Quantum deformations of the flat space superstring

Abstract

We discuss a quantum deformation of the Green-Schwarz superstring on flat space, arising as a contraction limit of the corresponding deformation of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$. This contraction limit turns out to be equivalent to a previously studied limit that yields the so-called mirror model---the model obtained from the light cone gauge fixed ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ string by a double Wick rotation. Reversing this logic, the ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ superstring is the double Wick rotation of a quantum deformation of the flat space superstring. This quantum deformed flat space string realizes symmetries of the timelike $\ensuremath{\kappa}$-Poincar\'e type and is T dual to ${\mathrm{dS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{H}}^{5}$, indicating interesting relations between symmetry algebras under T duality. Our results directly extend to ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{T}}^{6}$ and ${\mathrm{AdS}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{T}}^{4}$ and beyond string theory to many (semi)symmetric space coset sigma models, such as a deformation of the four-dimensional Minkowski sigma model with timelike $\ensuremath{\kappa}$-Poincar\'e symmetry. We also discuss possible null and spacelike deformations.