Abstract
In this letter we investigate the first law of thermodynamics of the two-dimensional conformal field theory (CFT) that is dual to black holes. We start from the Cardy formula and get the CFT thermodynamics with minimal reasonable assumptions. We use both the microcanonical ensemble and canonical ensemble versions of the Cardy formula. In the black hole/CFT correspondence the black hole is dual to a CFT with excitations, and the black hole mass M and charge N correspond to the energy and charge of the excited CFT. The CFT left- and right-moving central charges cL,R should be quantized, and so we assume that they are mass-independent. Also we assume the difference of the left- and right-moving sector levels NL–NR is mass-independent dual to level matching condition. The thermodynamics of two-dimensional CFT we get is universal and supports the thermodynamics method of black hole/CFT correspondence.
Highlights
In this note we investigate the first law of thermodynamics of the two-dimensional conformal field theory (CFT) that is dual to black holes
Supposing the validity of the canonical ensemble version of the Cardy formula we find that T+S+ = T−S− is equivalent to the equality of the left- and right-moving central charges cL = cR
In this note we investigated the thermodynamics of the 2D CFT that are dual to black holes in the CFT side
Summary
We investigate the thermodynamics in the CFT side and justify the assumptions in the last section. Our starting point is the microcanonical ensemble version of the Cardy formula [20]. NL,R are the levels of the excited state, and TLN,R are the dimensionless temperatures. The validity of (10) needs NL,R to be large NL,R ≫ cL,R, and the validity of (11) needs TLN,R to be large. Supposing that there are regions of parameters where both (10) and (11) are valid, and we can get [21]
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