TOPOLOGICAL PROPERTIES OF STATIC AND DYNAMIC DEFECT CONFIGURATIONS IN ORDERED LIQUIDS

The topology of linked disclinations is studied in uniaxial nematic liquids and in anisotropic liquids with an order parameter space SO(3)/P(i)(3). In these models {P(i)(3)} are the finite point symmetry groups of 3-space with applications to helium 3 (P(i)(3) = 1), a semi-classical approximation to the spin-1/2 Heisenberg antiferromagnet (SO(3) spin liquid, P(i)(3) = 1), biaxial nematic liquid (P(i)(3) = D2), and anisotropic super cooled liquids (P(i)(3) = groups of Platonic solids). The topological properties are studied via Hopf's invariant of the O(3) sigma-model and its relation to the Wess-Zumino term of the SO(3)/P(i)(3) sigma-model in an orthonormal drei-bein representation of SO(3). Dynamic processes are topology changing during intersection of disclinations and are studied via "magnetic" N-pole singularities and the instanton number-eta in an "electromagnetic" formalism. The connection with with tunneling amplitudes in a SO(3) Yang-Mills theory is indicated. Applications of the theory to topological fluid dynamics is worked out for the uniaxial nematic liquid and indicated for the SO(3) spin liquid.