Abstract
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) = I ), biaxial nematic liquid (P i (3) = D 2 ), 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) σ model and its relation to the Wess-Zumino term of the SO(3)/P i (3) σ 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 η in an “electromagnetic” formalism. The connection 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.
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