Kac--Moody groups and automorphic forms in low dimensional supergravity theories

TL;DR

This paper reviews the conjectural role of Kac--Moody groups as symmetry groups in low-dimensional supergravity and string theories, discussing their constructions, Eisenstein series, and potential quantum correction encodings.

Contribution

It introduces the conjectural symmetries of Kac--Moody groups in supergravity, details their constructions over real and integer rings, and extends Eisenstein series concepts to these infinite-dimensional groups.

Findings

Kac--Moody groups are conjectured to be symmetries in supergravity.

Eisenstein series on Kac--Moody groups relate to string theory corrections.

Constant terms of these series encode perturbative corrections.

Abstract

Kac--Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravity theories dimensionally reduced to dimensions less than 3, and their integral forms $G(\mathbb{Z})$ conjecturally encode quantized symmetries. In this review paper, we briefly introduce the conjectural symmetries of Kac--Moody groups in supergravity as well as the known evidence for these conjectures. We describe constructions of Kac--Moody groups over $\R$ and $\Z$ using certain choices of fundamental modules that are considered to have physical relevance. Eisenstein series on certain finite dimensional algebraic groups are known to encode quantum corrections in the low energy limit of superstring theories. We describe briefly how the construction of Eisenstein series extends to Kac--Moody groups. The constant terms of Eisenstein series on $E_9$, $E_{10}$ and $E_{11}$ are predicted to encode perturbative string theory corrections.