Gauge Symmetries from Nambu-Poisson Brackets

The gauge symmetry of volume-preserving diffeomorphism (VPD) is generated by the Nambu-Poisson bracket. We define the gauge potentials and field strengths, including their gauge transformation rules, in generic dimensions. Field strengths of other gauge symmetries are to be modified if they are required to be covariant under VPD. As an example, we con-sider the presence of an additional U(1) symmetry and construct its modified field strength. Interestingly, the gauge algebras of U(1) and VPD are mixed. In the two dimensional case when VPD means area-preserving diffeomorphism, the gauge symmetry can be deformed by the Moyal product. An extension of the deformed area-preserving diffeomorphism allows one to find the electromagnetic dual of the non-commutative U(1) gauge theory. 1