Abstract
We show that the “geometric models of matter” approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.
Highlights
It was shown in [12] that certain classes of 4-dimensional Riemannian manifolds with self-dual Weyl tensor behave in many ways like elementary particles, and can be used to provide geometric models of matter
We show that the “geometric models of matter” approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces
The geometric models of matter originally proposed in [12] have more recently been extended to models of nuclear physics and beta decay, in work of the first author and Nick Manton [10], using algebraic surfaces as geometric models of nuclei, with lepton and baryon numbers related to the topological invariants c2 and c21 and to the Enriques-Kodaira classification of compact complex surfaces
Summary
It was shown in [12] that certain classes of 4-dimensional Riemannian manifolds with self-dual Weyl tensor behave in many ways like elementary particles, and can be used to provide geometric models of matter These manifolds include gravitational instantons like the Taub-NUT manifold [80, 88] or the Atiyah-Hitchin manifold [9], as well as compact manifolds like CP2 or S4. In other recent work of the first author, [2], related to the famous question on the existence of complex structures on the sphere S6, odd and even modules for the quaternion group of order eight are considered, where the odd modules are faithful quaternionic representations, with value −1 on the center, while the even ones descend to abelian modules and have value +1 on the center We expect that these odd and even types will play a role in the geometric models of matter, where they may be related to topological insulators. We plan to investigate further possible connections to topological insulators and quantum computing aspects of the present work
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