In the context of two-time physics in $4+2$ dimensions we construct the most general $N=2$, 4 supersymmetric Yang-Mills gauge theories for any gauge group G. This builds on our previous work for $N=1$ supersymmetry (SUSY). The action, the conserved SUSY currents, and the SU(N) covariant SUSY transformation laws are presented for both $N=2$ and $N=4$. When the equations of motion are used the SUSY transformations close to the supergroup $\mathrm{SU}(2,2|N)$ with $N=1$, 2, 4. The $\mathrm{SU}(2,2)=\mathrm{SO}(4,2)$ subsymmetry is realized linearly on $4+2$ dimensional flat spacetime. All fields, including vectors and spinors, are in $4+2$ dimensions. The extra gauge symmetries in 2T field theory, together with the kinematic constraints that follow from the action, remove all the ghosts to give a unitary theory. By choosing gauges and solving the kinematic equations, the 2T field theory in $4+2$ flat spacetime can be reduced to various shadows in various $3+1$ dimensional (generally curved) spacetimes. These shadows are related to each other by dualities. The conformal shadows of our theories in flat $3+1$ dimensions coincide with the well known counterpart $N=1$, 2, 4 supersymmetric massless renormalizable field theories in $3+1$ dimensions. It is expected that our more symmetric new structures in $4+2$ spacetime may be useful for nonperturbative or exact solutions of these theories.

• Received 6 August 2008

DOI:https://doi.org/10.1103/PhysRevD.79.025001