Relation between the curvature-induced potential term and supersymmetric sine-Gordon theory

Determine whether the additional curvature-dependent term proportional to $\sin(\phi/2)$ in the deformed sine-Gordon equation is related to the corresponding fermion-dependent term in the supersymmetric sine-Gordon theory, potentially through a curvature-induced expectation value of the operator $\bar\psi\psi$.

Background

The deformed scalar equation contains an additional term proportional to sin⁡(ϕ/2)\sin(\phi/2), which the paper interprets as naturally arising from coupling to spacetime curvature. The supersymmetric sine-Gordon action contains a Yukawa interaction proportional to ψˉψcos⁡(ϕ/2)\bar\psi\psi\cos(\phi/2).

The authors speculate that, if curved spacetime dynamically generates an expectation value of ψˉψ\bar\psi\psi proportional to curvature, the scalar equation could acquire the observed extra term. Establishing this relationship would provide a supersymmetric interpretation of the curvature deformation.

References

Is there a relation between the extra term in the potential and the extra term that appears in the supersymmetric generalization of the sine-Gordon theory , with action:

— One more Sine-Gordon soliton in AdS  (2608.19859 - Diakonov, 20 Aug 2026) in Section 1, Introduction

One of our conjectures is that this class of theories can be embedded in the supersymmetric generalization of the sine-Gordon theory in $AdS$ spacetime, such that $q\sim \bar{\psi} \psi$ on such single solition solution.

— One more Sine-Gordon soliton in AdS  (2608.19859 - Diakonov, 20 Aug 2026) in Section 4, Conclusion and acknowledgments