Geometry and String Theory
The Dilogarithm Function (changes)

Showing changes from revision #5 to #6: 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) Li 2=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 = \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}

Bloch-Wigner Dilogarithm

D(z)ImLi 2(z)+arg(1z)log|z| D(z) \coloneqq \operatorname{Im} \operatorname{Li}_2(z) + \arg(1-z)\log|z|

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

D(e iθ)= n1sinnθn 2 D(z¯)=D(z) \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}.

D(z) =D(1z 1)=D((1z) 1) D(z 1)=D(1z)=D(z1z) \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 1()\mathbb{P}^1(\mathbb{C}) with a maximum at z=(1+3)/2z=(1+\sqrt{-3})/2: D(1+3)/2)=1.0149D(1+\sqrt{-3})/2)=1.0149\dots.

Define recursively

z n+1z n1=1z n z_{n+1}z_{n-1} = 1-z_n

then z n+5=z nz_{n+5}=z_n. If we call z 0=xz_0=x, z 1=yz_1=y, then we find

x,y,1yx,x+y1xy,1xy x,y,\frac{1-y}{x},\frac{x+y-1}{xy},\frac{1-x}{y}

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

The 5-term recursion relation is

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

This can be explained geometrically.

Layer 1 0 0 1 1 z z \infty
Layer 1 0 0 1 1 z z \infty \begin{svg} <svg width="803" height="302" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="94701"> <g> <title>Layer 1</title> <path id="svg_94701_5" d="m208.5,93.625l-208,208l594,0l208.140259,-209.14032" stroke="#000000" fill="#ffeeee"/> <path id="svg_94701_3" d="m323.25,11l0,207c18,-51 84,-54 99,0c10,-54.333344 33,-75.666656 51,-52l0,-152.857872" stroke-width="2" stroke="#000000" fill="#eeffff"/> <line id="svg_94701_6" y2="217.021844" x2="422.25" y1="11" x1="422.25" stroke-width="2" stroke="#000000" fill="none"/> <foreignObject height="18" width="14" font-size="16" id="svg_94701_7" y="218" x="316.25"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mn>0</mn> </mrow> <annotation encoding="application/x-tex">0</annotation> </semantics> </math> </foreignObject> <foreignObject id="svg_94701_8" height="18" width="14" font-size="16" y="222" x="415.25"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mn>1</mn> </mrow> <annotation encoding="application/x-tex">1</annotation> </semantics> </math> </foreignObject> <foreignObject id="svg_94701_14" height="20" width="14" font-size="16" y="171" x="467.25"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mi>z</mi> </mrow> <annotation encoding="application/x-tex">z</annotation> </semantics> </math> </foreignObject> <foreignObject id="svg_94701_20" height="18" width="14" font-size="16" y="0" x="304.25"> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math> </foreignObject> </g> </svg> \end{svg}\includegraphics[width=602]{tetrahedron}

In hyperbolic space, an ideal tetrahedron, with vertices at 0,1,,z0,1,\infty,z, has volume D(z)D(z). (z=1+32 z=\tfrac{}{} z=\tfrac{1+\sqrt{-3}}{2} is the regular tetrahedron, tetrahedron; more generally,zz is the cross ratio of the 4 vertices , which is invariant under PSL 2()=Isom()PSL_2(\mathbb{C})=\operatorname{Isom}(\mathbb{H})) The 5-term recursion relation comes from taking 5 points in 1()\mathbb{P}^1(\mathbb{C}) and constructing five tetrahedra by taking the points 4 at a time

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

The cancellation is the 3-2 Pachner move.

Layer 2 Layer 1 C C A A B B A A B B A ' A' A ' A' a a c c b b c c a a a ' a' a ' a' b ' b' b b c ' c' c ' c' B ' B' B ' B' b ' b' \begin{svg}<svg width="240" height="250" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="1192"> <g> <title>Layer 2</title> <path fill="none" stroke="#0000ff" stroke-width="2" d="m203.632599,155.024902l9.980377,7.562393l-6.336456,10.908295" id="svg_1192_9" transform="rotate(-143.842, 208.623, 164.259)"/> <path fill="none" stroke="#0000ff" stroke-width="2" d="m201.017426,71.585327l9.468994,8.20385l-7.262848,11.587776" id="svg_1192_11" transform="rotate(149.349, 205.752, 81.4812)"/> <path fill="none" stroke="#0000ff" stroke-width="2" d="m102.508148,34.318008l6.24678,6.765732l-4.141541,6.414501" id="svg_1192_10" transform="rotate(76.4296, 105.632, 40.9066)"/> <path fill="none" stroke="#0000ff" stroke-width="2" d="m46.074718,139.976669l8.902531,6.672256l-5.902531,9.327744" id="svg_1192_7" transform="rotate(2.43666, 50.5259, 147.977)"/> <path fill="none" stroke="#0000ff" stroke-width="2" d="m106.677193,204.135712l10.311234,6.958557l-6.748878,11.124786" id="svg_1192_8" transform="rotate(-64.8852, 111.832, 213.176)"/> </g> <g> <title>Layer 1</title> <path fill="none" stroke="#000000" stroke-width="2" d="m0.9997,83.9617c25.147851,99.815475 160.403376,144.391083 238.297877,68.521545" id="svg_1192_3"/> <path fill="none" stroke="#000000" stroke-width="2" d="m280.548462,128.498505c-67.919983,-77.345005 -208.61499,-55.296997 -243.310303,47.756989" transform="rotate(70.1023, 158.893, 129.232)" id="svg_1192_6"/> <path fill="none" stroke="#000000" stroke-width="2" d="m141.741013,1c-89.299133,51.198318 -95.646103,193.468353 -1.541626,247.949371" id="svg_1192_1"/> <path fill="none" stroke="#000000" stroke-width="2" d="m246.517212,125.435997c-67.919983,-77.344997 -208.61499,-55.296997 -243.310292,47.756996" id="svg_1192_4" transform="rotate(-19.7826, 124.861, 126.17)"/> <path fill="none" stroke="#000000" stroke-width="2" d="m257.517212,154.436005c-67.919983,-77.345001 -208.61499,-55.296997 -243.310292,47.756989" id="svg_1192_5" transform="rotate(145.579, 135.861, 155.17)"/> <foreignObject x="15.824219" y="140" id="svg_1192_12" font-size="16" width="20" height="20"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>C</mi> </mrow> <annotation encoding="application/x-tex">C</annotation> </semantics> </math> </foreignObject> <foreignObject x="54.324219" y="85" font-size="16" width="20" height="20" id="svg_1192_13"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>A</mi> </mrow> <annotation encoding="application/x-tex">A</annotation> </semantics> </math> </foreignObject> <foreignObject x="64.990885" y="172.333333" font-size="16" width="20" height="20" id="svg_1192_19"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>B</mi> </mrow> <annotation encoding="application/x-tex">B</annotation> </semantics> </math> </foreignObject> <foreignObject x="170.324219" y="146" font-size="16" width="20" height="20" id="svg_1192_25"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>A</mi> </mrow> <annotation encoding="application/x-tex">A</annotation> </semantics> </math> </foreignObject> <foreignObject x="176.990885" y="93.333333" font-size="16" width="20" height="20" id="svg_1192_31"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>B</mi> </mrow> <annotation encoding="application/x-tex">B</annotation> </semantics> </math> </foreignObject> <foreignObject x="137.324219" y="205" font-size="16" width="20" height="20" id="svg_1192_37"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>A</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">A'</annotation> </semantics> </math> </foreignObject> <foreignObject x="215.324219" y="93" font-size="16" width="20" height="20" id="svg_1192_43"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>A</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">A'</annotation> </semantics> </math> </foreignObject> <foreignObject x="47.824219" y="159" id="svg_1192_50" font-size="16" width="14" height="20"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>a</mi> </mrow> <annotation encoding="application/x-tex">a</annotation> </semantics> </math> </foreignObject> <foreignObject x="74.824219" y="128" font-size="16" width="14" height="20" id="svg_1192_51"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>c</mi> </mrow> <annotation encoding="application/x-tex">c</annotation> </semantics> </math> </foreignObject> <foreignObject x="41.824219" y="105" font-size="16" width="14" height="20" id="svg_1192_57"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>b</mi> </mrow> <annotation encoding="application/x-tex">b</annotation> </semantics> </math> </foreignObject> <foreignObject x="208.824219" y="112" font-size="16" width="14" height="20" id="svg_1192_69"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>c</mi> </mrow> <annotation encoding="application/x-tex">c</annotation> </semantics> </math> </foreignObject> <foreignObject x="186.824219" y="178" font-size="16" width="14" height="20" id="svg_1192_75"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>a</mi> </mrow> <annotation encoding="application/x-tex">a</annotation> </semantics> </math> </foreignObject> <foreignObject x="115.949219" y="167.875" font-size="16" width="14" height="20" id="svg_1192_81"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>a</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">a'</annotation> </semantics> </math> </foreignObject> <foreignObject x="168.449219" y="46.375" font-size="16" width="14" height="20" id="svg_1192_87"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>a</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">a'</annotation> </semantics> </math> </foreignObject> <foreignObject x="115.324219" y="76" font-size="16" width="14" height="20" id="svg_1192_94"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>b</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">b'</annotation> </semantics> </math> </foreignObject> <foreignObject x="190.324219" y="49.5" font-size="16" width="14" height="20" id="svg_1192_100"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>b</mi> </mrow> <annotation encoding="application/x-tex">b</annotation> </semantics> </math> </foreignObject> <foreignObject x="69.824219" y="55" font-size="16" width="14" height="20" id="svg_1192_106"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>c</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">c'</annotation> </semantics> </math> </foreignObject> <foreignObject x="76.824219" y="194.5" font-size="16" width="14" height="20" id="svg_1192_112"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>c</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">c'</annotation> </semantics> </math> </foreignObject> <foreignObject x="137.490885" y="27.333333" font-size="16" width="20" height="20" id="svg_1192_119"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>B</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">B'</annotation> </semantics> </math> </foreignObject> <foreignObject x="214.490885" y="132.333333" font-size="16" width="20" height="20" id="svg_1192_125"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>B</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">B'</annotation> </semantics> </math> </foreignObject> <path fill="none" stroke="#ff0000" stroke-width="2" d="m73.324219,106.5c-12.416668,9.833328 -26.583328,28.916672 -33,42.75c11.25,11.166672 32.25,24.833328 40.25,27.5c-6.583351,-14 -10.416672,-51 -7.25,-70.25z" id="svg_1192_132"/> <foreignObject id="svg_1192_2" x="166.324219" y="185" font-size="16" width="14" height="20"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>b</mi> <mo>'</mo> </mrow> <annotation encoding="application/x-tex">b'</annotation> </semantics> </math> </foreignObject> </g> </svg> \end{svg}\includegraphics[width=180]{pentagon}

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 π/2\pi/2.

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

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)(a_i, B_i, c_i, A_i, b_i), i=0,1,,4i=0,1,\dots,4, with (a 0,B 0,c 0,A 0,b 0)(a,B,c,A,b)(a_0, B_0, c_0, A_0, b_0)\equiv(a, B, c, A, b). Under a cyclic permutation

(a,B,c,A,A,b)(a 1,B 1,c 1,A 1,b 1)=(A,b,a,B,c) (a,B,c,A,A,b)\mapsto (a_1,B_1,c_1,A_1,b_1) = (A',b',a',B,c')

and

(a,B,c,A,b)(A,b,a,B,c) (a,B',c',A',b)\mapsto (A',b,a,B',c')

Triangle can be solved if any two parts are known

sina=tanbtanB=cosAcosc \sin a =\tan b \tan B' = \cos A' \cos c'

The surface

S:{1z 1=z 2z 0 1z 2=z 3z 1 1z 0=z 1z 4Aut(S)S 5 S: \left\{\begin{gathered}1-z_1=z_2z_0\\ 1-z_2=z_3z_1\\ \vdots\\ 1-z_0 = z_1z_4 \end{gathered}\right. \qquad \operatorname{Aut}(S)\simeq S_5

is a del Pezzo surface of degree 5.

j=0 4z j=3s,s= j=0 4z jSchöne Gleichung (1x)(1y)(1xy)sxy=0 \begin{gathered} \sum_{j=0}^4 z_j = 3-s,\quad -s=\prod_{j=0}^4 z_j\qquad\text{Schöne Gleichung}\\ (1-x)(1-y)(1-x-y) -s xy =0 \end{gathered}

universal elliptic curve with a 5-torsion point x 1(5)x_1(5).

Coxeter: 5-cycle

transmitted as mathematical gossip for a long time.

Number Theory

Li 2(1)=ζ(2)=π 26(Euler 1768) \operatorname{Li}_2(1)=\zeta(2) = \frac{\pi^2}{6}\qquad\text{(Euler 1768)}

More generally,

L(χ,2)= n1χ(n)n 2 χ:(/N) × ×Dirichlet character \begin{gathered} L(\chi,2) = \sum_{n\geq 1} \frac{\chi(n)}{n^2}\\ \chi\colon {(\mathbb{Z}/N\mathbb{Z})}^\times \to \mathbb{C}^\times \qquad\text{Dirichlet character} \end{gathered}