One-Loop Effective Action and Schwinger Effect in (Anti-) de Sitter Space

We study the Schwinger mechanism by a uniform electric field in ${\mathrm{d}\mathrm{S}}_2$ and ${\mathrm{A}\mathrm{d}\mathrm{S}}_2$ and the curvature effect on the Schwinger effect, and further propose a thermal interpretation of the Schwinger formula in terms of the Gibbons-Hawking temperature and the Unruh temperature for an accelerating charge in ${\mathrm{d}\mathrm{S}}_2$ and an analogous expression in ${\mathrm{A}\mathrm{d}\mathrm{S}}_2$. The exact one-loop effective action is found in the proper-time integral in each space, which is determined by the effective mass, the Maxwell scalar, and the scalar curvature, and whose pole structure gives the imaginary part of the effective action and the exact pair-production rate. The exact pair-production rate is also given the thermal interpretation.