  We obtain the analogue of the Crisanti-Sommers variational formula for
spherical spin glasses with vector spins. This formula is derived from the
discrete Parisi variational formula for the limit of the free energy of
constrained copies of spherical spin glasses. In vector spin models, the
variations of the functional order parameters must preserve the monotonicity of
matrix paths which introduces a new challenge in contrast to the derivation of
the classical Crisanti-Sommers formula.

Probability