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Motivated by the search for potentially exactly solvable time-dependent string backgrounds, we determine all homogeneous plane wave (HPW) metrics in any dimension and find one family of HPWs with geodesically complete metrics and another with metrics containing null singularities. The former generalises both the Cahen-Wallach (constant A ij ) metrics to time-dependent HPWs, A ij (x + ), and the Ozsvath-Schücking anti-Mach metric to arbitrary dimensions. The latter is a generalisation of the known homogeneous metrics with A ij ∼ 1/(x + ) 2 to a more complicated time-dependence. We display these metrics in various coordinate systems, show how to embed them into string theory, and determine the isometry algebra of a general HPW and the associated conserved charges. We review the Lewis-Riesenfeld theory of invariants of time-dependent harmonic oscillators and show how it can be deduced from the geometry of plane waves. We advocate the use of the invariant associated with the extra (timelike) isometry of HPWs for lightcone quantisation, and illustrate the procedure in some examples.

We present a Donaldson-Witten type field theory in eight dimensions on manifolds with Spin(7) holonomy. We prove that the stress tensor is BRST exact for metric variations preserving the holonomy and we give the invariants for this class of variations. In six and seven dimensions we propose similar theories on Calabi-Yau threefolds and manifolds of G 2 holonomy respectively. We point out that these theories arise by considering supersymmetric Yang-Mills theory defined on such manifolds. The theories are invariant under metric variations preserving the holonomy structure without the need for twisting. This statement is a higher dimensional analogue of the fact that Donaldson-Witten field theory on hyper-Kähler 4-manifolds is topological without twisting. Higher dimensional analogues of Floer cohomology are briefly outlined. All of these theories arise naturally within the context of string theory.

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)

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