Martin O'Loughlin

Homogeneous plane waves

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 Aij) metrics to time-dependent HPWs, Aij(x+), and the Ozsvath–Schucking anti-Mach metric to arbitrary dimensions. The latter is a generalisation of the known homogeneous metrics with Aij∼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.

Solvable models of strings in homogeneous plane-wave backgrounds

We solve closed string theory in all regular homogeneous plane-wave backgrounds with homogeneous NS three-form field strength and a dilaton. The parameters of the model are constant symmetric and anti-symmetric matrices kij and fij associated with the metric, and a constant anti-symmetric matrix hij associated with the NS field strength. In the light-cone gauge the rotation parameters fij have a natural interpretation as a constant magnetic field. This is a generalisation of the standard Landau problem with oscillator energies now being non-trivial functions of the parameters fij and kij. We develop a general procedure for solving linear but non-diagonal equations for string coordinates, and determine the corresponding oscillator frequencies, the light-cone Hamiltonian and level matching condition. We investigate the resulting string spectrum in detail in the four-dimensional case and compare the results with previously studied examples. Throughout we will find that the presence of the rotation parameter fij can lead to certain unusual and unexpected features of the string spectrum like new massless states at non-zero string levels, stabilisation of otherwise unstable (tachyonic) modes, and discrete but not positive definite string oscillator spectra.

Higher-dimensional analogues of Donaldson-Witten theory

Abstract 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-Kahler 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.

Euclidean D-branes and higher-dimensional gauge theory

Abstract We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane—that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory—is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an NT = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G2 holonomy.

Multiple M2-branes and plane waves

We propose a natural generalisation of the BLG multiple M2-brane action to membranes in curved plane wave backgrounds, and verify in two different ways that the action correctly captures the non-trivial space-time geometry. We show that the M2 to D2 reduction of the theory along a non-trivial direction in field space is equivalent to the D2-brane world-volume Yang-Mills theory with a non-trivial (null-time dependent) dilaton in the corresponding IIA background geometry. As another consistency check of this proposal we show that the properties of metric 3-algebras ensure the equivalence of the Rosen coordinate version of this action (time-dependent metric on the space of 3-algebra valued scalar fields, no mass terms) and its Brinkmann counterpart (constant couplings but time-dependent mass terms). We also establish an analogous result for deformed Yang-Mills theories in any dimension which, in particular, demonstrates the equivalence of the Rosen and Brinkmann forms of the plane wave matrix string action.

Aspects of U duality in matrix theory

We explore various aspects of implementing the full M-theory U-duality group Ed+1, and thus Lorentz invariance, in the finite N matrix theory (DLCQ of M-theory) describing toroidal IIA-compactifications on d-tori: (1) We generalize the analysis of Elitzur et al. (hep-th/9707217) from Ed to Ed+1 and identify the highest weight states unifying the momentum and flux Ed-multiplets into one Ed+1-orbit. (2) We identify the new symmetries, in particular the Weyl group symmetry associated to the (d + 1)th node of the Ed+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 Ed+1 for all d and show that in d = 5, 6 it reduces to the black hole entropy cubic E6- and quartic E7-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 1gs3 behaviour.

Self-duality in D{le}8-dimensional Euclidean gravity

In the context of D-dimensional Euclidean gravity, we define the natural generalization to D dimensions of the self-dual Yang-Mills equations as duality conditions on the curvature two-form of a Riemannian manifold. Solutions to these self-duality equations are provided by manifolds of SU(2), SU(3), G{sub 2}, and Spin(7) holonomy. The equations in eight dimensions are a master set for those in lower dimensions. By considering gauge fields propagating on these self-dual manifolds and embedding the spin connection in the gauge connection, solutions to the D-dimensional equations for self-dual Yang-Mills fields are found. We show that the Yang-Mills action on such manifolds is topologically bounded from below, with the bound saturated precisely when the Yang-Mills field is self-dual. These results have a natural interpretation in supersymmetric string theory. {copyright} {ital 1997} {ital The American Physical Society}

1/8 BPS States in AdS/CFT

We study a class of exact supersymmetric solutions of type IIB Supergravity. They have an SO(4) × SU(2) × U(1) isometry and preserve generically 4 of the 32 supersymmetries of the theory. Asymptotically AdS5 × S5 solutions in this class are dual to 1/8 BPS chiral operators which preserve the same symmetries in the = 4 SYM theory. They are parametrized by a set of four functions that satisfy certain differential equations. We analyze the solutions to these equations in a large radius asymptotic expansion: they carry charges with respect to two U(1) KK gauge fields and their mass saturates the expected BPS bound.

The universality of Penrose limits near space-time singularities

We prove that Penrose limits of metrics with arbitrary singularities of power-law type show a universal leading u−2-behaviour near the singularity provided that the dominant energy condition is satisfied and not saturated. For generic power-law singularities of this type the oscillator frequencies of the resulting homogeneous singular plane wave turn out to lie in a range which is known to allow for an analytic extension of string modes through the singularity. The discussion is phrased in terms of the recently obtained covariant characterisation of the Penrose limit; the relation with null geodesic deviation is explained in detail.

Singularities and closed time-like curves in type-IIB 1/2 BPS geometries

We study in detail the moduli space of solutions discovered in LLM relaxing the constraint that guarantees the absence of singularities. The solutions fall into three classes, non-singular, null-singular and time machines with a time-like naked singularity. We study the general features of these metrics and prove that there are actually just two generic classes of space-times — those with null singularities are in the same class as the non-singular metrics. AdS/CFT seems to provide a dual description only for the first of these two types of space-time in terms of a unitary CFT indicating the possible existence of a chronology protection mechanism for this class of geometries.