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2.1 The general action

We are interested in general theories describing Einstein gravity coupled to bosonic “matter” fields. The only known bosonic matter fields that consistently couple to gravity are p-form fields, so our collection of fields will contain, besides the metric, p-form fields, including scalar fields (p = 0). The action reads
∫ [ (p) ] [ (p)] D ∘ -(D)-- μ 1-∑ -eλ--φ-- (p) (p)μ1⋅⋅⋅μp+1 S gμν,φ, A = d x − g R − ∂ μφ∂ φ − 2 (p + 1)!F μ1⋅⋅⋅μp+1F + “more ”(,2.1 ) p
where we have chosen units such that 16πG = 1. The spacetime dimension is left unspecified. The Einstein metric g μν has Lorentzian signature (− ,+, ⋅⋅⋅ ,+ ) and is used to lower or raise the indices. Its determinant is (D ) g, where the index D is used to avoid any confusion with the determinant of the spatial metric introduced below. We assume that among the scalars, there is only one dilaton1, denoted φ, whose kinetic term is normalized with weight 1 with respect to the Ricci scalar. The real parameter λ(p) measures the strength of the coupling to the dilaton. The other scalar fields, sometimes called axions, are denoted A(0) and have dilaton coupling λ(0) ⁄= 0. The integer p ≥ 0 labels the various p-forms (p) A present in the theory, with field strengths (p) (p) F = dA,
(p) (p) Fμ1⋅⋅⋅μp+1 = ∂μ1Aμ2⋅⋅⋅μp+1 ± p permutations. (2.2 )
We assume the form degree p to be strictly smaller than D − 1, since a (D − 1)-form in D dimensions carries no local degree of freedom. Furthermore, if p = D − 2 the p-form is dual to a scalar and we impose also (D−2) λ ⁄= 0.

The field strength, Equation (2.2View Equation), could be modified by additional coupling terms of Yang–Mills or Chapline–Manton type [2029] (e.g., FC = dC (2) − C (0)dB (2) for two 2-forms C (2) and B (2) and a 0-form C (0), as it occurs in ten-dimensional type IIB supergravity), but we include these additional contributions to the action in “more”. Similarly, “more” might contain Chern–Simons terms, as in the action for eleven-dimensional supergravity [38].

We shall at this stage consider arbitrary dilaton couplings and menus of p-forms. The billiard derivation given below remains valid no matter what these are; all theories described by the general action Equation (2.1View Equation) lead to the billiard picture. However, it is only for particular p-form menus, spacetime dimensions and dilaton couplings that the billiard region is regular and associated with a Kac–Moody algebra. This will be discussed in Section 5. Note that the action, Equation (2.1View Equation), contains as particular cases the bosonic sectors of all known supergravity theories.


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