# 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?

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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 $$R_{mn}(g+\delta g)=0 \ ,$$ 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 $$\int_M J\wedge J\wedge J >0$$ 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 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 $B+iJ$.