On the Conformal Field Theories for Bosonic Strings in PP-Waves

TL;DR

This paper demonstrates that massless normal ordering (MNO) is the correct choice for bosonic strings in pp-wave backgrounds, as it yields the proper spectrum and consistent Virasoro algebra, unlike phase space normal ordering (PNO).

Contribution

The paper provides a covariant analysis showing MNO's validity over PNO for bosonic strings in pp-waves, clarifying normal ordering choices in curved backgrounds.

Findings

MNO produces the correct string spectrum in pp-wave backgrounds.

PNO leads to a spectrum with negative dimensions.

The quadratic space-time action is non-diagonal and reproduces known spectra.

Abstract

Recently Kazama and Yokoi (arXiv:0801.1561 [hep-th]) have used a phase-space method to study the Virasoro algebra of type IIB superstring theory in the maximally supersymmetric R-R plane wave background in a semi-light-cone gauge. Two types of normal ordering have been considered, namely "phase space normal ordering" (PNO) and "massless normal ordering" (MNO). The second one, which is the right one to choose in flat background, has been discarded with the argument that the Virasoro algebra closes only in the first case. To understand this issue better with a completely covariant treatment we consider the easiest case of bosonic strings propagating in an arbitrary pp-wave of the simplest kind. Using the phase-space method we show that MNO is in fact the right one to choose because of the following reason. For both types of normal ordering the energy-momentum tensor satisfies the desired Virasoro algebra up to anomalous terms proportional to the space-time equation of motion of the background. However, it is MNO which gives rise to the correct spectrum - we compute the quadratic space-time action by restricting the string field inside a transverse Hilbert space. This turns out to be non-diagonal. Diagonalizing this action reproduces the spectrum directly obtained in light-cone quantization. The same method with PNO gives rise to a spectrum with negative dimensions.