One-loop partition function from normal modes for N = supergravity in AdS ₃

With the recently discovered formula, which relates the off-shell Euclidean oneloop determinant to the on-shell quantities such as normal and quasinormal modes in real spacetime, we work out the one-loop partition function for N = 1 supergravity in AdS ₃ from scratch. In passing, we also provide the explicit expression for the one-loop determinant of a field with an arbitrary spin in AdS ₃. 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 AdS ₃. The whole procedure developed here turns out to be much simpler than the previous strategy.