How (if) is the gauge group for gravity incorporated in the Calabi-Yau manifold of string theory?

Question

The Calabi-Yau manifold is a stable complex 3D (or real 6D) manifold on the geometry of which information about strings can be stored as fibre bundles of tensors, more or less like the electromagnetic four-vector can be seen as a vector on the small circle in Kaluza-Klein theory (corresponding to $\operatorname{U}(1)$ symmetry), represented in the ecxtended 5d metric tensor as eight extra symmetric off-diagonal terms.

The Calabi-Yau manifold incorporates the all three gauge symmetries in particle physics, i.e. $\operatorname{SU}(3)\operatorname{SU}(2)_{l}\operatorname{U}(1)_{Y}$. These gauge symmetries correspond to Lie groups and gauge fields. The gauge transformations leave spacetime untouched, unlike the the gauge transformations used to derive general relativity (corresponding to the Poincaré Poincaré group) which only touch upon spacetime itself (the symmetry demand being obviously that spacetime transformations leave the GR defined Lagrangian unchanged), so not an a particle field in spacetime.

How is this symmetry incorporated in the Calabi-Yau manifold, and how does that introduce graviton modes?

Answer 1

1. The Yang-Mills type gauge groups of an effective theory in string theory do not arise solely from the compactification, but from "wrapping" D-branes (to which open strings can be attached) around non-trivial cycles in homology - that's the connection to the geometry of the compactification manifold. Non-Abelian gauge groups arise from coincident D-branes (for a discussion of why, see this question and its answers, and more generally this phenomenon is known as gauge enhancement), which in the geometry are typically modelled by taking some limit in which the compactification manifold becomes singular so that some of the distinct cycles "merge" at the singular point and the branes become coincident. Blowing up the singularity (i.e. returning to the non-singular version) corresponds to symmetry breaking (this is mostly an M-theory viewpoint).

2. In contrast, gravity - which as you say is not gauge theory like the others, for an exact discussion of what is "gauge" about gravity see this answer of mine - is not generated by D-branes or singularities or anything like that: The 10-dimensional uncompactified low-energy effective description of string theory/M-theory, i.e. the theory that is being compactified when we talk about Calabi-Yau manifolds and whatnot, is already a theory of supergravity, because the spectrum of states of the superstring already contains a massless spin-2 particle, and by arguments very similar to how massless spin-1 bosons are always associated with Yang-Mills type gauge theories (see this answer of mine and Weinberg's book for the full argument for spin-1 and this answer and its links for the spin-2 argument), such a massless spin-2 boson is always the graviton of a theory that looks like gravity.

So there is no need to get gravitons from the compactification - they are always there from the outset (and this is arguably the reason people started thinking about string theory as a potential theory of quantum gravity in the first place after it was initially designed to explain QCD).