Aspects of U-duality in matrix theory

Abstract

We explore various aspects of implementing the full M-theory U-duality group E d+1 , and thus Lorentz invariance, in the finite N matrix theory (DLCQ of M-theory) describing toroidal IIA-compactifications on dtori: (1) We generalize the analysis of Elitzur et al. (hep-th/9707217) from E d to E d+1 and identify the highest weight states unifying the momentum and flux E d -multiplets into one E d+1 -orbit. (2) We identify the new symmetries, in particular the Weyl group symmetry associated to the (d + 1)'th node of the E d+1 Dynkin diagram, with Nahm-duality-like symmetries (N-duality) exchanging the rank N of the matrix theory gauge group with other (electric, magnetic, . . . ) quantum numbers. (3) We describe the action of N-duality on BPS bound states, thus making testable predictions for the Lorentz invariance of matrix theory. (4) We discuss the problems that arise in the matrix theory limit for BPS states with no top-dimensional branes, i.e. configurations with N = 0. (5) We show that N-duality maps the matrix theory SYM picture to the matrix string picture and argue that, for d even, the latter should be thought of as an M-theory membrane description (which appears to be well defined even for d > 5). (6) We find a compact and unified expression for a U-duality invariant of E d+1 for all d and show that in d = 5, 6 it reduces to the black hole entropy cubic E 6 -and quartic E 7 -invariants respectively. (7) Finally, we describe some of the solitonic states in d = 6, 7 and give an example (a 'rolled-up' Taub-NUT 6-brane) of a configuration exhibiting the unusual 1/g 3 s -behaviour.1 E 3(3) = SL(3) × SL(2), E 4(4) = SL(5), E 5(5) = SO(5, 5)

Effective actions of nongeometric five-branes

“…These stringy symmetries lead to new extended states, which were discovered and classified in Refs. [3][4][5]. These were revisited recently in Ref.…”

“…Moreover, it was soon realized that the interplay between dualities and branes leads to a whole new family of extended objects, with Dp, NS5 and KKM being just a fraction of it [3][4][5][6]. These new BPS-objects have codimension 2 and are obtained as orbits of the U-duality group.…”

“…Further understanding emerges from the treatment in double field theory [31], also for analogous issue regarding the 5 2 2 -brane [12]. 4 The notation is explained in section 2.…”

A microscopical description of Giant gravitons

“…The fields C (2n−1) denote the R-R (2n − 1)-form potentials, B the Kalb-Ramond field and B its Hodge dual, the NS-NS 6-form potential. The field N (7) is the field that couples to the Type IIA Kaluza-Klein monopole, and N (8) to the so-called (6, 1 2 ; 3)-brane 3 , an "exotic" brane (not predicted by the Type IIA spacetime supersymmetry algebra), first mentioned in [14,15,16]. The currents I give the couplings between the Matrix (string) theory and the linearised background fields [11,12,13] and are functions of the matrix valued transverse coordinates X i , the world volume scalar A and their derivatives 4 .…”

Non-geometric five-branes in heterotic supergravity

“…The Uduality [1], which makes non-trivial connections among consistent superstring theories, relates various branes in each theory. It was shown that M-theory compactified on T d has the U-duality symmetry group E d(d) (R) in lower dimensions [2][3][4]. BPS branes in superstring theories form lower-dimensional multiplets under the U-duality group.…”