above to get as before the standard SUGRA frame. Also the relation between the dualized variables is now given by $$T^{i}+\bar{T}^{i}=\frac{1}{3}K_{U_{i}}-F_{U_{i}}=\Gamma^{i}(U,S,\phi,\bar{\phi})\label{eq:TUrelation}$$ Given a model for $K$ as a function of $U^{i}$ the differential equation can be solved. For instance in the case of type IIB string theory on a Calabi-Yau the (scalar components of) the linear multiplet variables are related to the two cycle volumes $v_{i}$ by $U_{i}=v_{i}/{\cal V}$ [@Grimm:2004uq] and $$\begin{aligned}
K & = & -\ln(S+\bar{S})-\hat{K}(z,\bar{z})-2\ln{\cal V},\label{eq:K1}\\
 & = & -\ln(S+\bar{S})-\hat{K}(z,\bar{z})+\ln\hat{{\cal V}}.\label{eq:K2}\end{aligned}$$ Here ${\cal V}=\frac{1}{6}\int J^{3}=\frac{1}{6}c^{ijk}v_{i}v_{j}v_{k}$, $\hat{{\cal V}}=1/{\cal V}^{2}=\frac{1}{6}c^{ijk}U_{i}U_{j}U_{k}$, with the indices running over $1,\ldots,h_{+}^{(1,1)}$ i.e. the Kähler moduli that survive the orientifold projection, $z$ stands for the set of complex structure moduli, and $S=e^{-\varphi}+iC_{0}$, where $\varphi$ is the dilaton and $C_{0}$ is the RR zero-form. We also have in this case $F=U_{i}V^{i}(\phi,\bar{\phi})$ where $V^{i}$ are a set of arbitrary real functions and[^2] $\phi=\{z,S\}$. It is important for future reference to note that these arbitrary functions are independent of the Kähler moduli. Using these we have from $$T^{i}+\bar{T}^{i}=\frac{2}{3}\tau^{i}-V^{i}(z,\bar{z};S,\bar{S})=\Gamma^{i}(U,S,\bar{S};z,\bar{z}),\label{eq:TtauGamma}$$ where $\tau^{i}=c^{ijk}v_{j}v_{k}=c^{ijk}U_{j}U_{k}/\hat{{\cal V}}$ is the volume of the $i$th 4-cycle.

The results of this subsection are summarized in the following table.

$$$$

                  

                                                 chiral                                                                  linear
  ------------------------ --------------------------------------------------- -------------------------------------------------------------------------------------------
            $K$                       $-2\ln{\cal V}(T+\bar{T}+V)$                                                 $\ln\hat{{\cal 