  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.
