Geometry and String Theory
The Dilogarithm Function (Rev #5, changes)

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Define

More generally, the polylogarithm $m=1,2,\dots$

Note that

and

So we get an analytic continuation

where the path from $0$ to $z$ is in $\mathbb{C}\setminus [1,\infty)$

Functional equations:

Monodromy (on $\operatorname{Li}_2(x),\log(x),1$)

generate a Heisenberg group

Bloch-Wigner Dilogarithm

is real-analytic in $\mathbb{C}\setminus\{0,1\}$ and continuous in $\mathbb{C}$.

hence vanishes on $\mathbb{R}$.

So we have a continuous real-vaued function on $\mathbb{P}^1(\mathbb{C})$ with a maximum at $z=(1+\sqrt{-3})/2$: $D(1+\sqrt{-3})/2)=1.0149\dots$.

Define recursively

then $z_{n+5}=z_n$. If we call $z_0=x$, $z_1=y$, then we find

(Laurent phenomenon). (Cremona transformation of order 5 on $\mathbb{P}^2(\mathbb{C})$ is $(x,y)\mapsto\left(y,\tfrac{1-y}{x}\right)$.)

The 5-term recursion relation is

This can be explained geometrically.

In hyperbolic space, an ideal tetrahedron, with vertices at $0,1,\infty,z$, has volume $D(z)$. ($z=\tfrac{}{}$ is the regular tetrahedron, more generally, $z$ is the cross ratio of the 4 vertices , which is invariant under $PSL_2(\mathbb{C})=\operatorname{Isom}(\mathbb{H})$) The 5-term recursion relation comes from taking 5 points in $\mathbb{P}^1(\mathbb{C})$ and constructing five tetrahedra by taking the points 4 at a time