Im}(\tau)$ is the dilaton. Therefore, in the large volume limit, the Kähler potential for the dilaton indeed reduces to just $-\log(\tau-\bar\tau)$, as expected from a Kaluza-Klein reduction.

Now on the torus the cubic expression is exact (as there are no worldsheet instanton corrections). We can then go to the $\zet_3\times \zet_3$ symmetric point $v^a=1/2$, and orbifold. Since orbifolding is a worldsheet operation it projects the worldsheet variables $v^a$ to their value at the orbifold point but does not touch the dilaton directly. It then follows easily from that after orientifold and orbifold the Kähler potential for the dilaton is $$K(\tau) = - 4\log(\tau-\bar\tau).$$ In a certain sense, one should view the factor of $4$ as a small volume correction to the usual expression and which emerges for the type IIB non-geometric models. This correction can only be derived by mirror symmetry and not by analytic continuation from a geometric Kaluza-Klein reduction in type IIB! In fact, [@grilo; @grimm] give a more general expression for the Kähler potential valid (in the supergravity approximation) throughout the hypermultiplet moduli space and this is what is used in [@wati] in their general analysis.

Notice that this [*non-geometric*]{} modification of the Kähler potential only affects 