Holomorphic sections that lead to the original parameterization of the gauged STU model

Question

The original action, given in 9903214, comes from the $\mathcal{N} = 2$ abelian truncation of the maximal $\mathcal{N} = 8$, $SO(8)$ gauged supergravity that can be obtained from $S^ 7$ reduction of 11d supergravity. Since I was wondering how this model fits into the general framework of $\mathcal{N} = 2$ gauged supergravity, I did a search and found 1712.01849. In appendix A they summarize some choices for the holomorphic sections:

A.1 Cveti$\mathrm{\check{c}}$ et al. gauge

A.2 Cacciatori-Klemm gauge

A.3 Pufu-Freedman gauge

A.4 Hristov-Vandoren gauge

They claim that using A.1 or A.2 the general action reduces to the gauged STU model action given in 9903214. But that's not true. The scalar potential is ok, but the resultant period matrix is different. After some minor modifications and trial and error I got these holomorphic sections that work: $$X^0=\sqrt{i z_1 z_2 z_3}, \quad\frac{X^1}{X^0}=-\frac{1}{z_2 z_3}, \quad\frac{X^2}{X^0}=-\frac{1}{z_1 z_3}, \quad\frac{X^3}{X^0}=-\frac{1}{z_1 z_2}$$ $$z_k=\chi_k+i e^{-\varphi_k}, \ \ \text{for } k\in\{1,2,3\}$$ I'm interested in knowing whether this choice has been considered before in the literature.

Has anyone seen the holomorphic sections I have found before?

Is there any reference with a choice of holomorphic sections that correctly reproduces the gauged STU model action given in eqn. (B.7) of 9903214?

This post imported from StackExchange Physics at 2023-02-15 12:48 (UTC), posted by SE-user Anthonny