  We propose that the structure of gauge theories, the $(2,0)$ and
little-string theories is encoded in a unique function on the real group
manifold $E_{10}(R)$. The function is invariant under the maximal compact
subgroup $K$ acting on the right and under the discrete U-duality subgroup
$E_{10}(Z)$ on the left. The manifold $E_{10}(Z)\backslash E_{10}(R) /K$
contains an infinite number of periodic variables. The partition function of
U(n), N=4 Super-Yang-Mills theory on $T^4$, with generic SO(6) R-symmetry
twists, for example, is derived from the $n^{th}$ coefficient of the Fourier
transform of the function with respect to appropriate periodic variables,
setting other variables to the R-symmetry twists and the radii of $T^4$. In
particular, the partition function of nonsupersymmetric Yang-Mills theory is a
special case, obtained from the twisted $(2,0)$ or little-string theories. The
function also seems to encode the answer to questions about M-theory on an
arbitrary $T^8$. The second conjecture that we wish to propose is that this
function is harmonic with respect to the $E_{10}(R)$ invariant metric. In a
similar fashion, we propose that there exists a function on the infinite
Kac-Moody group $DE_{18}$ that encodes the twisted partition functions of the
$E_8$ 5+1D theories as well as answers to questions about the heterotic string
on $T^7$.
