Automorphic forms: a physicist’s survey

Automorphic forms play an important rôle in physics, especially in the realm of string and M-theory dualities. Notably, the hidden symmetries of 11-dimensional supergravity compactifications, discovered by Cremmer and Julia, motivate the study of automorphic forms for exceptional arithmetic groups En(Z) (including their n ≤ 5 classical A andD instances). These Notes are a pedestrian introduction to these (seemingly abstract) mathematical objects, designed to offer a concrete footing for physicists. The basic concepts are introduced via the simple Sl(2) Eisenstein and Jacobi theta series. The general construction of continuous representations and of their accompanying Eisenstein series is detailed for Sl(3). Thereafter we present nilpotent representations and their theta series for arbitrary simply-laced groups based on our recent work with D. Kazhdan [1]. We include new results on the geometrical interpretation of minimal representations, implying e.g. that E6 acts on SO(5, 5) pure spinors. We close with some comments about the physical applications of automorphic forms which motivated our research. 3