SL(2, Z) symmetries, supermembranes and symplectic torus bundles

Abstract

We give the explicit formulation of the 11D supermembrane as a symplectic torus bundle with non trivial monodromy and non vanishing Euler class. This construction allows a classification of all supermembranes showing explicitly the discrete SL(2, Z) symmetries associated to dualities. It hints as the origin in M-theory of the gauging of the effective theories associated to string theories.

T-Duality Invariance of the Supermembrane

“…It is characterized by having a trivial class of H 2 (Σ 1 , Z). The Mass operator of the compactified supermembrane with winding and KK contribution [4], [3], is…”

“…In the following all transformed quantities under T-duality are denoted by a tilde. In order to define the T-duality transformation we introduce the following [4](47) dimensionless variables…”

“…where T 11 is the supermembrane tension, A = (2πR) 2 Imτ is the area of the target torus and Y = RImτ |qτ −p| . The tilde variables A, Y are the transformed quantities under the T-duality.…”

“…We notice that (p, q) may be interpreted as the wrapping of the membrane around the two cycles of the target torus. The corresponding change in the harmonic sector is [4] dX h = (qmd…”

“…As a new result, we show how in the String Theory limit, the T-duality transformation for the supermembrane becomes the standard T-duality transformation of the closed superstrings compactified on a circle. The compactified supermembrane on a target space M 9 × T 2 may be formulated in terms of sections on symplectic torus bundles [2]. There are two well-defined sectors: one in which a topological condition due to an irreducible wrapping is imposed, that corresponds to the so-called supermembrane with central charges [5], and a second one without that restriction.…”