Towards 2+4 formulation of M5-brane

Abstract: We present an attempt to formulate an action for the worldvolume theory of a single M5-brane, based on the splitting of the six worldvolume directions into 2+4, which breaks manifest Lorentz invariance from $SO(1,5)$ to $SO(1,1)\times SO(4)$. To this end, an action for the free six--dimensional (2,0) chiral tensor multiplet, and separately, a nonlinearly interacting chiral 2-form action are constructed. By studying the Lagrangian formulation for the chiral 2-form with 2+4 splitting, it is suggested that, if exists, the modified diffeomorphism of the theory on curved six--dimensional space--time is less trivial than its 1+5 and 3+3 counterpart, thus hindering the coupling of the chiral 2-form to the induced metric on the worldvolume of the M5-brane. We discuss difficulties of further generalisation of the theory. Finally, in terms of Hamiltonian analysis, we show that the naively gauge-fixed failed-PST-covariantised Lagrangian has the correct number of degrees of freedom, and satisfies the hyper--surface deformation algebra.