= Casimir energy of a two-dimensional conformal field theory
{title2=$E_0=-c/12$}
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 energy>
$$
E_0=-\frac{\pi\hbar v}{12L}(c_L+c_R).
$$
When $c_L=c_R=c$, the unit-radius convention $L=2\pi$, $\hbar=v=1$ 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 $H=(2\pi\hbar v/L)[L_0+\bar L_0-(c_L+c_R)/24]$. 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>.<String Theory by David Tong>[Section 4.4.1, cylinder/plane transformation and the Casimir shift; section 4.4.3, vacuum Hamiltonian normalization]{ref}
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