We present the tensor computer algebra package xTras, which provides functions and methods frequently needed when doing (classical) field theory. Amongst others, it can compute contractions, make Ansätze, and solve tensorial equations. It is built upon the tensor computer algebra system xAct, a collection of packages for Mathematica.

We show how, using different decompositions of E 11 , one can calculate the representations under the duality group of the so-called "de-form" potentials. Evidence is presented that these potentials are in one-to-one correspondence to the embedding tensors that classify the gaugings of all maximal gauged supergravities. We supply the computer program underlying our calculations.

We establish the correspondence between, on one side, the possible gaugings and massive deformations of half-maximal supergravity coupled to vector multiplets and, on the other side, certain generators of the associated very extended Kac-Moody algebras. The difference between generators associated to gaugings and to massive deformations is pointed out. Furthermore, we argue that another set of generators are related to the so-called quadratic constraints of the embedding tensor. Special emphasis is placed on a truncation of the Kac-Moody algebra that is related to the bosonic gauge transformations of supergravity. We give a separate discussion of this truncation when non-zero deformations are present. The new insights are also illustrated in the context of maximal supergravity. The Kac-Moody approach to supergravityThe spectrum of physical states of the different maximal supergravity theories can be obtained from the very extended Kac-Moody algebra E 11 [7][8][9]. This has been extended to the set of all possible deformation and top-form potentials in [13,14]. A similar analysis could be done for E 10 [34-36] except for the top-form potentials. In addition, non-maximal supergravity and the associated very extended Kac-Moody algebras have been discussed in [9,27,37]. In the present paper we will apply the "Kac-Moody approach" to extract the deformation and topform potentials of half-maximal supergravity. In general this approach breaks down into four steps:1. Reduce to D = 3 over a torus and determine the G/K(G) scalar coset sigma model. Take the very extension4. Read off the spectrum by means of a level decomposition.As steps 2, 3, and 4 can be automatically carried out on the computer [38], this approach is very simple to carry out in practice. We will now take a close look at each of these steps.The first step is to determine the G/K(G) scalar coset sigma model in three dimensions for the toroidally reduced supergravity in question, where K(G) is the maximal compact subgroup of G. If there is no such a sigma model, which often is the case for theories with less than 16 supercharges, the Kac-Moody approach comes to a standstill. But when the coset does exist, as is the case for maximal and half-maximal supergravity, we can go on and take the very extension G +++ /K(G +++ ). The first extension corresponds to the (untwisted) affine version of G, which has been shown to be the symmetry group of various supergravities in D = 2 [39]. Also the second (over) extension and the third (very) extension are conjectured to be symmetry groups of maximal supergravity: the former has been employed for a D = 1 coset [34][35][36] while the latter has been used for non-linear realisations of the higher-dimensional theory [7][8][9].Once G +++ /K(G +++ ) has been constructed, we are in the position to oxidize back to 3 ≤ D ≤ D max dimensions using group disintegrations. The valid disintegrations for G +++ are always of the type G D ⊗SL(D, R), where G D is the duality group in D dimensions and SL(D, R) refers to the space-time symme...

We analyze free conformal higher spin actions and the corresponding wave operators in arbitrary even dimensions and backgrounds. We show that the wave operators do not factorize in general, and identify the Weyl tensor and its derivatives as the obstruction to factorization. We give a manifestly factorized form for them on (A)dS backgrounds for arbitrary spin and on Einstein backgrounds for spin 2. We are also able to fix the conformal wave operator in d = 4 for s = 3 up to linear order in the Riemann tensor on generic Bach-flat backgrounds.

We present higher-derivative gravities that propagate an arbitrary number of gravitons of different mass on (A)dS backgrounds. These theories have multiple critical points, at which the masses degenerate and the graviton energies are non-negative. For six derivatives and higher there are critical points with positive energy.