From Higher Spins to Strings: A Primer

Abstract

A contribution to the collection of reviews Introduction to Higher Spin Theory edited by S. Fredenhagen, this introductory article is a pedagogical account of higher-spin fields and their connections with String Theory. We start with the motivations for and a brief historical overview of the subject. We discuss the Wigner classifications of unitary irreducible Poincaré-modules, write down covariant field equations for totally symmetric massive and massless representations in flat space, and consider their Lagrangian formulation. After an elementary exposition of the AdS unitary representations, we review the key no-go and yes-go results concerning higher-spin interactions, e.g., the Velo-Zwanziger acausality and its string-theoretic resolution among others. The unfolded formalism, which underlies Vasiliev's equations, is then introduced to reformulate the flat-space Bargmann-Wigner equations and the AdS massive-scalar Klein-Gordon equation, and to state the "central onmass-shell theorem". These techniques are used for deriving the unfolded form of the boundary-to-bulk propagator in AdS 4 , which in turn discloses the asymptotic symmetries of (supersymmetric) higher-spin theories. The implications for stringhigher-spin dualities revealed by this analysis are then elaborated.

On special quartic interaction of higher spin gauge fields with scalars and gauge symmetry commutator in the linear approximation

“…• From third and fourth lines we see that J (4) terms arose with different weight 1 5 and 2 15 . But we will see below that they should come with same weight to complete integration for interaction terms.…”

“…Using relation (A.1) we can bring improvement for discrepancy in numbers a front of J (4) terms in (2.15). The second relation (A.2) is suitable for cancellation of the fifth line of (2.15) connected with J (2) current.…”

“…In our previous articles [46], [47] we prove that in both AdS and flat backgrounds after corresponding field redefinition interaction of even spin s gauge field with spin s current constructed from scalar and derivatives could be written in the form supplemented by the whole tower of invariant actions for couplings of the same scalar with all gauge fields of smaller even spin. So the starting lagrangian for our task we take from [47] rewriting all terms in the flat background: S ΦΦh (4) (Φ, h (2) , h (4) ) = S 0 (Φ) + S 1 (Φ, h (2) ) + S 1 (Φ, h (4) ), (…”

“…From now on to avoid cumbersome notation and overlapping with symmetrization brackets we reserve notation h and ε for gauge field for spin four h (4) and corresponding gauge parameter ε (3) except the cases when we do not explicitly write out indices. In the case of other spin (rank) fields and parameters, we use these letters with an exact indication of rank.…”

Gauging the higher-spin-like symmetries by the Moyal product

“…It is not inconceivable that the degrees of freedom associated with these symmetries may account for dark matter, a missing puzzle in the standard cosmological model. The idea of gauging higher derivative symmetries is not new, and is at the core of the higher spin theory program (e.g., see reviews [2,3,4,5] and references therein). The standard approach is to introduce higher spin massless fields and construct higher spin gauge QFTs.…”