Three-dimensional N=8 conformal supergravity and its coupling to BLG M2-branes

This paper is concerned with the problem of coupling the N=8 superconformal Bagger-Lambert-Gustavsson (BLG) theory to N=8 conformal supergravity in three dimensions. We start by constructing the on-shell N=8 conformal supergravity in three dimensions consisting of a Chern-Simons type term for each of the gauge fields: the spin connection, the SO(8) R-symmetry gauge field and the spin 3/2 Rarita-Schwinger (gravitino) field. We then proceed to couple this theory to the BLG theory. The final theory should have the same physical content, i.e., degrees of freedom, as the ordinary BLG theory. We discuss briefly the properties of this "topologically gauged" BLG theory and why this theory may be useful.