Abstract
Abstract We consider Eisenstein series appearing as coefficients of curvature corrections in the low-energy expansion of type II string theory four-graviton scattering amplitudes. We define these Eisenstein series over all groups in the E n series of string duality groups, and in particular for the infinite-dimensional Kac-Moody groups E 9, E 10 and E 11. We show that, remarkably, the so-called constant term of Kac-Moody-Eisenstein series contains only a finite number of terms for particular choices of a parameter appearing in the definition of the series. This resonates with the idea that the constant term of the Eisenstein series encodes perturbative string corrections in BPS-protected sectors allowing only a finite number of corrections. We underpin our findings with an extensive discussion of physical degeneration limits in D < 3 space-time dimensions.
Highlights
Contained in the continuous symmetries of the low energy supergravities
The so-called constant term of Kac-Moody-Eisenstein series contains only a finite number of terms for particular choices of a parameter appearing in the definition of the series
In the present paper we have considered the perturbative sector of type II superstring fourgraviton scattering amplitudes in D dimensions that are expected to be invariant under the discrete Ed+1 duality groups of table 1 with d = 10 − D
Summary
Let us fix some general notation used in this paper. We denote the Lie algebra of a group G by g and the set of roots of the algebra is denoted by ∆. As already explained in the introduction, the duality groups appearing in reductions of type II string theory are discrete versions of the Ed+1 groups, which we will denote by Ed+1(Z) and take to be the associated Chevalley groups [45, 46]. These can be thought of as being generated by the integer exponentials of the (real root) generators of Ed+1 in the Chevalley basis. A convenient construction of parabolic subalgebras is obtained by selecting a subset Π1 of the set of simple roots Π. Where Mi∗ and Ni∗ are the groups associated with the subalgebras mi∗ and ni∗
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