Fractons are anyons classified into equivalence classes and they obey a specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension h. We consider this approach in the context of the Fractional Quantum Hall Effect ( FQHE ) and the concept of duality between such classes, defined byh = 3 − h shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes h and the modular group for the quantum phase transitions of the FQHE is also obtained. A β−function is defined for a complex conductivity which embodies the classes h. The thermodynamics is also considered for a gas of fractons (h, ν) with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes h.
We introduce the notion of fractal index associated with the universal class h of particles or quasiparticles, termed fractons, which obey specific fractal statistics. A connection between fractons and conformal field theory(CFT)quasiparticles is established taking into account the central charge c[ν] and the particle-hole duality ν ←→ 1 ν , for integer-value ν of the statistical parameter.In this way, we derive the Fermi velocity in terms of the central charge as v ∼ c[ν] ν+1 . The Hausdorff dimension h which labelled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.
We consider Farey series of rational numbers in terms of fractal sets labeled by the Hausdorff dimension with values defined in the interval 1 < h < 2 and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that the universality classes of the quantum Hall transitions are classified in terms of h. The connection between Number Theory and Physics appears naturally in this context.
We consider the fractal von Neumann entropy associated with the fractal distribution function and we obtain for some universal classes h of fractons their entropies. We obtain also for each of these classes a fractal-deformed Heisenberg algebra. This one takes into account the braid group structure of these objects which live in two-dimensional multiply connected space.
The invariant integration method for Chern-Simons theory for gauge group SU (2) and manifold Γ\H 3 is verified in the semiclassical approximation. The semiclassical limit for the partition function associated with a connected sum of hyperbolic 3-manifolds is presented. We discuss briefly L 2 − analytical and topological torsions of a manifold with boundary.
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