TITLE: what is the stringy Kähler moduli space?
QUESTION [14 upvotes]: 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?

REPLY [11 votes]: 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.