# How exactly are Calabi-Yau compactifications done?

To compactify 2 open dimensions to a torus, the method of identification written down for this example as

$$(x,y) \sim (x+2\pi R,y)$$

$$(x,y) \sim (x, y+2\pi R)$$

can be applied.

What are the methods to compactify 6 open dimensions to a Calaby-Yau manifold and how exactly do these methods work?

Instead, what is meant by "compactification" in physics is that you just choose a closed (and hence compact) manifold $Q$, then choose spacetime $X$ to be a $Q$-fiber bundle over space base space (often assumed to be just a product $X = Q \times Y$), and then describe the Kaluza-Klein mechanism for passing from physics on $X = Q \times Y$ to effective physics on just $Y$.
In particular for Calabi-Yau "compactifications" you just choose $Q$ to be a Calabi-Yau manifold, and then consider the Kaluza-Klein mechanism on spacetimes which are $Q$-fiber bundles. You don't actually obtain these spacetimes as compactifications of non-compact spacetimes in the sense of mathematics.
• I thought that "$Q$" was the Calabi-Yau manifold, not $Y$. For instance $X_{10} = CY_6 * M_{3,1}$. So I would have thought of $CY$-fiber bundles with a Minkowski base space $M_{3,1}$ – Trimok Jul 15 '13 at 9:37