Restrictions for n -point vertices in higher-spin theories
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Springer
Received: March 10, Revised: May 5, Accepted: May 22, Published: June 18,
2020 2020 2020 2020
Stefan Fredenhagen,a,b Olaf Kr¨ ugera and Karapet Mkrtchyanc a
University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria b Erwin Schr¨ odinger International Institute for Mathematics and Physics, Boltzmanngasse 9, 1090 Vienna, Austria c Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126 Pisa, Italy
E-mail: [email protected], [email protected], [email protected] Abstract: We give a simple classification of the independent n-point interaction vertices for bosonic higher-spin gauge fields in d-dimensional Minkowski spacetimes. We first give a characterisation of such vertices for large dimensions, d ≥ 2n − 1, where one does not have to consider Schouten identities due to over-antisymmetrisation of spacetime indices. When the dimension is lowered, such identities have to be considered, but their appearance only leads to equivalences of large-d vertices and does not lead to new types of vertices. We consider the case of low dimensions (d < n) in detail, where a large number of Schouten identities leads to strong restrictions on independent vertices. We also comment on the generalisation of our results to the intermediate region n ≤ d ≤ 2n − 2. In all cases, the independent vertices are expressed in terms of elementary manifestly gauge-invariant quantities, suggesting that no deformations of the gauge transformations are induced. Keywords: Higher Spin Gravity, Scattering Amplitudes, Higher Spin Symmetry, SpaceTime Symmetries ArXiv ePrint: 1912.13476
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)118
JHEP06(2020)118
Restrictions for n-point vertices in higher-spin theories
Contents 1 Introduction
1
2 Preliminaries 2.1 Vertex generating operators 2.2 Equivalence relations for vertex generating operators 2.3 Imposing gauge invariance
4 4 5 8 9 9 11
4 Lower dimension: dealing with Schouten identities
13
5 The 5.1 5.2 5.3 5.4 5.5
15 16 17 19 19 20 20 22
case n > d A minimal generating set of Schouten identities The choice of representative General restrictions from gauge invariance Restrictions for V Proofs 5.5.1 Proof of eq. (5.4) 5.5.2 Proof of eq. (5.7)
6 Parity-odd vertices
25
7 Discussion
27
1
Introduction
In this paper, we investigate a Lagrangian formulation of higher-spin (HS) theories in arbitrary dimensions. The aim of this work is, in particular, to obtain restrictions for all possible independent interaction vertices of order n ≥ 4 for massless higher-spin fields, extending the three-dimensional results of [1]. Together with the earlier results on the cubic vertices [2–9] (see also [10–15]), this work intends to complete the classification of all independent interacting deformations of free massless HS Lagrangians [16, 17] to the lowest order in the deformation parameters (coupling constants) in Minkowski spacetime of arbitrary dimensions d ≥ 3. HS Gravities [18–20] (see, e.g., [
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