Fermi Coordinates and Penrose Limits

We propose a formulation of the Penrose plane wave limit in terms of null Fermi coordinates. This provides a physically intuitive (Fermi coordinates are direct measures of geodesic distance in space-time) and manifestly covariant description of the expansion around the plane wave metric in terms of components of the curvature tensor of the original metric, and generalises the covariant description of the lowest order Penrose limit metric itself, obtained in hep-th/0312029. We describe in some detail the construction of null Fermi coordinates and the corresponding expansion of the metric, and then study various aspects of the higher order corrections to the Penrose limit. In particular, we observe that in general the first-order corrected metric is such that it admits a light-cone gauge description in string theory. We also establish a formal analogue of the Weyl tensor peeling theorem for the Penrose limit expansion in any dimension, and we give a simple derivation of the leading (quadratic) corrections to the Penrose limit of AdS_5 x S^5.Comment: 25 page

The Refractive Index of Curved Spacetime II: QED, Penrose Limits and Black Holes

This work considers the way that quantum loop effects modify the propagation of light in curved space. The calculation of the refractive index for scalar QED is reviewed and then extended for the first time to QED with spinor particles in the loop. It is shown how, in both cases, the low frequency phase velocity can be greater than c, as found originally by Drummond and Hathrell, but causality is respected in the sense that retarded Green functions vanish outside the lightcone. A "phenomenology" of the refractive index is then presented for black holes, FRW universes and gravitational waves. In some cases, some of the polarization states propagate with a refractive index having a negative imaginary part indicating a potential breakdown of the optical theorem in curved space and possible instabilities.Comment: 62 pages, 14 figures, some signs corrected in formulae and graph

Penrose Limits and Spacetime Singularities

We give a covariant characterisation of the Penrose plane wave limit: the plane wave profile matrix $A(u)$ is the restriction of the null geodesic deviation matrix (curvature tensor) of the original spacetime metric to the null geodesic, evaluated in a comoving frame. We also consider the Penrose limits of spacetime singularities and show that for a large class of black hole, cosmological and null singularities (of Szekeres-Iyer ``power-law type''), including those of the FRW and Schwarzschild metrics, the result is a singular homogeneous plane wave with profile $A(u)\sim u^{-2}$, the scale invariance of the latter reflecting the power-law behaviour of the singularities.Comment: 9 pages, LaTeX2e; v2: additional references and cosmetic correction

String Interactions in PP-Waves

We argue that string interactions in a PP-wave spacetime are governed by an effective coupling g_{eff}=g_s(\mu p^+\apm)f(\mu p^+ \apm) where f(\mu p^+ \apm) is proportional to the light cone energy of the string states involved in the interaction. This simply follows from generalities of a Matrix String description of this background. $g_{eff}$ nicely interpolates between the expected result ($g_s$) for flat space (small \mu p^+\apm) and a recently conjectured expression from the perturbative gauge theory side (large \mu p^+\apm).Comment: 8 page

Geometry of Schroedinger Space-Times II: Particle and Field Probes of the Causal Structure

We continue our study of the global properties of the z=2 Schroedinger space-time. In particular, we provide a codimension 2 isometric embedding which naturally gives rise to the previously introduced global coordinates. Furthermore, we study the causal structure by probing the space-time with point particles as well as with scalar fields. We show that, even though there is no global time function in the technical sense (Schroedinger space-time being non-distinguishing), the time coordinate of the global Schroedinger coordinate system is, in a precise way, the closest one can get to having such a time function. In spite of this and the corresponding strongly Galilean and almost pathological causal structure of this space-time, it is nevertheless possible to define a Hilbert space of normalisable scalar modes with a well-defined time-evolution. We also discuss how the Galilean causal structure is reflected and encoded in the scalar Wightman functions and the bulk-to-bulk propagator.Comment: 32 page

Goedel, Penrose, anti-Mach: extra supersymmetries of time-dependent plane waves

We prove that M-theory plane waves with extra supersymmetries are necessarily homogeneous (but possibly time-dependent), and we show by explicit construction that such time-dependent plane waves can admit extra supersymmetries. To that end we study the Penrose limits of Goedel-like metrics, show that the Penrose limit of the M-theory Goedel metric (with 20 supercharges) is generically a time-dependent homogeneous plane wave of the anti-Mach type, and display the four extra Killings spinors in that case. We conclude with some general remarks on the Killing spinor equations for homogeneous plane waves.Comment: 20 pages, LaTeX2

N=(4,4) Type IIA String Theory on PP-Wave Background

We construct IIA GS superstring action on the ten-dimensional pp-wave background, which arises as the compactification of eleven-dimensional pp-wave geometry along the isometry direction. The background geometry has 24 Killing spinors and among them, 16 components correspond to the non-linearly realized kinematical supersymmetry in the string action. The remaining eight components are linearly realized and shown to be independent of x^+ coordinate, which is identified with the world-sheet time coordinate of the string action in the light-cone gauge. The resultant dynamical N=(4,4) supersymmetry is investigated, which is shown to be consistent with the field contents of the action containing two free massive supermultiplets.Comment: latex, 15 pages; v2: typos corrected, polished, references adde

Penrose Limits of the Baryonic D5-brane

The Penrose limits of a D5-brane wrapped on the sphere of AdS_5 x S^5 and connected to the boundary by M fundamental strings, which is dual to the baryon vertex of the N=4 SU(M) super Yang-Mills theory, are investigated. It is shown that, for null geodesics that lead to the maximally supersymmetric Hpp-wave background, the resulting D5-brane is a 1/2-supersymmetric null brane. For an appropriate choice of radial geodesic, however, the limiting configuration is 1/4-supersymmetric and closely related to the Penrose limit of a flat space BIon.Comment: LaTeX, 1+18 pages, 1 figure; v2: obvious misquotation of the number of preserved supersymmetries correcte

Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation, previously employed for trivial bundles $\Sigma \times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $\Sigma$ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.Comment: 37 pages; v2: references adde

The Conformal Penrose Limit and the Resolution of the pp-curvature Singularities

We consider the exact solutions of the supergravity theories in various dimensions in which the space-time has the form M_{d} x S^{D-d} where M_{d} is an Einstein space admitting a conformal Killing vector and S^{D-d} is a sphere of an appropriate dimension. We show that, if the cosmological constant of M_{d} is negative and the conformal Killing vector is space-like, then such solutions will have a conformal Penrose limit: M^{(0)}_{d} x S^{D-d} where M^{(0)}_{d} is a generalized d-dimensional AdS plane wave. We study the properties of the limiting solutions and find that M^{(0)}_{d} has 1/4 supersymmetry as well as a Virasoro symmetry. We also describe how the pp-curvature singularity of M^{(0)}_{d} is resolved in the particular case of the D6-branes of D=10 type IIA supergravity theory. This distinguished case provides an interesting generalization of the plane waves in D=11 supergravity theory and suggests a duality between the SU(2) gauged d=8 supergravity of Salam and Sezgin on M^{(0)}_{8} and the d=7 ungauged supergravity theory on its pp-wave boundary.Comment: 20 pages, LaTeX; typos corrected, journal versio