Abstract
This paper considers the higher derivative terms in the effective action of type II string theory and in particular the behaviour of the automorphic forms they contain in all the different possible limits of the string parameters. The automorphic forms are thought to obey Poisson equations which contain the Laplacian defined on the coset space to which the scalars fields belong and we compute this Laplacian in all the possible string theory limits. We also consider these Poisson equations in the decompactification limit of a single dimension and by making two assumptions, one on the generic form of this equation and the other on the behaviour of the automorphic forms in this limit, we find strong constraints on the allowed form of this differential equation. We show that these constraints allow one to recover much of what was previously known about the automorphic forms corresponding to terms in the effective action that have fourteen or fewer space-time derivatives in a simple way.
Highlights
This paper considers the higher derivative terms in the effective action of type II string theory and in particular the behaviour of the automorphic forms they contain in all the different possible limits of the string parameters
Quite a number of these effects have been checked against known string corrections and this provides both strong evidence for these automorphic forms and strong evidence that the SL(2,R) symmetry of the IIB supergravity theory [4] really is a symmetry of string theory when discretised to SL(2,Z)
In this paper we have studied the behaviour of string theory in all possible limits of its parameters
Summary
We review how the parameters arise in string theory as discussed in [30] but we will use a slightly different definition of the parameters. In reference [30] we used the Planck length in d dimensions ld, but in this paper we have used the Planck length in the decompactified theory which leads to slightly different expressions in equations (2.4)–(2.9) in terms of the Chevalley fields. One may label the En+1 Dynkin diagram in terms of the parameters resulting from the dimensional reductions of both the type IIA and type IIB theories as shown in figures 1 and 2. The relations between the parameters of the d dimensional type IIA, type IIB string theories and M-theory may be derived through the dependence of the parameters on the Chevalley fields, for further details see reference [30].
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