T-Duality Invariance of the Supermembrane

Abstract

We show that the supermembrane theory compactified on a torus is invariant under Tduality. There are two different topological sectors of the compactified supermembrane (M2) classified according to a vanishing or nonvanishing second cohomology class. We find the explicit T-duality transformation that acts locally on the supermembrane theory and we show that it is an exact symmetry of the theory. We give a global interpretation of the T-duality in terms of bundles. It has a natural description in terms of the cohomology of the base manifold and the homology of the target torus. We show that in the limit when the torus degenerate into a circle and the M2 mass operator restricts to the stringlike configurations, the usual closed string T-duality transformation between the type IIA and type IIB mass operators is recovered. Moreover if we just restrict M2 mass operator to string-like configurations but we perform a generalized T-duality we find the SL(2,Z) non-perturbative multiplet of IIA.

Classification of M2-brane 2-torus bundles, U -duality invariance, and type II gauged supergravities

“…Let us recall [20,57], the relation between the radius modulus and its dual follows from (22), was obtained in:…”

“…with H = H. See [20] for further details. In [57] the authors showed that there always exists a T a parabolic matrix transformation of 'T-duality' given for any arbitrary value of the KK and winding charges. This parabolic transformation depends on the winding and KK momenta of the supermembrane bundle.…”

“…• The parabolic case T p = 1 t 0 1 has already being discussed [57], it corresponds to have a = 1, d = 1 with c = 0 and it is satisfied for any configuration of charges, windings and central charge, so in the following we will analyze the two other possibilities.…”

On global aspects of duality invariant theories: M2-brane vs DFT

“…The gauging of the R + scaling symmetry called trombone gives supergravities without lagrangians [29,30]. At quantum level this last symmetry corresponds to an SL(2, Z) non linearly realized and gives rise to a different symplectic torus bundle in comparison to the previous constructions in terms of linear representations [12,13]. Indeed, only when the monodromy is parabolic, the T-duality action is linear meanwhile for the elliptic and hyperbolic case, T-duality does not commute with the monodromy but forms a unique class of torus bundles in the type IIA side with an scaling symmetry A(1).…”

“…The supermembrane without the central charge condition corresponds to the type II maximal supergravity in nine dimensions. The mass operator and the hamiltonian are U-dual invariant [11] and when supermembrane is dimensionally reduced to string theory T-duality for supermembranes agree with the standard one in string theory compactified on a circle [13]. Moreover, for the supermembrane with nontrivial central charge, it was found all the inequivalent classes of torus bundles with SL(2, Z) monodromy that describes globally the theory, [12].…”