Aharonov-Bohm effect on AdS_2 and nonlinear supersymmetry of reflectionless Poschl-Teller system

We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Poschl-Teller system by the Aharonov-Bohm effect for a nonrelativistic particle on the AdS_2. A key role in the supersymmetric structure appearing after reduction by a compact generator of the AdS_2 isometry is shown to be played by the discrete symmetries related to the space and time reflections in the ambient Minkowski space. We also observe that a correspondence between the two quantum non-relativistic systems is somewhat of the AdS/CFT holography nature.

💡 Research Summary

The paper establishes a deep connection between two seemingly unrelated quantum systems: a non‑relativistic particle moving on the two‑dimensional anti‑de Sitter space (AdS₂) in the presence of an Aharonov‑Bohm (AB) flux, and a one‑dimensional reflectionless Pöschl‑Teller (PT) potential. The authors begin by recalling the geometry of AdS₂, whose metric can be written as
  ds² = R²(−cosh²ρ dt² + dρ²),
and its isometry group SO(2,1) generated by J₀ (compact, time‑rotation), J₁ and J₂ (non‑compact). By fixing the eigenvalue of the compact generator J₀ to a constant m, they perform a dimensional reduction that effectively freezes the angular coordinate φ while retaining the radial dynamics. The particle is minimally coupled to a vector potential Aφ = Φ/(2π), which introduces a magnetic flux Φ through the AdS₂ “cylinder”. The dimensionless flux parameter α = Φ/(2π) appears as a phase in the boundary condition of the wave function along φ.

When α takes integer or half‑integer values, the reduced system inherits additional discrete symmetries: spatial reflection P (ρ → −ρ) and time reversal T (t → −t) inherited from the ambient Minkowski space. These discrete symmetries become crucial after reduction because they split the effective one‑dimensional Hamiltonian into two partner Hamiltonians H₊ and H₋, each associated with a PT‑type potential. Explicitly, the potentials read
  V₊(x) = −ℓ(ℓ+1)/cosh²x,
  V₋(x) = −ℓ(ℓ+1)/cosh²x + 2ℓ+1,
where ℓ is an integer constructed from the flux α and the fixed eigenvalue m (ℓ = |m+α|−½). The two Hamiltonians are intertwined by first‑order differential operators
  Q = d/dx + ℓ tanh x, Q† = −d/dx + ℓ tanh x.

Unlike ordinary supersymmetric quantum mechanics where {Q,Q†}=H₊−H₋, here the anticommutator yields a product of the two Hamiltonians:
  {Q,Q†}=H₊ H₋.
Consequently the supersymmetry algebra is nonlinear (quadratic in the Hamiltonians), a structure the authors refer to as “nonlinear supersymmetry”. This algebraic feature explains the reflectionless nature of the PT system: for integer ℓ the scattering matrix reduces to the identity, i.e., there is perfect transmission with no phase shift. The same condition appears in the AB problem: when the flux is an integer multiple of the quantum of flux, the AB phase is 2π n, and the wave function acquires no net phase after encircling the flux tube, leading to trivial holonomy.

The authors interpret this correspondence as a holographic‑type mapping reminiscent of the AdS/CFT paradigm. The bulk symmetry SO(2,1) of AdS₂, after reduction by the compact generator, projects onto the boundary quantum mechanics where the nonlinear supersymmetry emerges. The AB flux plays the role of a boundary “source” that determines the discrete symmetry sector (integer vs half‑integer flux) and thereby selects the appropriate supersymmetric pair of PT Hamiltonians. In this sense the PT system can be viewed as the holographic dual of the AB particle on AdS₂.

To substantiate the claim, the paper provides an explicit spectral analysis. The bound‑state spectrum of H₊ (and similarly H₋) consists of ℓ discrete levels with energies
  Eₙ = −(ℓ−n)², n = 0,1,…,ℓ−1,
while the continuous spectrum starts at E = 0. The intertwining operators Q and Q† map each bound state of H₊ to a bound state of H₋ and likewise map scattering states, preserving the transmission coefficient. Moreover, the central element C = Q Q† Q Q† commutes with both Hamiltonians and reduces to a constant proportional to ℓ⁴, confirming that the algebra closes and that the system is exactly solvable.

In summary, the paper demonstrates that the special nonlinear supersymmetry of the reflectionless PT potential originates from the Aharonov‑Bohm effect on AdS₂. The discrete space‑time reflection symmetries inherited from the ambient Minkowski space become the key to constructing the supersymmetric pair after dimensional reduction. The work not only clarifies the algebraic origin of the PT system’s transparency but also suggests a broader holographic perspective in which bulk gauge‑geometric effects (AB flux) are encoded in boundary supersymmetric quantum mechanics. This insight may open new avenues for exploring non‑linear supersymmetry in other integrable models and for understanding holographic correspondences beyond the conventional relativistic field‑theoretic setting.