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| Paper IPM / P / 11552 |
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| Abstract: | |||||||
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As shown by Cardy [], modular invariance of the partition function
of a given unitary non-singular 2d CFT with left and right central
charges cL and cR, implies that the density of states in a
microcanonical ensemble, at excitations ∆ and ―∆ and in the
saddle point approximation, is ρ0(∆,―∆; cL, cR)=cL exp(2π√{cL∆/6} ) ·cR exp(2π√{cR―∆ / 6} ). In this paper, we extend Cardy's analysis and show that up to
contributions which are exponentially suppressed compared to the
leading Cardy's result, the density of states takes the form ρ(∆,―∆; cL, cR) = f(cL∆) f(cR―∆)ρ0 (∆,―∆; cL, cR), for a function f(x) which we specify. In
particular, we show that (i) ρ(∆,―∆; cL, cR)
is the product of contributions of left and right movers and hence, to this
approximation, the partition function of any modular invariant, non-singular
unitary 2d CFT is holomorphically factorizable, (ii)
ρ(∆,―∆; cL, cR)/(cLcR) is only a function of
cL∆ and cR―∆ and, (iii) treating ρ(∆,―∆; cL, cR) as the density of states of a microcanonical
ensemble, we compute the entropy of the system in the canonical counterpart
and show that the function f(c∆) is such that the canonical entropy,
up to exponentially suppressed contributions, is simply given by the
Cardy's result lnρ0(∆,―∆; cL, cR).
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