Membrane Duality Revisited

TL;DR

This paper explores the geometric structures underlying M-theory U-dualities, demonstrating how generalized geometry can manifest these symmetries for M2-branes in specific dimensions, and clarifying the limitations in higher dimensions.

Contribution

It shows how generalized geometry makes U-duality symmetries manifest for M2-branes in certain dimensions, addressing issues with nonintegrability and partial realization of the full symmetry group.

Findings

Generalized geometry renders $SL(3) imes SL(2)$ U-duality manifest for M2-branes in D=3.

In D=4, only a subgroup ${ m GL}(4,R) times R^4$ of $SL(5,R)$ U-duality is realized.

Identifies the nonintegrability problem in U-duality transformations related to the pull-back map.

Abstract

Just as string T-duality originates from transforming field equations into Bianchi identities on the string worldsheet, so it has been suggested that M-theory U-dualities originate from transforming field equations into Bianchi identities on the membrane worldvolume. However, this encounters a problem unless the target space has dimension $D = p + 1$. We identify the problem to be the nonintegrability of the U-duality transformation assigned to the pull-back map. Just as a double geometry renders manifest the $O(D,D)$ string T-duality, here we show in the case of the M2-brane in $D = 3$ that a generalised geometry renders manifest the $SL(3) \times SL(2)$ U-duality. In the case of M2-brane in $D=4$, with and without extra target space coordinates, we show that only the ${\rm GL}(4,R)\ltimes R^4$ subgroup of the expected $SL(5,R)$ U-duality symmetry is realised.