### 3.1 Cardy’s formula

In any unitary and modular invariant CFT, the asymptotic growth of states in the microcanonical
ensemble is determined only by the left and right central charge and the left and right eigenvalues ,
as
when , . This is known as Cardy’s formula derived originally in [61, 48] using modular
invariance of the CFT. A review can be found, e.g., in [62]. Transforming to the canonical ensemble using
the definition of the left and right temperatures,
we get
and, therefore, we obtain an equivalent form of Cardy’s formula,
valid when , .