Quantum Corrections to Central Charges and Supersymmetric Casimir Energy in AdS$_3$/CFT$_2$

We study the Casimir energy of bulk fields in AdS$_3$ and its relation to subleading terms in the central charge of the dual CFT$_2$. Computing both sides of the standard CFT$_2$ relation $E=-c/12$ independently we show that this relation is not necessarily satisfied at the level of individual bulk supergravity states, but in theories with sufficient supersymmetry it is restored at the level of bulk supermultiplets. Assuming only $(0,2)$ supersymmetry (or more), we improve the situation by relating quantum corrections to the central charge and the supersymmetric Casimir energy which in turn is related to an index. These relations adapt recent progress on the AdS$_5$/CFT$_4$ correspondence to AdS$_3$/CFT$_2$ holography. We test our formula successfully in several examples, including the $(0,4)$ MSW theory describing classes of 4D black holes and the large $(4,4)$ theory that is interesting for higher spin holography. We also make predictions for the subleading central charges in several recently proposed $(2,2)$ dualities where the CFT$_2$ is not yet well-understood.