# Wednesday, November 15


:::{.remark}
Last time: from a quiver $Q$, construct a cluster variety.
Letting $N = \ZZ^{Q_0}$ with basis $\ts{e_1, e_2}$, there is a pairing $\inp{e_i}{e_j} = a_{ij} - a_{ji}$ where $a_{ij}$ is the number of arrows from $e_i$ to $e_j$.
The $\mcx\dash$cluster variety is the union over mutation sequences of $\spec \CC[N] \cong (\cstar)^n$. 
The $\mca\dash$cluster variety is similarly the union of $\spec \CC[M]$.
Given the $\mcx\dash$cluster variables $X_i$, mutating along $e_k$ yields a map 
\[
(\cstar)^n &\to (\cstar)^n \\
(X_i) &\mapsto (X_i)'
\]
where 
\[
X_i = 
\begin{cases}
X_k\inv & i=k 
\\
X_i \qty{1 + X_k^{-\sgn \eps_{ij} }}^{-\eps_{ij}} & i\neq k.
\end{cases}
.\]

Similarly for the $\mca\dash$cluster variables $A_i$, mutating along $e_k$ yields $A_i\to A_i'$ where 
\[
A_i' =
\begin{cases}
A_j & i=k 
\\
{1\over A_k}\qty{ \prod_{j, \eps_{kj} > 0 } A_j^{\eps_{kj}} + \prod_{j, \eps_{kj} < 0 } A_j^{-\eps_{kj}} } & i\neq k.
\end{cases}
.\]
Note that the $\mcx, \mca$ cluster varieties are isomorphic if we've finitely many mutations, e.g. for ADE quivers.
:::


:::{.remark}
If you write the $\mca\dash$cluster variables in one chart in terms of variables in another chart, one always obtains Laurent polynomials.
This is note true for the $\mcx\dash$cluster variables.
This has something to do with GHK mirror symmetry and theta functions.
:::