Casimir energy of a two-dimensional conformal field theory ($E_0=-c/12$)

For a two-dimensional conformal field theory on a spatial circle of circumference $L$, with propagation speed $v$ and left/right central charges $c_L,c_R$, the plane-vacuum sector has Casimir energyWhen $c_L=c_R=c$, the unit-radius convention $L=2\pi$, $\hslash =v=1$ gives $E_0=-c/12$. A circle of unit circumference instead gives $E_0=-\pi c/6$ in these units. Specifying radius versus circumference prevents a factor-of-$2\pi$ ambiguity.

The exponential map from a cylinder to the complex plane shifts the stress-energy tensor by its Schwarzian derivative term. Equivalently, the cylinder Hamiltonian is $H=(2\pi \hslash v/L)[L_0+{\bar{L}}_0-(c_L+c_R)/24]$. Setting both plane-vacuum weights to zero yields the formula. Other sectors can have different lowest weights. In AdS/CFT correspondence, the unit-radius result matches the renormalized vacuum energy of global three-dimensional Anti-de Sitter spacetime.^[ref]