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Paper   IPM / Physic / 11552
School of Physics
  Title:   Beyond Logarithmic Corrections to the Cardy Formula
1.  F. Loran
2.  M.M. Sheikh-Jabbari
3.  M. Vincon
  Status:   Published
  Journal: JHEP
  Vol.:  1101
  Year:  2011
  Pages:   110
  Supported by:  IPM
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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