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A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian Gauge Symmetry G x G

The Chern-Simon action of the ABJM theory is not gauge invariant in the presence of a boundary. In the paper arXiv:0909.2333, this was shown to imply the existence of a Kac-Moody symmetry on the theory of multiple self-dual strings. In this paper we conjecture that the Kac-Moody symmetry induces a U(N)x U(N) gauge symmetry in the theory of N coincident M5-branes. As a start, we construct this G x G gauge symmetry algebra structure which naturally includes the tensor gauge transformation for a non-abelian 2-form tensor gauge field. This new G x G gauge structure allows us to write down a theory of a non-abelian tensor gauge field readily. For application to multiple M5-branes, we note that the field content of the G x G non-abelian tensor gauge theory can be fitted nicely as (1,0) supermultiplets; and we suggest a construction of the theory of multiple M5-branes with manifest (1,0) supersymmetry.