Toroidal compactifications of type IIB string Theory and $SO(5,5)/(SO(5)\times SO(5))$ invariant 6D sugra action

Question

It is usually stated that the compactification of (the bosonic part of the) type IIB ($D=10$, ${\cal N}=(2,0)$) supergravity on $\mathbb{T}^4$ gives a six-dimensional ${\cal N}=(4,4)$ supergravity whose scalar sector - the one I am interested in - can be written in a way that is manifestly invariant under the $$\frac{SO(5,5)}{SO(5)\times SO(5)}$$ transformations, that is, the 25 scalars $$h_{ij} \; (10), \; B_{ij} \; (6), \; C_{ij} \; (6) \; C_{ijkl} \;(1), \; \phi \; (1), \; C_0 \; (1) ,$$ corresponding to the metric, the Kalb-Ramond 2-form, and the RR 0-, 2-, and 4-forms with all legs on the $\mathbb{T}^4$ plus the dilaton, can be repacked into a 10$\times$10 matrix (with 5$\times$5 blocks) $M_{AB}$ whose kinetic term is $${\cal L} = \frac{1}{8} \, e\, {\rm Tr} \left[ \partial_\mu M^{AB} \partial^\mu M_{AB}\right] . $$

(I follow here the notation of Tanii's "Introduction to Supergravity" or Samtleben's "Lectures on Gauged Supergravity and Flux Compactifications")

In hep-th/9207016 Maharana and Schwarz show how to see that the bosonic part of $D=10$, ${\cal N}=(1,0)$ reduced on $\mathbb{T}^d$ are indeed invariant under $O(d,d)$.

My questions are now:

1. What is a nice reference for the Maharana - Schwarz computation for $D=10$, ${\cal N}=(2,0)$ theory reduced on $\mathbb{T}^4$, where it is explicitly shown the enlargement of the group $O(4,4)$ to $O(5,5)$ due to the inclusion of the scalars coming from the RR fields?
2. How I embed the 25 moduli into the 10$\times$10 matrix $M_{AB}$ with generic six-dimensional background geometry?
3. Bonus question: What is the complete $D=6, {\cal N}=(4,4)$ lagrangian and how I read the precise dictionary that relates 6D fields to 10D ones?

This post imported from StackExchange Physics at 2019-07-25 14:11 (UTC), posted by SE-user ifidamas