# Herbert Gangl, The Aomoto polylogarithm via iterated integrals (Wednesday, July 20)

:::{.remark}
The plan: 

- Iterated integrals and formal versions, leading to the "algebraic fingerprint" (i.e. symbols).
- Aomoto polylogarithms
- A new choice of coordinates on configuration spaces
- Relating Aomoto polylogarithms to iterated integrals via the symbol

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## Classical Polylogarithms

:::{.remark}
Recall
\[
\Li_n(z) = \sum_{k\geq 1} {z^k\over k^n }
.\]
Here $n$ is a *weight*, and the *depth* is 1 (the number of variables).
More generally, for example, $\Li_{m, n}(x, y)$ has weight $m+n$ and depth 2.
This can be written as an iterated integral:
\[
\mathrm{II}(a_0; a_1,\cdots, a_n: a_{n+1}) = \int_{\Delta_\gamma} {dt_1 \over t_1-a_1} \wedge \cdots\wedge {dt_n \over t_n - a_n}
\]
where $\gamma: [0, 1]\to \CC\smts{a_1,\cdots, a_n}$ with $\gamma(0) = a_0, \gamma(1) = a_{n-1}$ and $\Delta_\gamma$ is a certain simplex.
Some properties:

- Form an algebra under the shuffle product
- Path composition
- Path reversal
- An associated differential equation

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:::{.remark}
One can encode this formally as an algebra over a field $F$ with generators $I(a_0, \cdots, a_{n+1})$ and relations coming from the above properties.
A new feature: a "semicircle coproduct" due to Goncharov, whose terms are indexed by polygons involving any subset of the $a_i$ when arranged along a semicircle:

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![](figures/2022-07-20_09-14-37.png)

In general, the symbol for $I(\cdots)$ will take values in $(F\units)\tensorpower{?}{n}$.
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## Aomoto Polylogarithms

:::{.remark}
Idea: attach a period to any pair of simplices $(L, M)$ in $\PP^n(\CC)$ determined by coordinates $(\ell_0,\cdots, \ell_n; m_0, \cdots, m_n)$.

- To $L$, attach $\omega_L$ a differential form, $\dlog\qty{z_1\over z_0} \wedge \cdots\wedge \dlog\qty{z_n\over z_0}$.
- To $M$ attach an integral cycle.

This gives an assignment $(L, M) \mapsto \int_{\Delta_M} \omega_L$, which generalizes the classical polylogarithm.
Some properties:

- Degeneracies go to zero (?)
- Skew-symmetry: $(\sigma L, M) = (L, \sigma M) = \sign(\sigma)(L, M)$ for any $\sigma\in S_n$.
- Invariance for $g\in \PGL_{n+1}(\CC)$ in the sense that $(gL, gM) = (L, M)$.
- Additivity, a type of scissors congruence property: $\sum_{i=0}^{n+1} (-1)^i (\ell_0, \cdots, \hat \ell_i, \cdots, \ell_{n+1}; M ) = 0$.

BGSV ('89) formalized this in an $F\dash$algebra with generators $(L, M)$ and relations coming from these new properties, and found a good candidate for the category of mixed Tate motives over a field.
An issue is that the coproduct is only defined on generic such pairs.
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:::{.example title="?"}
Consider $n=1$ and $(L, M)$ with $(\ell_0, \ell_1, m_0, m_1)$, then
\[
A_1(L, M) = \int_{m_0}^{m_1} \dlog\qty{z_1 - \ell_1 \over z_0 - \ell_0}
.\]

One can extract a cross-ratio from this which lives in $F\units$.
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:::{.remark}
A surprise to BGSV: a certain expression can be written as a combination of Steinberg elements $a\tensor(1-a)$ which is related to motivic cohomology and the Bloch-Suslin complex.
Goncharov also writes $A_3(L, M)$ in terms of polylogarithms, and attaches a symbol to general $A_n(L, M)$ in terms of Plucker coordinates.
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:::{.theorem title="G-Charlton-Radchenko"}
The symbol of $A_n(\vector v)$ can be written as a sum of iterated integrals.
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