Special Lagrangian cones in C^3 and primitive harmonic maps

It is shown that every special Lagrangian cone in C3 determines, and is determined by, a primitive harmonic surface in the 6-symmetric space SU3/SO2. For cones over tori, this allows the classification theory of harmonic tori to be used to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some of the examples found recently by Joyce are put into this context