Abstract
ABSTRACT Micropolar continua are shown to be the generalisation of nematic liquid crystals through perspectives of order parameters, topological and geometrical considerations. Micropolar continua and nematic liquid crystals are recognised as antipodals of and in projective geometry. We show that position-dependent rotational axial fields in kinematic micropolar continua can be considered as solutions of anisotropic Higgs fields, characterised by integers N. We emphasise that the identical integers N are topological invariants through homotopy classifications based on defects of order parameters and a finite energy requirement. Magnetic monopoles and Skyrmions are investigated based on the theories of defects of continua in Riemann–Cartan manifolds.
Highlights
1.1 Background and motivationIf we can find a solution space for a given system, there might be a number of solutions that can be transformed continuously around the most stable solution
What physical system shall we put on S3 and RP 3 acted upon by these rotations respectively to see any physical correspondence? And what is the meaning of identifying the antipodals when one brings an actual physical system to the manifold? We will take such a state on S3 as Skyrmions and we will justify that we can put micropolar continua on the projective space RP 3
As an extension of this idea, we took the physical model on S3 as the Skyrmions based on the recognition that the spin-isospin symmetry for the pion field constitutes the transformation of the spinors acted by SU (2)
Summary
If we can find a solution space for a given system, there might be a number of solutions that can be transformed continuously around the most stable solution If these comparable and equivalent solutions form a distinct set under an internal symmetry, we might associate the set of solutions a group structure within the allowed finite energy of the system. One is based on the theory of defects in a given order parameter space and another approach is originated from the boundary conditions of field configurations These integer-valued assignments will yield the topologically invariant quantities through various physical models. We would like to investigate the consequences when we consider arbitrary position-dependent axial fields of SO(3), and its implications to physical systems that contain SO(3) as its symmetry (sub)group in relation with the assignment of the integers when we classify the solution space. We assume indices of vectors are naturally raised and indices of derivatives are naturally lowered, and a metric tensor with its signature (+1, −1, · · · , −1) in n spatial dimensions with one time component for tensors defined in (n + 1)-dimensional differentiable manifolds
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