The structures of boron clusters, such as flat clusters and fullerenes, resemble those of carbon. Various two‐dimensional (2D) borophenes have been proposed since the production of graphene. The recent successful fabrication of borophene sheets has prompted extensive researches, and some unique properties are revealed. In this review, the recent theoretical and experimental progress on the structure, growth, and electronic and thermal transport properties of borophene sheets is summarized. The history of prediction of boron sheet structures is introduced. Existing with a mixture of triangle lattice and hexagonal lattice, the structures of boron sheets have peculiar characteristics of polymorphism and show significant dependence on the substrate. Due to the unique structure and complex BB bonds, borophene sheets have many interesting electronic and thermal transport properties, such as strong nonlinear effect, strong thermal transport anisotropy, high thermal conductance in the ballistic transport and low thermal conductivity in the diffusive transport. The growth mechanism and synthesis of borophene sheets on different metal substrates are also presented. The successful prediction and synthesis will shed light on the exploration of new novel materials. Besides, the outstanding and peculiar properties of borophene make them tempting platform for exploring novel physical phenomena and extensive applications.

We construct linearized solutions to Vasiliev's four-dimensional higher spin gravity on warped AdS 3 × ξ S 1 which is an Sp(2) × U (1) invariant non-rotating BTZ-like black hole with R 2 × T 2 topology. The background can be obtained from AdS 4 by means of identifications along a Killing boost K in the region where ξ 2 ≡ K 2 ⩾ 0, or, equivalently, by gluing together two Bañados-Gomberoff-Martinez eternal black holes along their past and future space-like singularities (where ξ vanishes) as to create a periodic (non-Killing) time. The fluctuations are constructed from gauge functions and initial data obtained by quantizing inverted harmonic oscillators providing an oscillator realization of K and of a commuting Killing boost K. The resulting solution space has two main branches in which K star commutes and anti-commutes, respectively, to Vasiliev's twistedcentral closed two-form J. Each branch decomposes further into two subsectors generated from ground states with zero momentum on S 1 . We examine the subsector in which K anti-commutes to J and the ground state is U (1) K × U (1) K -invariant of which U (1) K is broken by momenta on S 1 and U (1) K by quasi-normal modes. We show that a set of U (1) K -invariant modes (with n units of S 1 momenta) are singularity-free as master fields living on a total bundle space, although the individual Fronsdal fields have membrane-like singularities at K 2 = 1. We interpret our findings as an example where Vasiliev's theory completes singular classical Lorentzian geometries into smooth higher spin geometries.5 It is worth mentioning that, from the point of view of the standard spin-2 geometry, there is no four-dimensional uplift of the three-dimensional rotating BTZ black hole, since, differently from the spinless case, the presence of an extra spatial dimension erases the horizon [38]. Since one of the issues to be studied in this paper is the resolution of singularities of fluctuation fields at the horizon within Vasiliev's higher spin gravity, we shall choose to investigate the linearized dynamics around a vacuum solution corresponding to the four-dimensional (topologically extended) non-rotating BTZ-like black hole.6 Rather, in constructing unfolded systems of equations it is usually assumed that if the frame field is invertible then the system must admit a dual interpretation as a complex for an algebraic differential whose cohomology in different degrees consists of the dynamical Fronsdal fields, their gauge parameters, and equations of motion and Bianchi identities [41]; for analogous treatment of mixed symmetry fields, see [43,44].

We present exact solutions to Vasiliev's bosonic higher spin gravity equations in four dimensions with positive and negative cosmological constant that admit an interpretation in terms of domain walls, quasi-instantons and Friedman-Robertson-Walker (FRW) backgrounds. Their isometry algebras are infinite dimensional higher-spin extensions of spacetime isometries generated by six Killing vectors. The solutions presented are obtained by using a method of holomorphic factorization in noncommutative twistor space and gauge functions. In interpreting the solutions in terms of Fronsdal-type fields in spacetime, a field-dependent higher spin transformation is required, which is implemented at leading order. To this order, the scalar field solves Klein-Gordon equation with conformal mass in (A)dS 4 . We interpret the FRW solution with de Sitter asymptotics in the context of inflationary cosmology and we expect that the domain wall and FRW solutions are associated with spontaneously broken scaling symmetries in their holographic description. We observe that the factorization method provides a convenient framework for setting up a perturbation theory around the exact solutions, and we propose that the nonlinear completion of particle excitations over FRW and domain wall solutions requires black hole-like states.

A 6th-order, but ghost-free, gauge-invariant action is found for a 4th-rank symmetric tensor potential in a three-dimensional (3D) Minkowski spacetime. It propagates two massive modes of spin 4 that are interchanged by parity, and is thus a spin-4 analog of linearized "new massive gravity". Also found are ghost-free spin-4 analogs of linearized "topologically massive gravity" and "new topologically massive gravity", of 5th-and 8th-order respectively. IntroductionThere is a well-developed theory of relativistic free-field spin-s gauge theories in a 4dimensional Minkowski spacetime (4D), based on symmetric rank-s gauge potentials. The topic was initiated by Fronsdal [1] and its geometric formulation was provided by de Wit and Freedman [2]. We refer the reader to [3] for a more recent review. The s ≤ 2 cases are standard; in particular, the s = 2 field equation is the linearized Einstein equation for a metric perturbation. This provides a model for integer "higherspin" (s > 2) where the gauge-invariant two-derivative field-strength is an analog of the linearized Riemann tensor. A feature of these higher-spin gauge theories of relevance here is that the gauge transformation parameter, a symmetric tensor of rank s − 1, is constrained to be trace-free. If this constraint on the parameter were to be relaxed then any gauge-invariant equation would be of higher than second order, and this would normally imply the propagation of ghost modes, i.e. modes of negative energy.The situation for 3-dimensional Minkowski spacetime (3D) is different, in many respects. One is that the standard "higher-spin" gauge field equations do not actually propagate any modes in 3D. One may take advantage of this simplification, and the fact that 3D gravity can be recast as a Chern-Simons (CS) theory [4,5], to construct CS models for higher-spin fields interacting with 3D gravity in an anti de Sitter (adS) background. The original model of this type [6] is analogous to Vasiliev's 4D theory of all integer higher-spins interacting in an adS background [7]. However, in 3D one can consider a "truncated" version describing only a finite number of higher-spin fields coupled to gravity [8,9]. Such models have recently yielded interesting insights [10][11][12] although the absence of propagating modes may limit their impact.Propagating modes arise in 3D when higher-derivative terms are included in the action. The best known case is "topologically massive gravity" (TMG) which involves the inclusion of a 3rd-order Lorentz-Chern-Simons term [13]. This is a parity-violating gravity model that propagates a single massive spin-2 mode, thereby illustrating another special feature of 3D: gauge-invariance is consistent with non-zero mass. TMG is ghost-free, despite the higher-derivative nature of the field equations, because one may choose the overall sign of the action to ensure that the one propagated mode has positive energy. Rather more surprising is the fact that there exists a parity-preserving unitary model with curvature-squared terms, and hence 4th-orde...