  We prove upper and lower bounds on the free energy in the
Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins in
terms of the variational inequalities based on the corresponding Parisi
functional. We employ the comparison scheme of Aizenman, Sims and Starr and the
one of Guerra involving the generalised random energy model-inspired processes
and Ruelle's probability cascades. For this purpose an abstract quenched large
deviations principle of the Gaertner-Ellis type is obtained. Using the
properties of Ruelle's probability cascades and the Bolthausen-Sznitman
coalescent, we derive Talagrand's representation of the Guerra remainder term
for our model. We study the properties of the multidimensional Parisi
functional by establishing a link with a certain class of the non-linear
partial differential equations. Solving a problem posed by Talagrand, we show
the strict convexity of the local Parisi functional. We prove the Parisi
formula for the local free energy in the case of the multidimensional Gaussian
a priori distribution of spins using Talagrand's methodology of the a priori
estimates.
