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 ℒ0, ¯ ℒ0 as
( ∘ ------ ∘ ------) cL-ℒ0 cR-¯ℒ0- 𝒮CFT = 2π 6 + 6 , (87 )
when ℒ0 ≫ cL, ¯ℒ0 ≫ cR. 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,
( ) ( ) ∂𝒮CFT-- = -1-, ∂𝒮CFT-- = -1-, (88 ) ∂ℒ0 ¯ℒ0 TL ∂ ¯ℒ0 ℒ0 TR
we get
π2- 2 ¯ π2- 2 ℒ0 = 6 cLTL, ℒ0 = 6 cRT R, (89 )
and, therefore, we obtain an equivalent form of Cardy’s formula,
π2 𝒮CFT = ---(cLTL + cRTR ), (90 ) 3
valid when TL ≫ 1, TR ≫ 1.
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