A theory of non-abelian tensor gauge field with non-abelian gauge symmetry G×G

The Chern-Simons action of the ABJM theory is not gauge invariant in the presence of a boundary. In Chu and Smith (2010) [1], this was shown to imply the existence of a Kac-Moody current algebra on the theory of multiple self-dual strings. In this paper we conjecture that the Kac-Moody symmetry induces a U(N)×U(N) gauge symmetry in the theory of N coincident M5-branes. As a start, we construct a G×G gauge symmetry algebra structure which naturally includes the tensor gauge transformation for a non-abelian 2-form tensor gauge field. The gauge covariant field strength is constructed. This new G×G gauge symmetry algebra allows us to write down a theory of a non-abelian tensor gauge field in any dimensions. The G×G gauge bosons can be either propagating, in which case the 2-form gauge fields would interact with each other through the 1-form gauge field; or they can be auxiliary and carry no local degrees of freedom, in which case the 2-form gauge fields would be self-interacting non-trivially. We finally comment on the possible application to the system of multiple M5-branes. We note that the field content of the G×G non-abelian tensor gauge theory can be fitted nicely into (1,0) supermultiplets; and we suggest a construction of the theory of multiple M5-branes with manifest (1,0) supersymmetry.