Twisted index theory on orbifold symmetric products and the fractional quantum Hall effect

Abstract

We extend the noncommutative geometry model of the fractional quantum Hall effect, previously developed by Mathai and the first author, to orbifold symmetric products. It retains the same properties of quantization of the Hall conductance at integer multiples of the fractional Satake orbifold Euler characteristics. We show that it also allows for interesting composite fermions and anyon representations, and possibly for Laughlin type wave functions.

Anyons in geometric models of matter

“…In the approach of [12] to geometric models of matter, the signature τ (M) is interpreted as a baryon number, while the electric charge is determined by the self-intersection number of the surface at infinity. The Euler characteristic, on the other hand, does not play a direct role as a quantum number in the geometric models of [12], unlike what typically happens in geometric models of the quantum Hall effect, where it is related to the noncommutative Kubo formula for the transport coefficient ( [19], [24], [74], [75], [76], [77]). In the more recent work of the first author and Nick Manton [10], for models of matter based on algebraic surfaces, baryon and lepton numbers are expressed in terms of both signature and Euler characteristic, with the signature measuring the difference between the number of protons and the number of neutrons.…”

“…In the case of a 4-dimensional orbifold geometry (M, Σ) with M reg = M Σ and 2-dimensional M sing = Σ, one can replace the braid groups B n (M reg ) = π 1 (M reg ) ≀ S n with the orbifold braid group as in [77], (3.9) B orb n (M) = π orb 1 (M) ≀ S n = π orb 1 (M) n ⋊ S n , with π orb 1 (M) as in (3.1). For example, in the case of the orbifold structure on M reg = S 4 RP 2 given by the Hitchin metrics of [45] with cone angle 2π/(k − 2), the orbifold braid groups are B orb n (M) = S n for k even and B orb n (M) = Z/2Z ≀ S n for k odd, while for the Atiyah-LeBrun orbifold structures on M reg = S 4 S 2 with cone angle 2π/ν the orbifold braid groups are B orb n (M) = Z/νZ ≀ S n .…”

“…It is well known that anyons arise only in two-dimensional systems, due to the topology of configuration spaces that allows for interesting braid groups, see [47], [60], [67]. Indeed, anyons and representations of braid groups have been considered in relation to quantum Hall systems, [49], [50], [77]. While anyons do not arise in higher dimensions, we will see that the presence of the 2-dimensional surfaces Σ of orbifold points allows for the existence of non-trivial anyon states and interesting braid group representations associated to the 4-dimensional orbifold geometries that describe our quasi-particle systems.…”

“…Orbifold braid groups. In the case of a 4-dimensional orbifold geometry (M, Σ) with M reg = M Σ and 2-dimensional M sing = Σ, one can replace the braid groups B n (M reg ) = π 1 (M reg ) ≀ S n with the orbifold braid group as in [77],…”

Fractional quantum numbers via complex orbifolds

“…We mention alternate approaches to (fractional) quantum numbers on hyperbolic space. For smooth surfaces [12,11,10], orbifolds [23,22,21] bulk-boundary correspondence [25], orbifold symmetric products [24]. These papers use operator algebras and noncommutative geometry methods, in contrast to the holomorphic geometry methods used in this paper.…”