Geometry and String Theory
The Dilogarithm Function

Define

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

Note that

and

So we get an analytic continuation

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

Functional equations:

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

generate a Heisenberg group

Bloch-Wigner Dilogarithm

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

hence vanishes on $ℝ$.

So we have a continuous real-vaued function on ${\mathbb{P}}^1(ℂ)$ 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 ${\mathbb{P}}^2(ℂ)$ is $(x,y)\mapsto (y,\frac{1-y}{x})$.)

The 5-term recursion relation is

This can be explained geometrically.

In hyperbolic space, an ideal tetrahedron, with vertices at $\mathrm{0,1},\mathrm{\infty },z$, has volume $D(z)$. ($z=\frac{1+\sqrt{-3}}{2}$ is the regular tetrahedron; more generally, $z$ is the cross ratio of the 4 vertices , which is invariant under ${\mathrm{PSL}}_2(ℂ)=Isom(ℍ)$) The 5-term recursion relation comes from taking 5 points in ${\mathbb{P}}^1(ℂ)$ and constructing five tetrahedra by taking the points 4 at a time

The cancellation is the 3-2 Pachner move.

Napier: Mirifici logorithorum canonis descriptio

Rule of circular parts

Spherical trigonometry (navigation mathematics): 6 quantities (3 angles + 3 lengths or, equivalently, 6 angles). Fix one to be $\pi /2$.

Parametrize parts wiith cross ratios of 4 points taken out of 5. The sides of the pentagon are $A,B,b\prime ,c,a\prime$.

Any equation among the parts remains valid after a cyclic permutation around the pentagon. Denote the five triangles by $(a_i,B_i,c_i,A_i,b_i)$, $i=\mathrm{0,1},\dots \mathrm{,4}$, with $(a_0,B_0,c_0,A_0,b_0)\equiv (a,B,c,A,b)$. Under a cyclic permutation

and

Triangle can be solved if any two parts are known

The surface

is a del Pezzo surface of degree 5.

universal elliptic curve with a 5-torsion point $x_1(5)$.

Coxeter: 5-cycle

transmitted as mathematical gossip for a long time.

Number Theory

More generally,