7/5 meeting
===

Recurrents and surperstable
---

**Definition** Let $c \in \text{Config}(G)$, and let $S \subseteq \tilde{V}$. Suppose $c'$ is the configuration obtained from $c$ by firing the vertices in $S$. Then $c \xrightarrow{S} c'$ is a *legal* set-firing if $c'(v) \geq 0$ for all $v \in S$.

**Definition** THn confinguration $c\in \text{Config}(G)$ is *superstable* is $c \geq 0$ and has no legal nonempty set-firings, i.e. for all nonempty $S \subseteq \tilde{V}$, there exists $v \in S$ such that
$$
c(v) < \text{outdeg}_S{v}
$$

We have the isomorphism :
$$
\begin{align}
\text{Config}(G)/\text{im}(\tilde{div}) &\xrightarrow{} \text{Jac}(G) \\
[c] &\xrightarrow{} [c-\text{deg}(c)q]
\end{align}
$$
So $c \sim c'$ as configurations exactly when $c -\text{deg}(c')q \sim c -\text{deg}(c')q$ as divisors. It follows form existence and uniqueness of q-reduced divisors that each $c \in \text{Config}(G)$.

The collection of surperstables of $G$ forms an belian group where group where addition is the usual addition of configurations followed by surperstabilization.

**Theorem 7.12**
Let $c \in \text{Config}(G)$ and $c \geq 0$ (i.e. $c$ is sandpile). Then,
+ $c$ is recurrent if and only if $c_\text{max} -c$ is surperstable.

::: spoiler example


:::

---

Monodromy pairing
===
[Ref](https://arxiv.org/pdf/0907.4764)
**Definition** Let $A$ be a matrix. Any matrix $L$ satisfying $ALA =A$ is called a generalized inverse of $A$.

---

For $D_1, D_2$ in $\text{Div}^0(G)$, let $m_1$ and $m_2$ be integers such that $m_1D_1 = \text{div}(f_1)$ and $m_2D_2 = \text{div}(f_2)$ are principal; these exist because $\text{Jac}(G)$ is a finite group. One can easily show that
$$
\frac{1}{m_2}\sum_{v \in V(G)}D_1(v)f_2(v) = \frac{1}{m_1}\sum_{v \in V(G)}D_2(v)f_1(v)
$$

**Definition** The monodromy pairing $\langle\cdot\,,\cdot\rangle : \text{Jac}(G) \times \text{Jac}(G) \xrightarrow{} \mathbb{Q} / \mathbb{Z}$ defined by
$$
\langle\overline{D_1},\overline{D_2}\rangle = \frac{1}{m_2} \sum_{v\in V(G)} D_1(v)f_2(2) (\mod{\mathbb{Z}})
$$
where $m_2D_2 = \text{div}(f_2)$. This is well-Defined, symmetric, bilnear, non-degenerate pairing on $\text{Jac}(G)$.


**Proposition** Let $L$ be any generalized inverse of the Laplacian matrix $Q$. Then the monodromy pairing is given by
$$
\langle\overline{D_1},\overline{D_2}\rangle = [D_1]^TL[D_2]^T (\text{mod}\mathbb{Z})
$$