what is the stringy Kähler moduli space? I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mirror manifold.
Could you suggest a nice paper/blog post where to read some heuristics about it? Some examples? Connections to stability conditions/curve-counting?
 A: I have written an very brief introduction to Calabi-Yau moduli. This is covered by standard textbooks of string theory. (See the chapter 9 of BBS.) In addition, there are many good reviews such as this on this topic.
The metric deformations $\delta g$ of a Calabi-Yau 3-fold $M$ are classified by the one of type $(1,1)$ and type $(2,1)$. The Ricci flat condition 
\begin{equation}
R_{mn}(g+\delta g)=0 \ ,
\end{equation}
requires the infinitesimal $(1,1)$-form $\delta g$ to be harmonic. Therefore the K\"ahler moduli space is locally described by the vector space of $H_{1,1}$. The K\"ahler metric defines the K\"ahler form $J = ig_{ij}dz^i\wedge dz^j$, and positivity of the metric is equivalent to
\begin{equation}
\int_M J\wedge J\wedge J >0
\end{equation}
The K\"ahler moduli space has a cone structure since if $J$ satises the positivity condition, so does $sJ$ for any positive number $s$.
In fact, in the study of string theory, the K\"ahler form of a Calabi-Yau 3-fold arises as one of the moduli fields of the associated 2d conformal field theory (Landau-Ginzburg model). Investigation of 2d CFT moduli space reveals that the corresponding geometrical description necessarily involves
configurations in which the (supposed) K\"ahler form lies outside of the K\"ahler cone of the particular Calabi-Yau being studied. This may be thought of as residing in a new K\"ahler cone which shares a common wall with the original one. It was first shown by Witten in the seminal paper that this is a flop transition as you move from the one cone to the next cone, which is a famous example of topology change in string theory.
In string theory, there is NS-NS two form $B$ which is an element of $H^{1,1}$. It is natural to combine the $B$-field
with the K\"ahler form $J$, providing the complexied K\"ahler form $B+iJ$.
As for the stability condition in the context of string theory, the paper by Aspinwall is a good point to start.
A: Let's see why the study of moduli space of complex structures is closely related to mirror pairs.
From the mirror map, we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$
where $M$ and $M^\vee$ are mirror to each other. 
We have the following relation for tangent space of moduli space of symplectic structures 
$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$
and from the BTT Theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for the tangent space of moduli of complex structures.
So from the mirror map, we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$
More generally if we take extended moduli space of complex and symplectic structures then 
$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$
and 
$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$
and hence from the extended mirror map we can identify
$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself a Calabi-Yau variety then 
$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$ 
See this Kontsevich paper.
