A note on a gauge-gravity relation and functional determinants

We present a refinement of a recently found gauge-gravity relation between one-loop effective actions: on the gauge side, for a massive charged scalar in 2d dimensions in a constant maximally symmetric electromagnetic field; on the gravity side, for a massive spinor in d-dimensional (Euclidean) anti-de Sitter space. The inclusion of the dimensionally regularized volume of AdS leads to a complete mapping within dimensional regularization. In even-dimensional AdS, we acquire a small correction to the original proposal, whereas in odd-dimensional AdS, the mapping is totally new and subtle, with the 'holographic trace anomaly' playing a crucial role.