Characters, quasinormal modes, and Schwinger pairs in $dS_2$ with flux

An integral representation of the 1-loop partition function for charged scalars and spinors, minimally coupled to a uniform $U(1)$ field on $S^2$, is given in terms of $SO(1,2)$ Harish-Chandra group characters and evaluated exactly in terms of Hurwitz $\zeta$-functions. Analytically continuing the $U(1)$ field, we interpret the path integrals as quasicanonical partition functions in $dS_2$ with an electric field. The character itself is obtained as a trace over states living at the future boundary of de Sitter and has a quasinormal mode expansion. The imaginary part of the partition function captures Schwinger pair creation in the static patch at finite temperature. The thermal enhancement is most noticeable for scalar masses below Hubble and leads to non-monotonicity of the current as a function of the field. This parameter range, when dimensionally reducing from a charged or rotating Nariai spacetime, is excluded by Swampland-inspired bounds. Around the $AdS_2$ black hole, in contrast to $dS_2$, there is a threshold to pair creation.