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Paper IPM / P / 11552  


Abstract:  
As shown by Cardy [], modular invariance of the partition function
of a given unitary nonsingular 2d CFT with left and right central
charges c_{L} and c_{R}, implies that the density of states in a
microcanonical ensemble, at excitations ∆ and ―∆ and in the
saddle point approximation, is ρ_{0}(∆,―∆; c_{L}, c_{R})=c_{L} exp(2π√{c_{L}∆/6} ) ·c_{R} exp(2π√{c_{R}―∆ / 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(∆,―∆; c_{L}, c_{R}) = f(c_{L}∆) f(c_{R}―∆)ρ_{0} (∆,―∆; c_{L}, c_{R}), for a function f(x) which we specify. In
particular, we show that (i) ρ(∆,―∆; c_{L}, c_{R})
is the product of contributions of left and right movers and hence, to this
approximation, the partition function of any modular invariant, nonsingular
unitary 2d CFT is holomorphically factorizable, (ii)
ρ(∆,―∆; c_{L}, c_{R})/(c_{L}c_{R}) is only a function of
c_{L}∆ and c_{R}―∆ and, (iii) treating ρ(∆,―∆; c_{L}, c_{R}) 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 lnρ_{0}(∆,―∆; c_{L}, c_{R}).
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