Relationship between compactification moduli and generations in standard model

Question

The situation I am describing is a $10D$ heterotic string theory which is compactified on a Calabi-Yau to get a $N=1$, $4D$ effective theory. It is mentioned in Ashoke Sen's notes on string compactification that the difference between the number of complex structure moduli and the Kahler moduli, i.e. $(h_{2,1}-h_{1,1})$ is the number of generations of the standard model. What is the exact argument for this?

Answer 1

In general in such heterotic compactifications on a Calabi-Yau manifold $X$, you have a structure group $G$ whose commutant is the effective low-energy structure group $H$, and you get effective supermultiplets transforming in representations $(R_i,r_i)$ of $G\times H$ by decomposing the adjoint of the heterotic $E_8$ into these irreducible representations of $G\times H$.

It turns out that the multiplicity $n_{r_i}$ of the representation $r_i$ is given by the dimension $h^1(X,V_{R_i})$ of the first cohomology $H^1(X,V_{R_i})$ of the bundle $V_{R_i}$ associated to the representation $R_i$ that occurs together with $r_i$, and the number of effective generations of particles is the difference between the number of generations $n_{r_i}$ and the number of anti-generations $n_{\bar{r}_i}$. Furthermore, $V_{\bar{r}_i} = V_{r_i}^\ast$, i.e. the anti-generation bundle is the dual of the generation bundle, so $h^1(X,V_{\bar{R}_i}) = h^2(X,V_{R_i})$, and the number of effective generations is $h^2(X,V) - h^1(X,V)$, which, incidentally, is the Euler characteristic of $V$ if $h^0$ and $h^3$ vanish, as they often do.

In the case you are asking about, the $V_{R_i}$ is simply taken to be the tangent bundle of the Calabi-Yau, so the $h^i(X,V)$ are simply the standard cohomologies associated to $X$. Therefore, the number of effective generations is $h^2 - h^1$.

See chapters 5.6 through 5.8 of "Heterotic and M-theory Compactifications for String Phenomenology" and the references therein for the full argument.