Let us consider the the quiver with 2 nodes and 1 arrow (we can think of some specific physical system, e.g. two points of the moduli space of the Coulomb branch of a $\mathcal{N}=2$ theory). Then, we can write this as $1 \to 2$. Now let us consider some representations of this quivers.

- $F_0=(\{0,0\}, 0)$
- $F_1=(\{ \mathbb{C},0\}, 0)$
- $F_2=(\{0, \mathbb{C} \}, 0)$
- $F_3=(\{ \mathbb{C}, \mathbb{C} \}, 0)$
- $F_4=(\{ \mathbb{C}, \mathbb{C} \}, z)$

where $z\in \mathbb{C}$. Of course the notation $(\{ A, B\}, x)$ means that the vector space of the node 1 is $A$ the vector space of node 2 is $B$ and the map between them is $x$. Ok, now, using King's parameters $(\theta_1, \theta_2)$ the slope is defined as

$$ \mu(F_i) = \frac{ \theta_1 d_1 + \theta_2 d_2 }{ d_1 + d_2} $$

where $d_i$ is the dimension of the vector space of node $i$. Now a representation is called $semi-stable$ for $(\theta_1,\theta_2)$ if for any sub representation $F' \subset F$ we have $\mu(F') \leq \mu(F)$. Now, there exists a second stability condition arising from extended supersymmetric gauge theories. There, we have a central charge function. For the specific quiver it is a map

$$ Z : \mathbb{Z}^{2} \to \mathbb{C} $$

such that the central charges associated to the nodes $Z(d_1,d_2)$ lie in a half-plane $H_{\phi}$ defined by

$$ H_{\phi} = \{ z \in \mathbb{C}^{*} | \, \phi < \text{Arg}(z) \leq \phi + \pi \} $$

Then a representation is called semi-stable if $\text{Arg}(Z(d_1(F'), d_2(F'))) \leq \text{Arg}(Z(d_1(F), d_2(F))) $. Let me mention I am already a bit lost here, with this definition of semi-stability.

Now it is possible to choose King's parameters as a function of the central charge as following

$$ \theta_1 = \frac{Z(\delta_1)}{Z(d_1)}, \,\,\,\,\,\,\, \theta_2 = \frac{Z(\delta_2)}{Z(d_2)}$$

where $\vec{\delta}_i = (0, \ldots, 1 , \ldots, 0) $, i.e. 1 at position i. In our case, with 2 nodes of course we only have the vectors $\delta_1 = (1,0)$ and $\delta_2 = (0,1)$. Now, and this is my second and bigger confusion is how can we determine from the above that for $\theta_1 < \theta_2$ the semi-stable representations are $F_0, F_1, F_2$ while for $\theta_1 > \theta_2 $ the semi-stable representations are $F_0,F_1,F_2,F_4$. How do we get this?

This is taken from this set of notes pages 3,4,5.