Solvable model of strings in a time-dependent plane-wave background

TL;DR

The paper presents a solvable time-dependent plane-wave background with metric $ds^2=2du\,dv-\frac{k}{u^2}x^2du^2+dx^2$ and analyzes its geometric, group-theoretic, and string-theoretic structure. By supplementing the background with a $u$-dependent dilaton, it becomes an exact string solution, and the classical string equations reduce to linear, solvable forms via Bessel functions, enabling full light-cone quantization. The work studies scalar and string propagation, showing dilaton-driven suppression of backreaction near the $u=0$ singularity, and demonstrates that string modes can be propagated through the singularity using analytic continuation, with a caveat for zero modes that may require boundary conditions or brane-like sources. Overall, the model provides a controlled setting to probe null cosmology, singularities in string theory, and the role of dilaton in ensuring UV finiteness and smooth evolution across a curvature singularity, offering a platform for exploring global definitions and backreaction in time-dependent backgrounds.

Abstract

We investigate a string model defined by a special plane-wave metric ds^2 = 2dudv - l(u) x^2 du^2 + dx^2 with l(u) = k/u^2 and k=const > 0. This metric is a Penrose limit of some cosmological, Dp-brane and fundamental string backgrounds. Remarkably, in Rosen coordinates the metric has a ``null cosmology'' interpretation with flat spatial sections and scale factor which is a power of the light-cone time u. We show that: (i) This spacetime is a Lorentzian homogeneous space. In particular, like Minkowski space, it admits a boost isometry in u,v. (ii) It is an exact solution of string theory when supplemented by a u-dependent dilaton such that its exponent (i.e. effective string coupling) goes to zero at u=infinity and at the singularity u=0, reducing back-reaction effects. (iii) The classical string equations in this background become linear in the light-cone gauge and can be solved explicitly in terms of Bessel's functions; thus the string model can be directly quantized. This allows one to address the issue of singularity at the string-theory level. We examine the propagation of first-quantized point-particle and string modes in this time-dependent background. Using certain analytic continuation prescription we argue that string propagation through the singularity can be smooth.

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