  In a previous paper we conjectured that the structure of various gauge
theories as well as M-theory on $T^8$ is encoded in a unique function $\Xi$ on
the coset $E_{10}(Z)\backslash E_{10}(R)/K$ and that this function is harmonic
with respect to the $E_{10}(R)$ invariant metric. In this paper we elaborate on
the conjecture. We discuss various mass deformations of the D-instanton
integral and their realizations in $\Xi$. We then present a conjectured
prescription for extracting partition functions of the twisted little-string
theory out of $\Xi$. We also study various effects of combinations of branes
such as D0-branes near D4-branes with 2-form flux, D-instantons near Taub-NUT
metrics, and more, in terms of harmonic functions on $E_d(R)/K$. We propose
tests of the conjecture that are related to BPS states of global symmetries in
gauge theories.
