Two Conjectures on Gauge Theories, Gravity, and Infinite Dimensional Kac-Moody Groups

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 E10(R). The function is invariant under the maximal compact subgroup K acting on the right and under the discrete U-duality subgroup E10(Z) on the left. The manifold E10(Z)\E10(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 E10(R) invariant metric. In a similar fashion, we propose that there exists a function on the infinite Kac-Moody group DE18 that encodes the twisted partition functions of the E8 5+1D theories as well as answers to questions about the heterotic string on T 7. [email protected]