U-duality transformation of membrane on $T^{n}$ revisited

Understand

The problem with the U-duality transformation of membrane on $T^{n}$ is recently addressed in [arXiv:1509.02915 [hep-th]].

• We will consider the U-duality transformation rule of membrane on $T^{n}\times R$.
• It turns out that winding modes on $T^{n}$ should be taken into account, since the duality transformation may bring the membrane configuration without winding modes into the one with winding modes.
• With the winding modes added, the membrane worldvolume theory in lightcone gauge is equivalent to the $n+1$ dimensional super-Yang-Mills (SYM) theory in $\tilde{T}^{n}$, which has $SL(2,Z)\times SL(3,Z)$ and $SL(5,Z)$ symmetries for $n=3$ and $n=4$, respectively.