Quantum Corrections to Supergravity on AdS$_2\times S^2$

We compute the off-shell spectrum of supergravity on AdS$_2\times S^2$ by explicit diagonalization of the equations of motion for an effective AdS$_2$ theory where all fields are dualized to scalars and spin-${1\over 2}$ fermions. Classifying all bulk modes as physical, gauge violating, and pure gauge let us identify boundary modes as physical fields on $S^2$ that are formally pure gauge but with gauge function that is non-normalizable on AdS$_2$. As an application we compute the leading quantum correction to AdS$_2\times S^2$ as a sum over physical fields including boundary states.