Covariant M5-brane action with self-dual 3-form

In this work, we extend a theory which describes a linearly self-dual exact 3-form field in six dimensional spacetime within the formalism of arXiv:1511.08220 and arXiv:1903.12196 to an action which fully describes an M5-brane in the eleven-dimensional supergravity background. The action we obtain has all the required symmetries including kappa-symmetry. Although the derivation of the action and the proof the symmetries do not require one to express the action explicitly in terms of the original field contents, we also give an initial attempt to give this expression.