Higher Spin Gravity Research Articles

Finiteness of 3D higher spin gravity Landscape

Abstract We give Swampland constraints on the three dimensional Landscape of Anti-de Sitter higher spin gravity in the Chern-Simons formulation with connection valued in various split real forms of Lie algebras. We derive the finiteness conjecture by computing the upper bound on the rank of possible gauge groups then we refine it using the AdS distance conjecture. We discuss the implications of this Swampland constraint on the spectrum of higher spin gravity theories and we compare it with the gravitational exclusion principle required from BTZ black hole consideration to excerpt a constraint on the Chern-Simons level k.

Fractional spins, unfolding, and holography. Part II. 4D higher spin gravity and 3D conformal dual

This paper completes the analysis initiated in the companion work arXiv:2403.02283 — referred to as Paper I — by showing how Vasiliev’s 4D higher-spin gravity (HSG) and 3D coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields (coloured conformal HSG, or CCHSG) emerge as consistent reductions of a common parent model. The latter is a Frobenius-Chern-Simons model with superconnection valued in a fractional-spin extension of Vasiliev’s higher-spin algebra, and was defined and studied in Paper I. Here, we i) realize HSG as a subcase of a more general 4D reduction, describing HSG coupled to coloured, fractional-spin matter, which we refer to as 4D fractional-spin gravity; ii) study the CCHSG model, in particular exhibiting the crucial role played by novel colour gauge fields in coupling conformal matter to conformal HSG, thereby completing the models due to Vasiliev and Nilsson; iii) extract conformal currents and composite coloured sources in an expansion of the CCHSG model around 3D Minkowskian leaves; and iv) compare our results with Vasiliev’s holography proposal of arXiv:1203.5554. The common origin of HSG and CCHSG, besides relating the two models directly, is the starting point for establishing the holographic correspondence between the two models via overlap conditions, to be presented separately.

A chiral limit for Chern-Simons-matter theories

Large N quasi-fermionic Chern-Simons-matter theories have an approximate higher-spin symmetry that strongly constrains their correlation functions. In particular, the 3-point functions for generic spins are combinations of 3 structures (with specific dependence on the positions and helicities), and the coupling-dependence of the coefficient of each structure is uniquely determined. In the past few years, several relations between different structures were found. In this paper we show that all the relations between the structures follow from (or, conversely, they imply) a specific form written by Skvortsov for the vertices of the dual higher-spin gravity theory on four-dimensional anti-de Sitter space, when written in spinor-helicity variables. The dual bulk theory has a specific limit where it simplifies and becomes a “chiral higher-spin gravity theory”, and we discuss what can be said about this limit in the dual Chern-Simons-matter theories, where it involves an analytic continuation to complex couplings.

Fractional spins, unfolding, and holography. Part I. Parent field equations for dual higher-spin gravity reductions

In this work and in the companion paper arXiv:2403.02301, we initiate an approach to holography based on the AKSZ formalism. As the first step, we refine Vasiliev’s holography proposal in arXiv:1203.5554 by obtaining 4D higher-spin gravity (HSG) and 3D coloured conformal higher-spin gravity (CCHSG) — i.e., coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields — as two distinct and classically consistent reductions of a single parent theory. The latter consists, on-shell, of a flat superconnection valued in a fractional-spin extension of Vasiliev’s higher-spin algebra. The HSG and CCHSG reductions are characterized by dual structure groups and two-form cohomology elements, and their embedding in a common parent model provides a rationale for deriving holographic relations from multi-dimensional AKSZ partition functions on cylinders with dual boundary conditions, to appear separately. In this work we i) construct the underlying non-commutative geometry as a metaplectic operator algebra represented in a Hermitian module of a pair of conformal particles; ii) identify a discrete modular group, arising from twisted boundary conditions of the first-quantized system, and connecting different boundary conditions of the second-quantized system; and iii) identify the holonomies, structure groups and two-form cohomology elements that characterize the HSG and CCHSG reductions, and equate the dual second Chern classes.

Low spin solutions of higher spin gravity: BPST instanton

Higher spin gravities do not have a low energy limit where higher-spin fields decouple from gravity. Nevertheless, it is possible to construct fine-tuned exact solutions that activate low-spin fields without sourcing the higher-spin fields. We show that BPST (Belavin-Polyakov-Schwartz-Tyupkin) instanton is an exact solution of Chiral Higher Spin Gravity, i.e. it is also a solution of the holographic dual of Chern-Simons matter theories. This gives an example of a low-spin solution. The instanton sources the opposite helicity spin-one field and a scalar field. We derive an Effective Field Theory that describes the coupling between an instanton and the other two fields, whose action starts with the Chalmers-Siegel action and has certain higher derivative couplings.

Strong homotopy algebras for chiral higher spin gravity via Stokes theorem

Chiral higher spin gravity is defined in terms of a strong homotopy algebra of pre-Calabi-Yau type (noncommutative Poisson structure). All structure maps are given by the integrals over the configuration space of concave polygons and the first two maps are related to the (Shoikhet-Tsygan-)Kontsevich Formality. As with the known formality theorems, we prove the A∞-relations via Stokes’ theorem by constructing a closed form and a configuration space whose boundary components lead to the A∞-relations. This gives a new way to formulate higher spin gravities and hints at a construct encompassing the known formality theorems.

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} = 2 supersymmetry in the twistor description of higher-spin holography

We study the holographic duality between higher-spin (HS) gravity in 4d and free vector models in 3d, with special attention to the role 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} = 2 supersymmetry (SUSY). For the type-A bosonic bulk theory, dual to spin-0 fields on the boundary, there exists a twistor-space description; this maps both single-trace boundary operators and linearized bulk fields to spacetime-independent twistor functions, whose HS-algebra products compute all boundary correlators. Here, we extend this description to the type-B bosonic theory (dual to spin-1/2 fields on the boundary), and to the supersymmetric theory containing both. A key role is played by boundary bilocals, which in type-A are dual to the Didenko-Vasiliev 1/2-BPS “black hole”. We extend this to an infinite family of linearized 1/2-BPS “black hole” solutions. Remarkably, the full supersymmetric theory (along with the SUSY generators) fits in the same space of twistor functions as the type-A theory. Instead of two sets of bosonic bulk fields, the formalism sees one set of linearized fields, but with both types of boundary data allowed.

W(0,b) algebra and the dual theory of 3D asymptotically flat higher spin gravity

Bondi-Metzner-Sachs (BMS) algebra in three spacetime dimensions can be deformed into a two parameter family of algebra known as W(a,b) algebra. For a=0, we show that other than W(0,−1), no other W(0,b) algebra admits a nondegenerate bilinear and thus one cannot have a Chern-Simons gauge theory formulation with them. However, they may appear in a three-dimensional gravity description, where we also need to have a spin 2 generator, that comes from the (a=0,b=−1) sector. In the present work, we have demonstrated that the asymptotic symmetry algebra of a spin-3 gravity theory on flat spacetime has both the W(0,−1) and W(0,−2) algebras as subalgebras. We have also constructed a dual boundary field theory for this higher spin gravity theory by using the Chern-Simons/Wess-Zumino-Witten correspondence. Published by the American Physical Society 2024

Geometric tool kit for higher spin gravity (Part II): An introduction to Lie algebroids and their enveloping algebras

These notes provide a self-contained introduction to Lie algebroids, Lie-Rinehart algebras and their universal envelopes. This review is motivated by the speculation that higher-spin gauge symmetries should admit a natural formulation as enveloping algebras of Lie algebroids since rigid higher-spin algebras are enveloping algebras of Lie algebras. Nevertheless, the material covered here may be of general interest to anyone interested in the description of gauge symmetries, connections and covariant derivatives, in terms of Lie algebroids. In order to be self-contained, a concise introduction to the algebraic characterization of vector bundles as projective modules over the algebra of functions on the base manifold is provided.

Covariant action for conformal higher spin gravity

Conformal higher spin (HS) gravity is a HS extension of Weyl gravity and is a family of local HS theories, which was put forward by Segal and Tseytlin. We propose a manifestly covariant and coordinate-independent action for these theories. The result is based on an interplay between HS symmetries and deformation quantization: a locally equivalent but manifestly background-independent reformulation, known as the parent system, of the off-shell multiplet of conformal HS fields (Fradkin–Tseytlin fields) can be interpreted in terms of Fedosov deformation quantization of the underlying cotangent bundle. This brings into the game the invariant quantum trace, induced by the Feigin–Felder–Shoikhet cocycle of Weyl algebra, which extends Segal’s action into a gauge invariant and globally well-defined action functional on the space of configurations of the parent system. The same action can be understood within the worldline approach as a correlation function in the topological quantum mechanics on the circle.

Microscopic entanglement wedges

We study the holographic duality between the free O(N) vector model and higher spin gravity. Conserved spinning primary currents of the conformal field theory (CFT) are dual to spinning gauge fields in the gravity. Reducing to independent components of the conserved CFT currents one finds two components at each spin. After gauge fixing the gravity and then reducing to independent components, one finds two components of the gauge field at each spin. Collective field theory provides a systematic way to map between these two sets of degrees of freedom, providing a complete and explicit identification between the dynamical degrees of freedom of the CFT and the dual gravity. The resulting map exhibits many features expected of holographic duality: it provides a valid bulk reconstruction, it reproduces insights expected from the holography of information and it provides a microscopic derivation of entanglement wedge reconstruction.

Quartic locality of higher-spin gravity in de Sitter and Euclidean anti-de Sitter space

We consider higher-spin gravity in (A)dS4, defined as the holographic dual of a free O(N) or Sp(N) vector model. At the quartic level, this theory has been judged non-local at distances greater than the (A)dS radius, due to a mismatch of massless (twist=1) exchange-type terms in its boundary OPE behavior. We review the non-locality argument, and note that it relies on a double-lightcone limit, which requires a Lorentzian boundary. In the Euclidean OPE limit, we demonstrate the absence of massless exchange-type non-localities of any spin, by inspecting a known formula for the bulk exchange diagrams in Euclidean AdS, and constructing upper bounds in which the spin-dependence factorizes from the position-dependence. Our results suggest that higher-spin theory is local (at distances greater than the curvature radius) in spacetimes with Euclidean boundary signature. For Lorentzian bulk, this implies locality in de Sitter space, as opposed to anti-de Sitter.

Asymptotic dynamics of three dimensional supergravity and higher spin gravity revisited

We reconsider the Hamiltonian reduction of the action for three dimensional AdS supergravity and W3 higher spin AdS gravity in the Chern-Simons formulation under asymptotically anti-de Sitter boundary conditions. We show that the reduction gives two copies of chiral bosons on the boundary. In particular, we take into account the holonomy of the Chern-Simons connection which manifests itself as zero mode of the momentum of the boundary chiral boson. We provide an equivalent formulation of the boundary action which we claim to be the geometric action on symplectic leaves of a (super-)Virasoro or a higher spin WN Poisson manifold in the case of supergravity or higher spin gravity respectively, where the intersection of leaves (given in terms of leaves representatives) can be identified as the bulk holonomy. This concludes the extension to non-linear algebras where the notion of coadjoint representation is not well-defined. The boundary Hamiltonian depends on a choice of boundary conditions and is equivalent to the Schwarzian action for corresponding Brown-Henneaux boundary conditions. We make this connection explicit in the extended supersymmetric case. Moreover, we discuss the geometric action in the case of W3 AdS3 gravity in both mathfrak{sl} (3) highest weight representations based on principal and diagonal mathfrak{sl} (2) embeddings.

Chiral higher spin gravity and convex geometry

Chiral Higher Spin Gravity is the minimal extension of the graviton with propagating massless higher spin fields. It admits any value of the cosmological constant, including zero. Its existence implies that Chern-Simons vector models have closed subsectors and supports the 3d3d bosonization duality. In this letter, we explicitly construct an A_\inftyA∞-algebra that determines all interaction vertices of the theory. The algebra turns out to be of pre-Calabi-Yau type. The corresponding products, some of which originate from Shoikhet-Tsygan-Kontsevich formality, are given by integrals over the configuration space of convex polygons.

Complex saddles of three-dimensional de Sitter gravity via holography

We determine complex saddles of three-dimensional gravity with a positive cosmological constant by applying the recently proposed holography. It is sometimes useful to consider a complexified metric to study quantum gravity as in the case of the no-boundary proposal by Hartle and Hawking. However, there would be too many saddles for complexified gravity, and we should determine which saddles to take. We describe the gravity theory by three-dimensional SL$(2,\mathbb{C})$ Chern-Simons theory. At the leading order in the Newton constant, its holographic dual is given by Liouville theory with a large imaginary central charge. We examine geometry with a conical defect, called a de Sitter black hole, from a Liouville two-point function. We also consider geometry with two conical defects, whose saddles are determined by the monodromy matrix of Liouville four-point function. Utilizing Chern-Simons description, we extend the similar analysis to the case with higher-spin gravity.

Higher-spin gravity in AdS3 and folds on Fermi surface

In this paper, we introduce new sets of boundary conditions for higher spin gravity in $AdS_3$ where the boundary dynamics of spin two and other higher spin fields are governed by the interacting collective field theory Hamiltonian of Avan and Jevicki. We show that the time evolution of spin two and higher spin fields can be captured by the classical dynamics of folded fermi surfaces in the similar spirit of Lin, Lunin and Maldacena. We also construct infinite sequences of conserved charges showing the integrable structure of higher spin gravity (for spin 3) under the boundary conditions we considered. Further, we observe that there are two possible sequences of conserved charges depending on whether the underlying boundary fermions are non-relativistic or relativistic.

Chiral approach to partially-massless fields

We propose a new (chiral) description of partially-massless fields in 4d, including the partially-massless graviton, that is similar to the pure connection formulation for gravity and massless higher spin fields, the latter having a clear twistor origin. The new approach allows us to construct complete examples of higher spin gravities with (partially-)massless fields that feature Yang-Mills and current interactions.

On (spinor)-helicity and bosonization in AdS4/CFT3

Helicity is a useful concept both for AdS4 and CFT3 studies. We work out the complete AdS4/CFT3 dictionary for spinning fields/operators in the spinor-helicity base that allows one to scalarize any n-point contact vertex. AdS4-vertices encode correlation functions of conserved currents, stress-tensor and, more generally, higher spin currents in a simple way. We work out the dictionary for Yang-Mills- and gravity-type theories with higher derivative corrections as well as some higher spin examples and exemplify the relation to the three-dimensional bosonization duality. The bosonization can be understood as a simple surgery: vertices/correlators are built via an EM-duality transformation by sewing together (anti)-Chiral higher spin gravities, to whose existence the three-dimensional bosonization duality can be attributed (up to the proof of uniqueness).

Geometric tool kit for higher-spin gravity (Part I): An introduction to the geometry of differential operators

This paper provides an introduction to the algebra and geometry of differential operators and jet bundles. Its point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalizations of Lie derivatives and diffeomorphisms. Nevertheless, the material covered here may be of general interest to anyone working on topics where geometrical (coordinate-free, global, generic) and mathematically rigorous definitions of differential operators are required.

More on Chiral Higher Spin Gravity and convex geometry

Recently, a unique class of local Higher Spin Gravities with propagating massless fields in 4d – Chiral Higher Spin Gravity – was given a covariant formulation both in flat and (A)dS4 spacetimes at the level of equations of motion. We unfold the corresponding homological perturbation theory as to explicitly obtain all interaction vertices. The vertices reveal a remarkable simplicity after an appropriate change of variables. Similarly to formality theorems the A∞/L∞ multi-linear products can be represented as integrals over a configuration space, which in our case is the space of convex polygons. The A∞-algebra underlying Chiral Theory is of pre-Calabi–Yau type. As a consequence, the equations of motion have the Poisson sigma-model form.