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

TL;DR

This paper revisits the U-duality transformation of membranes on tori, emphasizing the importance of winding modes and their role in connecting membrane configurations to super-Yang-Mills theories with specific duality symmetries.

Contribution

It clarifies the U-duality transformation rules for membranes on $T^{n}$, incorporating winding modes, and explores the realization of duality symmetries at classical and quantum levels.

Findings

Winding modes are essential for correct U-duality transformations.

Membrane worldvolume theory maps to super-Yang-Mills with specific duality groups.

Classical $SL(2,Z)\times SL(3,Z)$ transformations are realizable, but $SL(5,Z)$ may only appear at the quantum level.

Abstract

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. The $SL(2,Z)\times SL(3,Z)$ transformation can be realized classically, making the on-shell field configurations transformed into each other. However, the $SL(5,Z)$ symmetry may only be realized at the quantum level, since the classical $5d$ SYM field configurations cannot form the representation of $SL(5,Z)$.