Higher Spin Gauge Theories Research Articles

Interactions in the IKKT matrix model on covariant quantum spacetime

We study the interactions of the higher-spin gauge theory arising from the IKKT matrix model on a covariant quantum FLRW quantum space-time MJ1,3, denoted as HS-IKKT. In particular, we elaborate some of the vertices and observe that they are not manifestly Lorentz invariant in the unitary formulation. We argue that Lorentz invariance of HS-IKKT can be recovered since Lorentz transformations are part of the gauge invariance in the covariant formulation. This statement is verified for some vertices with lowest number of derivatives. The lowest-derivative sector of this theory is expected to be governed by an “almost”-Lorentz-invariant Yang-Mills theory coupled to emergent gravity.

Toward higher-spin symmetry breaking in the bulk

We present a new vacuum of the bosonic higher-spin gauge theory in d+1 dimensions, which has leftover symmetry of the Poincaré algebra in d dimensions. Its structure is very simple: the space-time geometry is that of anti–de Sitter space, while the only nonzero field is a scalar. The scalar extends along the Poincaré radial coordinate z and is shown to be linearly exact for an arbitrary mixture of its two Δ=2 and Δ=d−2 conformal branches. The obtained vacuum breaks the global higher-spin symmetry, leading to a broken phase that lives in the Minkowski space-time. Published by the American Physical Society 2024

Quantum [formula omitted]-Yang-Mills from the IKKT matrix model

We study the one-loop effective action of the higher-spin gauge theory induced by the IKKT matrix model on a M1,3×K background, where M1,3 is an FLRW cosmological spacetime brane and K are compact fuzzy extra dimensions. In particular, we show that all non-abelian (hs-valued) gauge fields in this model acquire mass via quantum effects, thus avoiding no-go theorems. This leads to a massive non-abelian quantum hs-Yang-Mills theory, whose detailed structure depends on K. The stabilization of K at one loop is understood as a result of the coupling between K and the U(1)-flux bundle on space-time. This flux stabilization induces the KK scale into the N=4 SYM sector of the model, which break superconformal symmetry.

Spinorial description for Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT

We introduce a novel spinorial description for the higher-spin gauge theory induced by the IKKT matrix model on an FLRW spacetime with Lorentzian signature, called Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT theory. The new description is based on Weyl spinors transforming under the space-like isometry subgroup SL(2, ℂ) of the structure group SO(2, 4) ≃ SU(2, 2). It allows us to exploit the full power of the spinor formalism in Lorentzian signature, in contrast to a previous formalism based on the compact subgroup SU(2)L × SU(2)R of SU(2, 2). Some cubic vertices of the Yang-Mills sector and the corresponding scattering amplitudes are computed. We observe that the n-point (for n ≥ 4) tree-level amplitudes are typically non-trivial on-shell, but exponentially suppressed in the late-time regime. While Lorentz invariance of the higher-spin amplitudes is not manifest, it is expected to be restored by higher-spin gauge invariance.

Moderately non-local ηη¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta {\\bar{\\eta }}$$\\end{document} vertices in the AdS4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AdS_4$$\\end{document} 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 is softer than those resulting from the shifted homotopy approach available in the literature so far, is worked out in the mixed ηη¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta {\\bar{\\eta }}$$\\end{document} sector of the Vasiliev higher-spin theory. To calculate moderately non-local vertices Υηη¯(ω,C,C,C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Upsilon ^{\\eta \\bar{\\eta }}(\\omega , C,C,C)$$\\end{document} for all ordering of the fields ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} and C we apply an interpolating homotopy, that respects the moderate non-locality in the perturbative analysis of the higher-spin equations.

Spinorial higher-spin gauge theory from IKKT model in Euclidean and Minkowski signatures

We explore the semi-classical relation between the fuzzy 4-hyperboloid HN4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {H}_N^4 $$\\end{document} and non-compact quantized twistor space ℙN1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{P}}_N^{1,2} $$\\end{document} at large N. This provides two backgrounds of the IKKT matrix model via two natural stereographic projections, leading to higher-spin gauge theories with Euclidean and Minkowski signature denoted by HS-IKKT. The resulting higher-spin gauge theory can be understood as an uplift of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills to twistor space. The action of HS-IKKT is written using a spinor formalism in both in Euclidean and Minkowski signature. We then compute the tree-level amplitudes of the massless sector within the Yang-Mills part of the HS-IKKT model in the flat limit in Euclidean signature. All n-point tree-level scattering amplitudes for n ≥ 4 of this sector are found to vanish in the flat limit.

A twistorial description of the IKKT-matrix model

We consider the fuzzy 4-sphere {S}_N^4 as a background in the IKKT matrix model, and explore the relation between {S}_N^4 and fuzzy twistor space in the semi-classical limit. A novel description for the IKKT-matrix model in terms of spinorial indices is given, which is reminiscent of mathcal{N} = 4 super-symmetric Yang-Mills (SYM) in 4d. On fuzzy twistor space, the interactions of the IKKT model are of gravitational type. The higher-spin (HS) gauge theory emerging in this limit from the IKKT model, denoted as HS-IKKT, on fuzzy twistor space is shown to be a higher-spin extension of mathcal{N} = 4 SYM, with vertices that have more than two derivatives. We obtain its (Euclidean) spacetime action using the Penrose transform. Although this is a gravitational theory, it shares many features with the higher-spin extensions of Yang-Mills in 4d flat space obtained in [1, 2]. The tree-level amplitudes of the HS-IKKT are studied in the semi-classical flat limit. The self-dual gauge sector of the IKKT model is obtained by dropping some parts of the cubic- and the quartic interactions, which is shown to reduce to a mathcal{BF} -type action on commutative deformed projective twistor space.

Carrollian and Galilean conformal higher-spin algebras in any dimensions

We present higher-spin algebras containing a Poincaré subalgebra and with the same set of generators as the Lie algebras that are relevant to Vasiliev’s equations in any space-time dimension D ≥ 3. Given these properties, they can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Carrollian conformal higher-spin symmetries in one less dimension. We build these Lie algebras as quotients of the universal enveloping algebra of mathfrak{iso}left(1,D-1right) and we show how to recover them as Inönü-Wigner contractions of the rigid symmetries of higher-spin gauge theories in Anti de Sitter space or, equivalently, of relativistic conformal higher-spin symmetries. We use the same techniques to also define higher-spin algebras with the same set of generators and containing a Galilean conformal subalgebra, to be interpreted as non-relativistic limits of the conformal symmetries of a free scalar field. We begin by showing that the known flat-space higher-spin algebras in three dimensions can be obtained as quotients of the universal enveloping algebra of mathfrak{iso}left(1,2right) and then we extend the analysis along the same lines to a generic number of space-time dimensions. We also discuss the peculiarities that emerge for D = 5.

Unconstrained massless higher spin supermultiplet in AdS4 spacetime

We consider unconstrained formulation of the higher spin gauge theory in anti--de Sitter (AdS) spacetime, given by actions [A. Y. Segal, arXiv:0103028; M. Najafizadeh, Phys. Rev. D 98, 125012 (2018)], and provide on-shell supersymmetry transformations for the $\mathcal{N}=1$ unconstrained massless higher spin supermultiplet in four-dimensional ${\mathrm{AdS}}_{4}$. Such an irreducible supermultiplet $({\mathrm{\ensuremath{\Phi}}}_{1},{\mathrm{\ensuremath{\Phi}}}_{2};{\mathrm{\ensuremath{\Psi}}}_{1},{\mathrm{\ensuremath{\Psi}}}_{2})$ contains a pair of bosonic fields, with opposite parity, which are generating functions (infinite collection of totally symmetric real tensor fields of all integer rank $s=0,1,\dots{},\ensuremath{\infty}$), as well as two fermionic fields, which have opposite signs of the AdS radius, that are spinorial generating functions (infinite tower of totally symmetric Majorana spinor-tensor fields of all half-integer spin $s=\frac{1}{2},\frac{3}{2},\dots{},\ensuremath{\infty}$).

Extended superconformal higher-spin gauge theories in four dimensions

Using the off-shell formulation for mathcal{N} = 2 conformal supergravity in four dimensions, we describe superconformal higher-spin multiplets of conserved currents in a curved background and present their associated unconstrained gauge prepotentials. The latter are used to construct locally superconformal chiral actions, which are demonstrated to be gauge invariant in arbitrary conformally flat backgrounds. The main mathcal{N} = 2 results are then generalised to the mathcal{N} -extended case. We also present the gauge-invariant field strengths for on-shell massless higher-spin mathcal{N} = 2 supermultiplets in anti-de Sitter space. These field strengths prove to furnish representations of the mathcal{N} = 2 superconformal group.

Curvature tensors of higher-spin gauge theories derived from general Lagrangian densities

Curvature tensors of higher-spin gauge theories have been known for some time. In the past, they were postulated using a generalization of the symmetry properties of the Riemann tensor (curl on each index of a totally symmetric rank-n field for each spin-n). For this reason they are sometimes referred to as the generalized “Riemann” tensors. In this article, a method for deriving these curvature tensors from first principles is presented; the derivation is completed without any a priori knowledge of the existence of the Riemann tensors or the curvature tensors of higher-spin gauge theories. To perform this derivation, a recently developed procedure for deriving exactly gauge invariant Lagrangian densities from quadratic combinations of N order of derivatives and M rank of tensor potential is applied to the N = M = n case under the spin-n gauge transformations. This procedure uniquely yields the Lagrangian for classical electrodynamics in the N = M = 1 case and the Lagrangian for higher derivative gravity (“Riemann” and “Ricci” squared terms) in the N = M = 2 case. It is proven here by direct calculation for the N = M = 3 case that the unique solution to this procedure is the spin-3 curvature tensor and its contractions. The spin-4 curvature tensor is also uniquely derived for the N = M = 4 case. In other words, it is proven here that, for the most general linear combination of scalars built from N derivatives and M rank of tensor potential, up to N = M = 4, there exists a unique solution to the resulting system of linear equations as the contracted spin-n curvature tensors. Conjectures regarding the solutions to the higher spin-n N = M = n are discussed.

Scalar fields in 3D asymptotically flat higher-spin gravity

In this work we construct a novel associative algebra and use it to define a theory of higher-spin gravity in (2 + 1)-dimensional asymptotically flat spacetimes. Our construction is based on a quotient of the universal enveloping algebra of with respect to the ideal generated by its Casimir elements, the mass squared and the three-dimensional analogue of the square of the Pauli–Lubanski vector and propose to call the resulting associative algebra . We provide a definition of its generators and even though we are not yet able to provide the complete set of multiplication rules of this algebra our analysis allows us to study many interesting and relevant sub-structures of . We then show how to consistently couple a scalar field to an higher-spin gauge theory.

A connection between linearized Gauss–Bonnet gravity and classical electrodynamics II: Complete dual formulation

In a recent publication, a procedure was developed which can be used to derive completely gauge invariant models from general Lagrangian densities with [Formula: see text] order of derivatives and [Formula: see text] rank of tensor potential. This procedure was then used to show that unique models follow for each order, namely classical electrodynamics for [Formula: see text] and linearized Gauss–Bonnet gravity for [Formula: see text]. In this paper, the nature of the connection between these two well-explored physical models is further investigated by means of an additional common property; a complete dual formulation. First, we give a review of Gauss–Bonnet gravity and the dual formulation of classical electrodynamics. The dual formulation of linearized Gauss–Bonnet gravity is then developed. It is shown that the dual formulation of linearized Gauss–Bonnet gravity is analogous to the homogenous half of Maxwell’s theory; both have equations of motion corresponding to the (second) Bianchi identity, built from the dual form of their respective field strength tensors. In order to have a dually symmetric counterpart analogous to the nonhomogenous half of Maxwell’s theory, the first invariant derived from the procedure in [Formula: see text] can be introduced. The complete gauge invariance of a model with respect to Noether’s first theorem, and not just the equation of motion, is a necessary condition for this dual formulation. We show that this result can be generalized to the higher spin gauge theories, where the spin-[Formula: see text] curvature tensors for all [Formula: see text] are the field strength tensors for each [Formula: see text]. These completely gauge invariant models correspond to the Maxwell-like higher spin gauge theories whose equations of motion have been well explored in the literature.

Higher-rank tensor field theory of non-abelian fracton and embeddon

We formulate a new class of tensor gauge field theories in any dimension that is a hybrid class between symmetric higher-rank tensor gauge theory (i.e., higher-spin gauge theory) and anti-symmetric tensor topological field theory. Our theory describes a mixed unitary phase interplaying between gapless and gapped topological order phases (which can live with or without Euclidean, Poincaré, or anisotropic symmetry, at least in ultraviolet high or intermediate energy field theory, but not yet to a lattice cutoff scale). The “gauge structure” can be compact, continuous, abelian or non-abelian. Our theory sits outside the paradigm of Maxwell electromagnetic theory in 1865 and Yang–Mills isospin/color theory in 1954. We discuss its local gauge transformation in terms of the ungauged vector-like or tensor-like higher-moment global symmetry. The non-abelian gauge structure is caused by gauging the non-commutative symmetries: a higher-moment symmetry and a charge conjugation (particle–hole) symmetry. Vector global symmetries along time direction may exhibit time crystals. We explore the relation of these long-range entangled matters to a non-abelian generalization of Fracton order in condensed matter, a field theory formulation of foliation, the spacetime embedding and Embeddon that we newly introduce, and possible fundamental physics applications to dark matter or dark energy.

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 → −∞ limiting shifted homotopy proposed recently in [1] where all ω2C2 higher-spin vertices were shown to be spin-local. For the β → −∞ 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 [2] to the β-shifted contracting homotopy and the relation underlying locality of the ω2C2 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 con- stituent 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.

Chaos in three-dimensional higher spin gravity

We discuss SL(N, ℂ) Chern-Simons higher spin gauge theories in Euclidean AdS3. With appropriate boundary term, we derive the higher spin generalization of Schwarzian on-shell action. We investigate gravitationally dressed bi-locals, and we study the soft higher spin mode expansion to obtain soft mode eigenfunctions. We also derive the spin-s eigenfunction from Ward identity of mathcal{W} -algebra and a recursion relation. Using the on-shell action, we evaluate the contributions of the soft higher spin modes to the out-of-time-order correlators, and the corresponding Lyapunov exponent of SL(N) higher spin gravity is found to be frac{2pi }{beta}left(N - 1right) .

Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model

We discuss a (3+1)-dimensional covariant quantum space-time describing a FLRW cosmology with Big Bounce, obtained by a projection of the fuzzy hyperboloid Hn4. This provides a background solution of the IKKT matrix model with mass term. We characterize the bosonic fluctuation spectrum, which consists of a tower of higher-spin modes, truncated at n. The modes are organized in terms of an underlying SO(4, 2) structure group, which is broken to the SO(3, 1) isometry of the background. The resulting higher-spin gauge theory includes all degrees of freedom required for gravity, and should be well suited for quantization. All modes propagate with the same speed of light, even though local boost invariance is not manifest. The propagating metric perturbation modes comprise those of a massless graviton, as well as a scalar mode. Gauge invariance allows to obtain the analog of the linearized Einstein-Hilbert action, which is expected to be induced upon quantization.

Conformal geometry and (super)conformal higher-spin gauge theories

We develop a manifestly conformal approach to describe linearised (super)conformal higher-spin gauge theories in arbitrary conformally flat backgrounds in three and four spacetime dimensions. Closed-form expressions in terms of gauge prepotentials are given for gauge-invariant higher-spin (super) Cotton and (super) Weyl tensors in three and four dimensions, respectively. The higher-spin (super) Weyl tensors are shown to be conformal primary (super)fields in arbitrary conformal (super)gravity backgrounds, however they are gauge invariant only if the background (super) Weyl tensor vanishes. The proposed higher-spin actions are (super) Weyl-invariant on arbitrary curved backgrounds, however the appropriate higher-spin gauge invariance holds only in the conformally flat case. We also describe conformal models for generalised gauge fields that are used to describe partially massless dynamics in three and four dimensions. In particular, generalised higher-spin Cotton and Weyl tensors are introduced.

The fuzzy 4-hyperboloid [formula omitted] and higher-spin in Yang–Mills matrix models

We consider the SO(4,1)-covariant fuzzy hyperboloid Hn4 as a solution of Yang–Mills matrix models, and study the resulting higher-spin gauge theory. The degrees of freedom can be identified with functions on classical H4 taking values in a higher-spin algebra associated to so(4,1), truncated at spin n. We develop a suitable calculus to classify the higher-spin modes, and show that the tangential modes are stable. The metric fluctuations encode one of the spin 2 modes, however they do not propagate in the classical matrix model. Gravity is argued to arise upon taking into account induced gravity terms. This formalism can be applied to the cosmological FLRW space-time solutions of [1], which arise as projections of Hn4. We establish a one-to-one correspondence between the tangential fluctuations of these spaces.

N=2 supersymmetric higher spin gauge theories and current multiplets in three dimensions

We describe several families of primary linear supermultiplets coupled to three-dimensional ${\cal N}=2$ conformal supergravity and use them to construct topological $BF$-type terms. We introduce conformal higher-spin gauge superfields and associate with them Chern-Simons-type actions that are constructed as an extension of the linearised action for ${\cal N}=2$ conformal supergravity. These actions possess gauge and super-Weyl invariance in any conformally flat superspace and involve a higher-spin generalisation of the linearised ${\cal N}=2$ super-Cotton tensor. For massless higher-spin supermultiplets in (1,1) anti-de Sitter (AdS) superspace, we propose two off-shell Lagrangian gauge formulations, which are related to each other by a dually transformation. Making use of these massless theories allows us to formulate consistent higher-spin supercurrent multiplets in (1,1) AdS superspace. Explicit examples of such supercurrent multiplets are provided for models of massive chiral supermultiplets. Off-shell formulations for massive higher-spin supermultiplets in (1,1) AdS superspace are proposed.