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

TL;DR

This paper proposes a novel framework where gauge theories and string theories are encoded in a special invariant function on the infinite-dimensional group manifold of $E_{10}(R)$, linking various theories through Fourier transforms and harmonic properties.

Contribution

It introduces the idea that gauge and string theories are represented by a unique invariant function on $E_{10}(R)$, connecting partition functions and dualities in a unified infinite-dimensional group setting.

Findings

Partition function of N=4 SYM on $T^4$ derived from the Fourier transform of the proposed function.

The function encodes answers to M-theory questions on $T^8$ and heterotic string theories on $T^7$.

The conjecture that this function is harmonic with respect to the $E_{10}(R)$ metric.

Abstract

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 $T^7$.