Minimal curves in six dimensions

Abstract

Classical strings and minimal surfaces in a flat spacetime correspond to pairs of 'minimal' (i.e. null) curves in its complexification. The authors show how to find null curves in dimension six by use of twistor methods. The twistor space for C6 is the space of pure spinors for C8. They review the geometry of this space and establish a correspondence between 'pure' curves in twistor space and null ruled 2-surfaces in C6. This correspondence generates an explicit parameterisation of null curves in C6, thereby generalising the Weierstrass-Eisenhart parametrisation of null curves in C3 or C4. They also describe covariant methods for finding the periodic real null curves for string theory in real 6-space of any signature.