One-loop partition function from normal modes for supergravity in AdS3

Abstract

With the recently discovered formula, which relates the off-shell Euclidean one-loop determinant to the on-shell quantities such as normal and quasinormal modes in real spacetime, we work out the one-loop partition function for supergravity in AdS3 from scratch. In passing, we also provide the explicit expression for the one-loop determinant of a field with an arbitrary spin in AdS3. To achieve this, we first derive the determinant expression for the one-loop partition function in question using the powerful decomposition technique, and then we construct the normal modes in a purely algebraic way by demonstrating that the space of normal modes falls into the representation of SL(2,R) Lie algebra associated with AdS3. The whole procedure developed here turns out to be much simpler than the previous strategy.