K(E9) from K(E10)

TL;DR

This paper explores the structure and representations of the infinite-dimensional symmetry group K(E9) in the context of reduced supergravity, revealing how certain finite-dimensional holonomy groups emerge as quotients of K(E9).

Contribution

It provides a detailed analysis of K(E9) representations, their relation to ideals, and how finite holonomy groups arise as quotients, extending previous work on K(E10).

Findings

Finite-dimensional holonomy groups are quotients of K(E9).

Unfaithful representations relate to ideals within K(E9).

Evaluation maps connect K(E9) quotients to supergravity fields.

Abstract

We analyse the M-theoretic generalisation of the tangent space structure group after reduction of the D=11 supergravity theory to two space-time dimensions in the context of hidden Kac-Moody symmetries. The action of the resulting infinite-dimensional `R symmetry' group K(E9) on certain unfaithful, finite-dimensional spinor representations inherited from K(E10) is studied. We explain in detail how these representations are related to certain finite codimension ideals within K(E9), which we exhibit explicitly, and how the known, as well as new finite-dimensional `generalised holonomy groups' arise as quotients of K(E9) by these ideals. In terms of the loop algebra realisations of E9 and K(E9) on the fields of maximal supergravity in two space-time dimensions, these quotients are shown to correspond to (generalised) evaluation maps, in agreement with previous results of Nicolai and Samtleben (hep-th/0407055). The outstanding question is now whether the related unfaithful representations of K(E10) can be understood in a similar way.