Nonstandard signature of spacetime, superstrings, and the split composition algebras

Abstract

It has been established that the covering group of the Lorentz group in the dimensions D=3, 4, 6 and 10, can be expressed in a unified way, based on the four composition-division algebras ℝ, ℂ, ℚ and $\mathbb{O}$. If the division algebras in the construction of the covering groups of the Lorentz groups in D=3, 4, 6 and 10 are replaced by the split composition algebras, then the sequence of groups $\widetilde{{\text{SO}}}{\text{(2,2)}}$, $\widetilde{{\text{SO}}}{\text{(3,3)}}$ and $\widetilde{{\text{SO}}}{\text{(5,5)}}$ results. Classical superstrings embedded in such spacetimes can be defined, and the split composition algebras provide a natural framework for their description.