### 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 -form fields, so our collection of
fields will contain, besides the metric, -form fields, including scalar fields (). The action reads
where we have chosen units such that . The spacetime dimension is left unspecified. The
Einstein metric has Lorentzian signature and is used to lower or raise the indices. Its
determinant is , where the index 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
dilaton,
denoted , whose kinetic term is normalized with weight 1 with respect to the Ricci scalar. The real
parameter measures the strength of the coupling to the dilaton. The other scalar fields, sometimes
called axions, are denoted and have dilaton coupling . The integer
labels the various -forms present in the theory, with field strengths ,
We assume the form degree to be strictly smaller than , since a -form in
dimensions carries no local degree of freedom. Furthermore, if the -form is dual to a scalar
and we impose also .
The field strength, Equation (2.2), could be modified by additional coupling terms of Yang–Mills or
Chapline–Manton type [20, 29] (e.g., for two 2-forms and and a
0-form , 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 -forms. The billiard
derivation given below remains valid no matter what these are; all theories described by the
general action Equation (2.1) lead to the billiard picture. However, it is only for particular
-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 (2.1), contains as particular cases the bosonic sectors of all known supergravity
theories.