N = 2 Supersymmetric Yang–mills and the Quantum Hall Effect

Abstract

It is argued that there are strong similarities between the infra-red physics of N=2 supersymmetric Yang-Mills and that of the quantum Hall effect, both systems exhibit a hierarchy of vacua with a sub-group of the modular group mapping between them. The coupling flow for pure SU (2) N = 2 supersymmetric Yang-Mills in 4-dimensions is reexamined and an earlier suggestion in the literature, that was singular at strong coupling, is modified to a form that is well behaved at both weak and strong coupling and describes the crossover in an analytic fashion. Similarities between the phase diagram and the flow of SUSY Yang-Mills and that of the quantum Hall effect are then described, with the Hall conductivity in the latter playing the role of the θparameter in the former. Hall plateaux, with odd denominator filling fractions, are analogous to fixed points at strong coupling in N=2 SUSY Yang-Mills, where the massless degrees of freedom carry an odd monopole charge.

Meromorphic scaling flow of supersymmetric Yang–Mills with matter

“…In terms of τ the relevant sub-groups of Sl(2, Z) for determining the β-functions are: Γ 0 (2) for N f = 0; Γ(1) for N f = 1; Γ 0 (2) for N f = 2; and Γ 0 (4) for N f = 3. 1 For N f = 0 and N f = 2 these are larger than the monodromy group. The N f = 1 case realises the full modular group, so the self-dual point τ = i is a fixed point of the element sending τ → − 1 τ which is in the monodromy group.…”

“…Two appendices give a summary of the conventions concerning Jacobi ϑ-functions and the technical aspects of the strong coupling instanton expansion used for the analysis in section 6. 2. N f = 0 This case was treated in [1] using the normalisation appropriate to the adjoint representation of SU(2),τ = θ 2π + 4πi g 2 , but when matter in the fundamental representation of SU(2) is included it is better to define τ = θ π + 8πi g 2 , [9]. In order to set the notation and illustrate the method for N f = 0 the derivation of the β-function in [1] is given here using the original techniques of [2], adapted to the present notation.…”

“…However it was pointed out in [6] that (2.10) has a pathology in that it is singular at strong coupling, in particular at τ = 0 (u = Λ 2 0 ) and τ = 2 (u = −Λ 2 0 ). A remedy was proposed in [1], based on [4]. The idea is to tame the singularities at strong coupling, without disturbing the asymptotic properties at u → ∞, by defining…”

Holomorphic and anti-holomorphic conductivity flows in the quantum Hall effect

“…This holomorphic flow was compared with experimental data for temperature flow in the integral quantum effect in [31] and the fractional effect in [32]: figure 2 is taken from [31] and figures 3 and 4 are from [32]. For a short review see [36].…”

Modular Symmetry and Fractional Charges in N = 2 Supersymmetric Yang-Mills and the Quantum Hall Effect

“…It is further argued in [48] that the correct form of the Callan-Symanzik β-function at τ = i∞, τ = 0 and τ = 1, up to a constant factor, is obtained by using the scaling function…”