World Sheet Commuting βγ CFT and Non-Relativistic String Theories

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\cite{Gomis:2000bd} and Danielsson et. al. \cite{Danielsson:2000gi, Danielsson:2000mu}. Our theory is much simpler than their theory and does not assume a compact coordinate. This non-relativistic string theory has a bosonic matter $\beta\gamma$ CFT with the conformal weight of $\beta$ as 1. It is natural to identify time as a linear combination of $\gamma$ and $\bar{\gamma}$ through an explicit realization of the Galilean boost symmetry. The angle between $\gamma$ and $\bar{\gamma}$ parametrizes one parameter family of selection sectors. These selection sectors are responsible for having a non-relativistic dispersion relation without a nontrivial topology in the non-relativistic setup, which is one of the major differences from the previous works\cite{Gomis:2000bd, Danielsson:2000gi, Danielsson:2000mu}. This simple theory is the non-relativistic 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 $\beta\gamma$ CFT. We also mention supersymmetric generalizations of these theories.