# Mirror symmetry in String Theory?

1. Jul 28, 2013

### johne1618

From looking at the Wikipedia entry on string theory I gather that it is found that any given physical model implies two Calabi-Yau spaces.

Perhaps one space gives rise to a sector of particles with left-handed weak interactions and the other gives rise to a sector with right-handed weak interactions.

The right-handed sector might be the dark matter.

http://en.wikipedia.org/wiki/String_theory#Mirror_symmetry

Last edited: Jul 28, 2013
2. Jul 28, 2013

### fzero

Mirror symmetry in string theory is a type of duality, which is an equivalence between two theories. This type of duality means that the string theory on the CY manifold X is equivalent to another string theory on the mirror manifold Y. One does not add the matter computed from the theory on X to that computed from Y.

Instead, we note that the string theory on X has certain parameters, including the string coupling, as well as the size and shape parameters associated with X. For a certain range of parameters, the string theory on X is weakly-coupled and well-behaved. For other ranges, the string theory on X could be strongly coupled or otherwise poorly behaved. For instance when X develops a singularity, there are new light states appearing that are not easily described by the perturbative CFT description of X. For certain types of singularities, the description via the perturbative theory on Y is a better way to describe the physics.

Also the "mirror" term in mirror symmetry does not refer to spacetime parity. To understand it, one really needs to know some differential topology. But suffice to say, there is a certain type of topological data about manifolds, known as Hodge numbers. For a Calabi-Yau 3-manifold, the only Hodge numbers that can be different from 0 or 1 are $h^{1,1}$ and $h^{1,2}$ (while $h^{2,1} = h^{1,2}$). When physicists plotted $h^{1,1}$ vs $h^{1,2}$ for the then known CY manifolds, they found a symmetry around the line $h^{1,1}=h^{1,2}$. Namely, when there was a CY with Hodge numbers $(h^{1,1},h^{1,2})=(a,b)$, there was a corresponding CY with numbers $(h^{1,1},h^{1,2})=(b,a)$. These are the mirror pairs, and the mirror symmetry refers to the mirror reflection in the $h^{1,1},h^{1,2}$ plane when we plot the Hodge numbers of all CY 3-manifolds.

3. Jul 29, 2013