World sheet commutingβγconformal field theory and nonrelativistic string theories

Abstract

We construct a sigma model in two dimensions with Galilean symmetry in flat target space similar to the sigma model of the critical string theory with Lorentz symmetry in 10 flat spacetime dimensions. This is motivated by the works of Gomis and Ooguri [J. Math. Phys. (N.Y.) 42, 3127 (2001)] and Danielsson et al. [J. High Energy Phys. 10 (2000) 020; J. High Energy Phys. 03 (2001) 041.]. Our theory is much simpler than their theory and does not assume a compact coordinate. This nonrelativistic string theory has a bosonic matter $\ensuremath{\beta}\ensuremath{\gamma}$ conformal field theory with the conformal weight of $\ensuremath{\beta}$ as 1. It is natural to identify time as a linear combination of $\ensuremath{\gamma}$ and $\overline{\ensuremath{\gamma}}$ through an explicit realization of the Galilean boost symmetry. The angle between $\ensuremath{\gamma}$ and $\overline{\ensuremath{\gamma}}$ parametrizes one parameter family of selection sectors. These selection sectors are responsible for having a nonrelativistic dispersion relation without a nontrivial topology in the nonrelativistic setup, which is one of the major differences from the previous works of Gomis and Ooguri and of Danielsson and co-workers. This simple theory is the nonrelativistic analogue of the critical string theory, and there are many different avenues ahead to be investigated. We mention a possible consistent generalization of this theory with different conformal weights for the $\ensuremath{\beta}\ensuremath{\gamma}$ conformal field theory. We also mention supersymmetric generalizations of these theories.