M Theory from F Theory

Abstract

We write down a $GL(D+1)$ (D for the dimension of string theory) manifest fundamental brane worldvolume current algebra description of M theory, which consists of a pair of vector field $X^m$ and dual 2-form field $X_{mn}$, compositing together to parametrize the spacetime, with a selfduality condition for sectioning. The worldvolume of the brane itself is a (D+2) dimensional object, and the background spacetime after sectioning has dimension (D+1). We summarize the features of the algebra. The field contents of the corresponding background geometry, the usual vielbein $e_a{}^m$ and the 3-form $A_{mnp}$, could be identified as different blocks of the composite spacetime vielbein, by solving the orthogonality condition. Their behaviour under gauge transformation are also determined by the corresponding rules of the composite vielbein. Then by solving F theory $\mathcal{V}$ constraints, we reduce the number of worldvolume and show how to recover M theory from F theory.