Geometry and String Theory
The Dilogarithm Function (Rev #4, changes)

Showing changes from revision #3 to #4: Added | ~~Removed~~ | ~~Chan~~ged

Define

\operatorname{Li}_2(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^2},\qquad |z|\lt 1

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

\operatorname{Li}_m(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^m},\qquad |z|\lt 1

Note that

\operatorname{Li}_1(z) = -\log(1-z)

and

\frac{d}{d z} \operatorname{Li}_m(z) = \operatorname{Li}_{m-1}(z)

So we get an analytic continuation

\operatorname{Li}_2(z) = -\int_0^z \log(1-u) \frac{d u}{u}

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

Functional equations:

\begin{gathered} \operatorname{Li}_1(1-x y) = \operatorname{Li}_1(1-x) + \operatorname{Li}_1(1-y)\\ \operatorname{Li}_2 = \text{5 terms (Spence 1809, Abel 1828, ...)} \end{gathered}

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

\gamma_0=\begin{pmatrix}1&0&0\\0&1&2\pi i\\0&0&1\end{pmatrix}, \gamma_1=\begin{pmatrix}1&-2\pi i&0\\0&1&0\\0&0&1\end{pmatrix}

generate a Heisenberg group

\begin{pmatrix}1&\mathbb{Z}(1)&\mathbb{Z}(2)\\ 0&1&\mathbb{Z}(1)\\0&0&1\end{pmatrix}

Bloch-Wigner Dilogarithm

D(z) \coloneqq \operatorname{Im} \operatorname{Li}_2(z) + \arg(1-z)\log|z|

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

\begin{gathered} D\left(e^{i\theta}\right) = \sum_{n\geq 1} \frac{\sin n\theta}{n^2}\\ D(\overline{z}) = - D(z) \end{gathered}

hence vanishes on $\mathbb{R}$ .

\begin{split} D(z)&= D\left(1-z^{-1}\right)= D\left({(1-z)}^{-1}\right)\\ & - D\left(z^{-1}\right) = - D(1-z) = -D\left(-\frac{z}{1-z}\right) \end{split}

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

z_{n+1}z_{n-1} = 1-z_n

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

x,y,\frac{1-y}{x},\frac{x+y-1}{xy},\frac{1-x}{y}

(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

\sum_{j=0}^4 D(z_j)=0

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

0 = \sum_{j=0}^4 {(-1)}^{j}\operatorname{Vol}((w_0,\dots,\hat{w}_j,\dots,w_4))

Revision from October 3, 2010 15:11:54 by Jacques Distler

Geometry and String Theory The Dilogarithm Function (Rev #4, changes)

Bloch-Wigner Dilogarithm

Geometry and String Theory
The Dilogarithm Function (Rev #4, changes)