# Relationship between compactification moduli and generations in standard model

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?

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 $$h^2 - h^1$$.