Towards 2+4 formulation of M5-brane

TL;DR

This paper explores a 2+4 splitting approach to formulate an action for the M5-brane's worldvolume theory, revealing challenges in maintaining Lorentz invariance and coupling to curved space, with a Hamiltonian analysis confirming correct degrees of freedom.

Contribution

It introduces a novel 2+4 splitting framework for the M5-brane, constructing related chiral tensor actions and analyzing their gauge and geometric properties.

Findings

The 2+4 split breaks manifest Lorentz invariance from SO(1,5) to SO(1,1)×SO(4).

The modified diffeomorphism in curved space is more complex than in other formulations.

The Hamiltonian analysis confirms the correct degrees of freedom and algebraic structure.

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.