ABSTRACT We investigate plane-parallel wave metrics from the point of view of their (Poisson-Lie) T-dualizability. For that purpose we reconstruct the metrics as backgrounds of nonlinear sigma models on Lie groups. For construction of dual backgrounds we use Drinfel’d doubles obtained from the isometry groups of the metrics. We find dilaton fields that enable to satisfy the vanishing beta equations for the duals of the homogenous plane-parallel wave metric. Torsion potentials or B-fields, invariant w.r.t. the isometry group of Lobachevski plane waves are obtained by the Drinfel’d double construction. We show that a certain kind of plurality, different from the (atomic) Poisson-Lie T-plurality, may exist in case that metrics admit several isometry subgroups having the dimension of the Riemannian manifold. An example of that are two different backgrounds dual to the homogenous plane-parallel wave metric.
Cite this paper
L. Hlavatý and M. Turek, "Nonabelian Dualization of Plane Wave Backgrounds," Journal of Modern Physics, Vol. 3 No. 9, 2012, pp. 1088-1095. doi: 10.4236/jmp.2012.39143.
 C. Klim?ík and P. ?evera, “Dual Non-Abelian Duality and the Drinfeld Double,” Physics Letters B, 1995, pp. 455-462.
 M. A. Lledo and V. S. Varadarajan, “SU(2) Poisson-Lie T-Duality,” Letters in Mathematical Physics, Vol. 45, No. 3, 1998, pp. 247-257. doi:10.1023/A:1007498803198
 K. Sfetsos, “Poisson-Lie T-Duality beyond the Classical Level and the Renormalization Group,” Physics Letters B, Vol. 432, No. 3-4, 1998, pp. 365-375.
 L. Hlavaty and L. ?nobl, “Poisson-Lie T-Plurality of Three- Dimensional Conformally Invariant Sigma Models II: Nondiagonal Metrics and Dilatonpuzzle,” Journal of High Energy Physics, No. 10, 2004.
 G. Papadopoulos, J. G. Russo and A. A. Tseytlin, “Solvable Model of Strings in a Time-Dependent Plane-Wave Background,” Classical and Quantum Gravity, pp. 969- 1016, [hep-th/0211289].
 M. Blau and M. O’Loughlin, “Homogeneous Plane Waves,” Nuclear Physics B, Vol. 652, No. 1-2, 2003, pp. 135-176.
 S. T. C. Siklos, “Lobatchewski Plane Gravitational Waves in Galaxies, Axisymmetric Systems and Relativity,” M. A. H. MacCallum, Ed., Cambridge University Press, Cambridge, 1985, p. 247.
 J. Podolsky, “Interpretation of the Siklos Solutions as Exact Gravitational Waves in the Anti-De Sitter Universe,” Classical and Quantum Gravity, Vol. 15, No. 3, 1998, pp. 719-733. doi:10.1088/0264-9381/15/3/019
 C. Klim?k, “Poisson-Lie T-Duality,” Nuclear Physics A, 1996, pp. 116-121, [hepth9509095].
 R. von Unge, “Poisson-Lie T-Plurality,” Journal of High Energy Physics, 2002, [hepth0205245].
 J. Patera, P. Winternitz and H. Zassenhaus, “Continuous Subgroups of the Fundamental Groups of Physics. I. General Method and the Poincaré Group,” Journal of Mathematical Physics, Vol. 16, No. 8, 1975, pp. 1597-1614.
 V. R. Kaigorodov, “Einstein Spaces of Maximum Mobility,” Soviet Physics Doklady, Vol. 7, 1963, p. 893.