I'm trying to understand some of the important properties of KK-compactifications of 10-dim heterotic string supergravity on 6-dim Calabi-Yau ($CY_3$) manifolds to a 4-dim theory with ${\cal N}=1$ supersymmetry (4 supercharges).
Towards the end of this final introductory lecture on string theory by Freddy Cachazo, a couple of S-dualities between compactifications of type IIA and heterotic $E_8 \times E_8$ supergravity are mentioned:
- type IIA on $K3$ $\;\;\longleftrightarrow\;\;$ heterotic $E_8 \times E_8$ on $T^4$ (6-dim, 16 supercharges)
- type IIA on $CY_3$ $\;\;\longleftrightarrow\;\;$ heterotic $E_8 \times E_8$ on $T^2 \times K3$ (4-dim, 8 supercharges)
By compactifying heterotic $E_8 \times E_8$ on $CY_3$, we then get a 4-dim theory with 4 supercharges.
- Continuing the pattern above, is this compactification of heterotic $E_8 \times E_8$ dual to some compactification of type IIA (there isn't one mentioned here)?
- Based on answers such as this, is it true to say that the particular choice of $CY_3$ determines how the $E_8 \times E_8$ gets spontaneously broken to a smaller group?
