On the Sugawara Current Algebra Proposal for M-Theory

TL;DR

This paper investigates the possibility of formulating M-theory using a Sugawara-type current algebra based on $E_{11} imes_s l_1$, analyzing its feasibility and mathematical consistency.

Contribution

It demonstrates a construction of such a current algebra for a rigid $E_{11}$ model with generalized coordinates treated as inert, and discusses issues with the bilinear form extension.

Findings

Constructed a Sugawara-type current algebra for a rigid $E_{11}$ model.

Identified degeneracy issues in extending the Cartan-Killing form to $E_{11} imes_s l_1$.

Highlighted differences between the model and E-theory regarding generalized coordinates.

Abstract

We examine the proposal of [29] that M-theory may admit a Sugawara-type current algebra formulation based on $E_{11} \otimes_s l_1$. Motivated by the role of generalized coordinates in E-theory, we ask whether current algebra relations of this type can be derived in a setting that includes those coordinates systematically. We show that such a construction can indeed be carried out for a rigid $E_{11}$ model in which the generalized coordinates are treated as inert under the rigid symmetry, in contrast with E-theory. We also argue that the bilinear form entering the Schwinger term requires closer scrutiny, since any natural ad-invariant extension of the $E_{11}$ Cartan-Killing form to $E_{11} \otimes_s l_1$ is degenerate.