Geometry and String Theory
The Dilogarithm Function (Rev #2, changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

Define

Li 2(z) n1z nn 2,|z|<1 \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,m=1,2,\dots

Li m(z) n1z nn m,|z|<1 \operatorname{Li}_m(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^m},\qquad |z|\lt 1

Note that

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

and

ddzLi m(z)=Li m1(z) \frac{d}{d z} \operatorname{Li}_m(z) = \operatorname{Li}_{m-1}(z)

So we get an analytic continuation

Li 2(Z)= 0 zlog(1u)duu \operatorname{Li}_2(Z) = -\int_0^z \log(1-u) \frac{d u}{u}

where the path from 00 to zz is in [1,)\mathbb{C}\setminus [1,\infty)

Functional equations:

Li 1(1xy)=Li 1(1x)+Li 1(1y) LiLi 22=5 terms (Spence 1809, Abel 1828, ...) \begin{gathered} \operatorname{Li}_1(1-x y) = \operatorname{Li}_1(1-x) + \operatorname{Li}_1(1-y)\\ \operatorname{Li}-2 \operatorname{Li}_2 = \text{5 terms (Spence 1809, Abel 1828, ...)} \end{gathered}

Monodromy (on Li 2(x),log(x),1\operatorname{Li}_2(x),\log(x),1)

γ 0=(1 0 0 0 1 2πi 0 0 1),γ 1=(1 2πi 0 0 1 0 0 0 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

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