Thomas Basile

A bstract: Based on an efficient technology for the excavation of entire open string trajectories of physical states, we propose an algorithmic method of constructing the largely unknown closed string trajectories. Due to combinatorial complexity, the “double copy” of open strings is limited in efficiency as a means of building closed string trajectories. We bypass this technical difficulty by employing one open string as the fundamental seed and then dressing it by a suitable selection of generators of a symplectic algebra that acts on both the left and the right sector. By applying Howe duality, the dressed seeds amount to closed string trajectory candidates, finding the physical subset of which is possible by solving systems of equations involving Diophantine-like recursion relations, which we also illustrate with examples.

The Alexandrov–Kontsevich–Schwarz–Zaboronsky (AKSZ) construction encodes all the data of a topological sigma model in a finite-dimensional symplectic Q -manifold. Relaxing the nondegeneracy condition, i.e. considering a presymplectic form instead, extends the construction to non-topological models. The gauge-invariant action functional of (presymplectic) AKSZ sigma models is written in terms of space-time differential forms and can be seen as a covariant multidimensional analogue of the usual 1st order Hamiltonian action. In this work, we show that the AKSZ construction admits a natural generalisation where the target space Q -manifold is equipped with a form of arbitrary degree Ω (possibly inhomogeneous) which is ( d + L Q ) -closed. This data defines a higher-derivative generalisation of the AKSZ action which is still invariant under the natural gauge transformations determined by Q and which is efficiently formulated in terms of a version of the Chern–Weil map introduced by Kotov and Strobl. It turns out that a variety of interesting gauge theories, including higher-dimensional Chern–Simons theory, MacDowell–Mansouri–Stelle–West and self-dual gravity as well as the higher spin extension of the latter, can be concisely reformulated as such multisymplectic AKSZ models. We also present a version of the construction in the setup of PDE geometry and demonstrate that the counterpart of the multisymplectic AKSZ action is precisely the standard multisymplectic formulation, where the Chern–Weil map corresponds to the usual pullback map.

Massless chiral fields of arbitrary spin in six spacetime dimensions, also known as higher spin singletons, admit a simple formulation in terms of SU^*(4) \cong SL(2,\mathbb{H}) S U * ( 4 ) ≅ S L ( 2 , ℍ ) tensors. We show that, paralleling the four-dimensional case, these fields can be described using a 0 0 -form and a gauge 2 2 -form, taking values in totally symmetric tensors of SU^*(4) S U * ( 4 ) . We then exhibit an example of interacting theory that couples a tower of singletons of all integer spin to a background of \mathfrak{g} ???? -valued higher spin fields, for \mathfrak{g} ???? an arbitrary Lie algebra equipped with an invariant symmetric bilinear form. Finally, we discuss the formulation of these models in arbitrary even dimensions, as well as their partially-massless counterpart.

A bstract: Conformal higher-spin gravity is the log-divergent part of the effective action of the scalar field coupled to background fields via higher-spin currents, as was defined by Segal and Tseytlin, which can be worked out over the flat space background. We revisit the problem of the scalar field in a higher-spin background and propose a manifestly covariant version thereof. The construction utilizes the Fedosov quantization of the cotangent bundle and the action is written with the help of the trace on a curved phase space that is provided by the Feigin-Felder-Shoikhet cocycle. The same construction allows one to formulate quantum mechanics on a curved space, the phase space being the cotangent bundle.

A bstract: We propose a realisation of partially-massless higher spin algebras in four dimensions in terms of bosonic and fermionic oscillators, using Howe duality between sp (4, ℝ) ≅ so (2, 3) and osp (1|2( ℓ − 1), ℝ). More precisely, we show that the centraliser of osp (1|2( ℓ − 1), ℝ) in the Weyl-Clifford algebra generated by 4 bosonic and 8( ℓ − 1) fermionic symbols, modulo osp (1|2( ℓ − 1), ℝ) generators, is isomorphic to the higher spin algebra of the type-A _ℓ theory whose spectrum contains partially-massless fields of all spins and depths t = 1, 3,…, 2 ℓ − 1. We also discuss the possible existence of a deformation of this algebra, which would encode interaction for the type-A _ℓ theory.

We propose a covariant method of constructing entire trajectories of physical states in superstring theory in the critical dimension. It is inspired by a recently developed covariant technology of excavating bosonic string trajectories, that is facilitated by the observation that the Virasoro constraints can be written as linear combinations of lowering operators of a bigger algebra, namely a symplectic algebra, which is Howe dual to the spacetime Lorentz algebra. For superstrings, it is the orthosymplectic algebra that appears instead, with its lowest weight states forming the simplest class of physical trajectories in the NS sector. To construct the simplest class in the R sector, the lowest weight states need to be supplemented with other states, which we determine. Deeper trajectories are then constructed by acting with suitable combinations of the raising operators of the orthosymplectic algebra, which we illustrate with several examples.