Covariant action for a D = 11 five-brane with the chiral field

We propose a complete Born-Infeld-like action for a bosonic 5-brane with a worldvolume chiral field in a background of gravitational and antisymmetric gauge fields of D = 11 supergravity. When the five-brane couples to a three-rank antisymmetric gauge field, local symmetries of the five-brane require the addition to the action of an appropriate Wess-Zumino term. To preseve general coordinate and Lorentz invariance of the model we introduce a single auxiliary scalar field. The auxiliary field can be eliminated by gauge fixing a corresponding local symmetry at the price of the loss of manifest d = 6 worldvolume covariance. The double dimensional reduction of the five-brane model results in the Born-Infeld action with the Wess-Zumino term for a D = 10 four-D-brane.