Classical propagation of strings across a big crunch/big bang singularity

One of the simplest time-dependent solutions of M theory consists of nine-dimensional Euclidean space times $1+1$-dimensional compactified Milne space-time. With a further modding out by $Z_2$, the space-time represents two orbifold planes which collide and re-emerge, a process proposed as an explanation of the hot big bang [J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Phys. Rev. D 64, 123522 (2001).][P. J. Steinhardt and N. Turok, Science 296, 1436 (2002).][N. Turok, M. Perry, and P. J. Steinhardt, Phys. Rev. D 70, 106004 (2004).]. When the two planes are near, the light states of the theory consist of winding M2-branes, describing fundamental strings in a particular ten-dimensional background. They suffer no blue-shift as the M theory dimension collapses, and their equations of motion are regular across the transition from big crunch to big bang. In this paper, we study the classical evolution of fundamental strings across the singularity in some detail. We also develop a simple semiclassical approximation to the quantum evolution which allows one to compute the quantum production of excitations on the string and implement it in a simplified example.