Moduli space of Bridgeland semistable objects: what is it? I usually meet this kind of moduli space in recent papers on Bridgeland stability conditions:

the moduli space $M_{\sigma}(v)$ of $\sigma$-semistable objects of $\mathcal{T}$ with certain numerical class $v$.

Here $\mathcal{T}=D^b(\textbf{Coh}(X))$ for a complex smooth projective variety $X$ or $\mathcal{T}\subset D^b(\textbf{Coh}(X))$ an admissible subcategory and $\sigma$ is a Bridgeland stability condition on $\mathcal{T}$.
However, I could not find the precise definition for this moduli space in these papers.
Do we have a nice treatmeant for this notion? Which moduli problem does it represent? Is the moduli space coarse or fine?
 A: I recommend D. Huybrechts' Introduction to stability conditions. Moduli spaces, 179–229, London Math. Soc. Lecture Note Ser. 411, Cambridge Univ. Press (2014).
A: As far as I can find, the earliest paper treating this notion is Moduli stacks and invariants of semistable objects on K3 surfaces.
Consider a complex algebraic K3 surface $X$ and a Bridgeland stability condition $\sigma$ on $D(X)$, Toda defines the coarse moduli space
$$\mathfrak{M}_{(\alpha,\phi)}(\sigma)$$
of $\sigma$-objectswith class $\alpha$ and phase $\phi$. The space $\mathfrak{M}_{(\alpha,\phi)}(\sigma)$ is an Artin stack of finite type over $\mathbb{C}$. This construction is based on Lieblich's work, where the moduli stack
$$\mathfrak{M}$$
of objects $E$ with $\text{Ext}^{i<0}(E,E)=0$ is constructed. Then $\mathfrak{M}_{(\alpha,\phi)}(\sigma)$ is the substack of $\mathfrak{M}$. I believe that Toda's method can be generalized to other smooth varieties. More discussions can be found in the expository of E. Macrì and B. Schmidt.
So in this case, we might glue these stacks together to get what we want.
Anyway, I still want a reference for this because it is used without reference in some recent papers. I guess it should be known to experts.
