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

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?