Bazhanov–Kirillov–Reshetikhin dilogarithm identity for simply laced Y-systems

Let $X_r$ be a simply laced Dynkin diagram of finite type with rank $r$ and index set $I$, and let $ell a geq 2$ be an integer. Let $L(x)$ be the Rogers dilogarithm function. For positive real numbers $Y^{(a)}_m$, with $a i I$ and $1 i m i \ell-1$, assume

$$(Y^{(a)}_m)^2=\frac{\displaystyle\prod_{b:b\sim a}(1+Y^{(b)}_m)}{(1+Y^{(a)}_{m-1}{}^{-1})(1+Y^{(a)}_{m+1}{}^{-1})},$$

where $b\sim a$ means adjacency in $X_r$ and $Y^{(a)}_0{}^{-1}=Y^{(a)}_\ell{}^{-1}=0$ when they occur. Dilogarithm identities. One has

$$\frac{6}{\pi^2}\sum_{a\in I}\sum_{m=1}^{\ell-1}L\left(\frac{Y^{(a)}_m}{1+Y^{(a)}_m}\right)=\frac{\ell\dim\mathfrak{g}}{h+\ell}-r,$$

where $h$ and $\mathfrak{g}$ are respectively the Coxeter number and the simple Lie algebra of type $X_r$. The identity expresses the central charge of the corresponding simply laced conformal field theory in terms of the Rogers dilogarithm. The source describes this family of identities as conjectured by Bazhanov, Kirillov, and Reshetikhin and says they had been partly proved; the paper proves them in full generality.