Eisenstein series for infinite-dimensional U-duality groups

TL;DR

This paper extends the definition of Eisenstein series to all E_n duality groups, including infinite-dimensional Kac-Moody groups, revealing finite constant terms that relate to perturbative string corrections in BPS sectors.

Contribution

It introduces Eisenstein series for infinite-dimensional Kac-Moody groups E9, E10, and E11, and demonstrates their finite constant terms, linking to string theory perturbative corrections.

Findings

Finite constant terms in Kac-Moody Eisenstein series for specific parameters.

Connection between constant terms and perturbative string corrections.

Analysis of physical degeneration limits in various dimensions.

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 E9, E10 and E11. 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.