Reference for geometry in heterotic string compactifications

Question

I'm looking for a good reference - including articles, reviews, talks, even blog posts - that go into the gory details of the compactification of the $E_8 \times E_8$ heterotic string on a Calabi-Yau.

Specifically, a text which describes the relation between the vector bundle $V$ (whose structure group $H$ breaks one of the $E_8$ factors into the maximal subgroup $G \times H$) and the supposed "$E_8$ bundle" that should exist in 10 dimensions. None of this is particularly clear in the second volumes of Green-Schwarz-Witten or Polchinski, which don't really go beyond the standard embedding where $V=TX$, the tangent bundle to the Calabi-Yau.

My understanding is that one constructs the bundle $V$ and identifies various portions of the fibers of the $E_8$ bundle with copies of $V$, $V^*$, $V\otimes V^*$ etc. But I don't really see this description in the literature, so I'm not sure this is correct. And it's not clear where the four dimensional picture arises out of this either.

Is there any more detailed reference regarding these issues around?

This post imported from StackExchange Physics at 2017-10-27 21:09 (UTC), posted by SE-user nonreligious