Supersymmetric Yang-Mills Theory in Eleven Dimensions

TL;DR

This paper develops a Lorentz-invariant formulation of supersymmetric Yang-Mills theory in eleven dimensions, introducing a space-like vector to maintain covariance at the lagrangian level and exploring its implications for supermembrane actions.

Contribution

It presents a novel Lorentz-invariant lagrangian for 11D supersymmetric Yang-Mills theory using a space-like vector, with implications for supermembrane models.

Findings

Lorentz symmetry is spontaneously broken at the field equation level.

The formulation reproduces 10D SYM equations with additional constraints.

A $ ext{-symmetric supermembrane action with SYM backgrounds is constructed.

Abstract

We present a Lorentz invariant lagrangian formulation for a supersymmetric Yang-Mills vector multiplet in eleven dimensions (11D). The Lorentz symmetry is broken at the field equation level, and therefore the breaking is spontaneous, as in other formulations of supersymmetric theories in 12D or higher dimensions. We introduce a space-like unit vector formed by the gradient of a scalar field, avoiding the problem of Lorentz non-invariance at the lagrangian level, which is also an analog of non-commutative geometry with constant field strengths breaking Lorentz covariance. The constancy of the space-like unit vector field is implied by the field equation of a multiplier field. The field equations for the physical fields are formally the same as those of 10D supersymmetric Yang-Mills multiplet, but now with some constraints on these fields for supersymmetric consistency. This formulation also utilizes the multiplier fields accompanied by the bilinear forms of constraints, such that these multiplier fields will not interfere with the physical field equations. Based on this component result, we also present a $\k$-symmetric supermembrane action with the supersymmetric Yang-Mills backgrounds.