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?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.