Universal BPS Structure of Scalar Kinks in Static Geometries

Abstract

We present a geometric extension of the Bogomolny-Prasad-Sommerfield (BPS) construction for scalar kinks in (1+1) dimensions embedded in static curved spacetimes. By introducing a nonminimal coupling between the scalar prepotential and the extrinsic curvature of the static foliation, the flat-space first-order Bogomolny equation remains exactly valid for arbitrary static backgrounds. As a consequence, the kink profile is unchanged, while the effective potential and vacuum structure acquire a controlled geometric dependence. We show that these curved-space BPS kinks are always linearly stable. However, the existence of the translational zero mode is not guaranteed: its normalizability depends on the competition between the intrinsic length scale of the kink and the asymptotic curvature scale of the geometry. When the geometric scale dominates, the zero mode is removed and the soliton becomes geometrically pinned, despite remaining an exact BPS solution. Explicit realizations in AdS2 demonstrate how different static slicings of the same spacetime lead to qualitatively distinct physical outcomes, ranging from preserved translational invariance to its complete removal by horizons. These results establish geometry as a precise mechanism for controlling solitonic moduli without compromising linear stability.

Figures (5)

• Figure 1: Geometric interpretation of a static BPS configuration in curved spacetime. A preferred spatial path, defined by the unit normal vector of a chosen static foliation, is selected so that the soliton can remain static despite curvature.
• Figure 2: Geometric deformation of the scalar potential induced by curvature. A nonminimal coupling between the scalar prepotential and the extrinsic curvature reshapes the effective potential in a position-dependent manner, while preserving the flat-space BPS equation.
• Figure 3: Global static foliation of the $\mathrm{AdS}_2$ hyperboloid and BPS kink profile for $h(x)=\cosh(\kappa x)$. The translational zero mode remains normalizable despite the absence of translational isometries.
• Figure 4: Horizon-type slicing of the $\mathrm{AdS}_2$ hyperboloid with $h(x)=\sinh(\kappa x)$ and the corresponding BPS kink profile. Although the kink remains an exact solution of the Bogomolny equation, the presence of a horizon truncates the spatial domain.
• Figure 5: Poincaré slicing of the $\mathrm{AdS}_2$ hyperboloid defined by $h(x)=e^{\kappa x}$ and the associated BPS kink profile. Geometrically, this slicing covers one half of the $\mathrm{AdS}_2$ hyperboloid, extending from an asymptotic boundary to a Killing horizon associated with the chosen static foliation.