# Monday, October 23

## Cluster Varieties

:::{.remark}
Fomin-Zelevisnky formulated the notion of cluster algebras.
Fock-Goncharov define cluster varieties.
These are built in a process: quiver $\to$ seed $\to$ cluster variety.
A quiver is a finite oriented graph, possibly with self-loops.
We write $Q_0$ for the set of vertices, $Q_1$ for the set of edges.
A seed is that data of a lattice $N \cong \ZZ^n$ with a basis $\ts{e_i}$ and a skew-symmetric form $N\times N\to \ZZ$.
The seed associated to a quiver $Q$: write $N = \ZZ^{\abs{Q_0}}$ with basis \( \ts{ e_i } \) and $w:N\times N\to \ZZ$ defined by $w(e_i, e_j) \da a_{ij} - a_{ji}$ where $a_{ij}$ is the number of arrows from $v_i$ to $v_j$.
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:::{.example title="?"}
Consider the following quiver: 

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![](figures/2023-10-23_14-08-19.png)

Here $\abs{Q_0} = 2$ and $N\cong \ZZ^2 = \gens{e_1, e_2}_\ZZ$.
One can check $w(e_1, e_2) = 1-0 = 1$ and $w(e_2, e_1) = 0-1 = -1$.
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:::{.example title="?"}
For Dynkin diagrams of type $A,D,E$, orienting the arrows yields a quiver.
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:::{.remark}
Mutating a seed: write $Q_0 = \ts{v_i}$ with $v_i \to e_i$ a choice of basis.
Define a new basis 
\[
e_i' \da 
\begin{cases}
e_i + \max\ts{ w(e_i, e_k), 0} e_k &  i\neq k
\\
-e_k & i=k.
\end{cases}
.\]
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:::{.example}
Examples of mutating seeds.
For the quiver $\bullet_1 \to \bullet_2$, one can mutate along $e_1$ to obtain $e_1' = -e_1$ and $e_2' = e_2$.
Here $w(e_1', e_2') = w(-e_1, e_2)$.
To mutate along $e_2$, one gets $e_1' = e_1 + e_2$ and $e_2' = -e_2$.
Mutating this again along $e_2$ yields $e_1'' = e_1 + 2e_2$ and $e_2'' = e_2$.
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:::{.remark}
Mutating entire quivers: given $Q$, mutate at $e_i$ by doing the following: 

1. Given $v_{i-1}\to v_{i} \to v_{i+1}$, insert a new arrow $v_{i-1} \to v_{i+1}$. If this forms an oriented 2-cycle $\bullet \mapstofrom \bullet$, delete these two edges.
2. Reverse all oriented arrows connected to $v_i$, so e.g. $v_1\to v_2$ becomes $v_1\from v_2$.

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:::{.remark}
In this case, $w(e_1', e_2') = w(-e_1, e_2) = -w(e_1, e_2) = -1$.
There is not a bijective correspondence between seed mutations and quiver mutations.
E.g. consider the $A_3$ quiver $v_1\to v_2\to v_3$ and mutate along $e_2$ twice.
One obtains the following: 

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![](figures/2023-10-23_14-38-54.png)

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:::{.remark}
$\chi\dash$cluster varieties: take $\union \spec \CC[N]$ where the union is over all possible mutations of an initial seed $\ts{e_1,\cdots, e_n}$.
Note that $\spec \CC[N]$ is a torus $(\cstar)^n$, so this is a gluing over many tori.
The gluing data is given by the rational maps
\[
(\cstar)^n &\to (\cstar)^n \\
z^n &\mapsto z^n (1 + z^{e_k})^{(n, \eta_k)}, \qquad \eta_k \da w(e_k, \wait)
.\]
For any $A\dash$type cluster variety, take $\union \spec \CC[M]$ and 
\[
(\cstar)^n &\to (\cstar)^n \\
z^m &\mapsto z^m(1 + z^{\eta_k} )^{-(e_k, m)}
.\]
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