenko V and Skvortsov E 2013 J. High Energy Phys. JHEP04(2013)158 using an alternative method amenable to the computation of subleading corrections obtained by perturbation theory in normal order. On the free CFT side, our proof involves the explicit computation of the separate cyclic building blocks of the correlation functions of n conserved currents in arbitrary dimension using polarization vectors, which is an original result. It is shown to agree, for , with the results obtained in Gelfond O A and Vasiliev M A 2013 Nucl. Phys. B 876 871–917 in various dimensions and where polarization spinors were used.

Invariant functionals in higher-spin theory

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.

Holography and quantum states in elliptic de Sitter space

We outline a program for interpreting the higher-spin dS/CFT model in terms of physics in the causal patch of a dS observer. The proposal is formulated in "elliptic" de Sitter space dS_4/Z_2, obtained by identifying antipodal points in dS_4. We discuss recent evidence that the higher-spin model is especially well-suited for this, since the antipodal symmetry of bulk solutions has a simple encoding on the boundary. For context, we test some other (free and interacting) theories for the same property. Next, we analyze the notion of quantum field states in the non-time-orientable dS_4/Z_2. We compare the physics seen by different observers, with the outcome depending on whether they share an arrow of time. Finally, we implement the marriage between higher-spin holography and observers in dS_4/Z_2, in the limit of free bulk fields. We succeed in deriving an observer's operator algebra and Hamiltonian from the CFT, but not her S-matrix. We speculate on the extension of this to interacting higher-spin theory.

BRST analysis of the supersymmetric higher spin field models

We develop the BRST approach for all massless integer and half-integer higher spins in 4D Minkowski space, using the two component spinor nota- tion and develop the Lagrangian formulation for supersymmetric higher spin models. It is shown that the problem of second class constraints disappears and the BRST procedure becomes much more simple than in tensorial nota- tion. Furthermore, we demonstrate that the BRST procedure automatically provides extra auxiliary components that belong in the set of supersymmetry auxiliary components. Finally, we demonstrate how supersymmetry transfo- rmations are realized in such an approach. As a result, we conclude that the BRST approach to higher spin supersymmetric theories allows to derive both the Lagrangian and the supersymmetry transformations. Although most part of the work is devoted to massless component supersymmetric models, we also discuss generalization for massive component supersymmetric models and for superfield models.

A calculus for higher spin interactions

Higher spin theories can be efficiently described in terms of auxiliary Stückelberg or projective space field multiplets. By considering how higher spin models couple to scale, these approaches can be unified in a conformal geometry/tractor calculus framework. We review these methods and apply them to higher spin vertices to obtain a generating function for massless, massive and partially massless three-point interactions.

Bosonic conformal higher-spin fields of any symmetry

Free Lagrangians are found both for gauge and non-gauge bosonic conformal fields of any symmetry type and in any space–time dimension. Conformal gauge fields of various types, their gauge transformations and gauge invariant field strengths (generalized Weyl tensors), which are derived by the σ − cohomology technics in the frame-like formulation, are shown to correspond to supersymmetric vacua of certain supersymmetric matrix mechanics. The correspondence between conformal and AdS d higher-spin models, that turn out to have identical generalized Weyl tensors, is discussed.

Transverse invariant higher-spin fields

It is shown that a symmetric massless bosonic higher-spin field can be described by a traceless tensor field with reduced (transverse) gauge invariance. The Hamiltonian analysis of the transverse gauge invariant higher-spin models is used to control a number of degrees of freedom.

Higher Spin Superalgebras in any Dimension and their Representations

Fock module realization for the unitary singleton representations of the d?1 dimensional conformal algebra o(d?1,2), which correspond to the spaces of single-particle states of massless scalar and spinor in d?1 dimensions, is given. The pattern of the tensor product of a pair of singletons is analyzed in any dimension. It is shown that for 3$>d>3 the tensor product of two boson singletons decomposes into a sum of all integer spin totally symmetric massless representations in AdSd, the tensor product of boson and fermion singletons gives a sum of all half-integer spin symmetric massless representations in AdSd, and the tensor product of two fermion singletons in 4$>d>4 gives rise to massless fields of mixed symmetry types in AdSd depicted by Young tableaux with one row and one column together with certain totally antisymmetric massive fields. In the special case of o(2,2), tensor products of 2d massless scalar and/or spinor modules contain infinite sets of 2d massless conformal fields of different spins. The obtained results extend the 4d result of Flato and Fronsdal [1] to any dimension and provide a nontrivial consistency check for the recently proposed higher spin model in AdSd [2]. We define a class of higher spin superalgebras which act on the supersingleton and higher spin states in any dimension. For the cases of AdS3, AdS4, and AdS5 the isomorphisms with the higher spin superalgebras defined earlier in terms of spinor generating elements are established.

Ruling out a higher spin field solution to the cosmological constant problem

We consider the modification of Newton's gravity law in Dolgov's higher spin models designed to compensate the cosmological constant. We find that the effective Planck mass is unacceptably large in these models. We also point out that the properties of gravitational waves are entirely different in these models as compared to general relativity.

Higher-spin gauge interactions for massive matter fields in 3D AdS space-time

A remarkable feature of the models with interactions exhibiting higher-spin (HS) gauge symmetries in d > 2 is that their most symmetric vacua require the (anti)-de Sitter (AdS) geometry rather than the flat geometry. In striking parallelism to what might be expected of M-theory, HS gauge theories describe infinite towers of fields of all spins and possess naturally space-time SUSY and Chan-Paton type inner symmetries. In this paper, we analyze at the level of the equations of motion the simplest non-trivial HS model which describes HS gauge interactions (on the top of the usual supergravitational and (Chern-Simons) Yang-Mills interactions) of massive spin-0 and spin- 1 2 matter fields in d = 2 + 1 AdS space-time. The parameter of mass of the matter fields is identified with the vev of a certain auxiliary field in the model. The matter fields are shown to be arranged into 3d N = 2 massive hypermultiplets in certain representations of U( n) × U( m) Yang-Mills gauge groups. Discrete symmetries of the full system are studied, and the related N = 1 supersymmetric truncations with O( n) and Sp( n) Yang-Mills symmetries are constructed. The simplicity of the model allows us to elucidate some general properties of the HS models. In particular, a new result, which can have interesting implications to the higher-dimensional models, is that our model is shown to admit an “integrating” flow that proves existence of a non-local Bäcklund-Nicolai-type mapping to the free system.

Diagonal solutions to reflection equations in higher spin models

A general fusion method to find solutions to the reflection equation in higher spin representations starting from the fundamental one is shown. The method is illustrated by applying it to obtaining the K diagonal boundary matrices in an alternating spin 1 2 and spin 1 chain. The Hamiltonian is also given. The applicability of the method to higher rank algebras is shown by obtaining the K diagonal matrices for a spin chain in the {3 ∗} representation of su(3) from the {3} representation.

Critical theory of quantum spin chains.

The relation between quantum spin chains and conformal field theories is reexamined. Using a generalized Hubbard model representation it is argued that the critical theory for generic half-odd-integer spin antiferromagnets is the Wess-Zumino-Witten model (WZW model) with topological coupling, k=1, whereas generic integer spin antiferromagnets have an energy gap. The higher-k WZW models (which describe integrable higher spin models) are multicritical points in the space of all spin Hamiltonians. The k=1 WZW model represents a stable fixed point for many theories including WZW models of arbitrary odd k with relevant operators added, generalized Hubbard or Thirring models with an odd number of colors and the O(3) \ensuremath{\sigma} model at topological angle \ensuremath{\theta}=\ensuremath{\pi}.