This paper gives a precise count of the massless E6 singlet states in a class of heterotic string compactifications built from Calabi–Yau orbifolds. The author works in a solvable conformal field theory (CFT) description known as the free-field construction of Berglund–Hübsch orbifolds. Using that exact CFT data, the paper computes the full spectrum of singlets, including subtle “descendant” states that earlier lattice or geometric counts missed.

What the researcher did is combine three ingredients. First, the internal CFT is written as a product of N=2 minimal models, so each factor has a finite set of representations. Second, the content of descendants inside those representations is read from the ranks of Shapovalov matrices. A Shapovalov matrix is the Gram matrix of inner products among descendant states; its rank gives the number of independent states at a given level. Third, the orbifold twisted sectors and the required projection conditions are implemented in the usual free-field way. Putting these pieces together gives a simple rule: count states representation by representation, and for each allowed grade use the Shapovalov rank to get the number of independent descendants.

This rule reproduces known results and gives new ones. For the well-studied quintic orbifold with Hodge numbers (17,21) the method reproduces 17 generations, 21 antigenerations and 234 singlets. For the quintic Calabi–Yau itself the computation at the Gepner point yields 330 singlets. The paper points out that this 330 is the correct count at that symmetric CFT point and agrees with an earlier Landau–Ginzburg computation by Kachru and Witten. The frequently quoted geometric number 326 is the count at a generic complex structure and differs from 330 by four states that are charged under extra U(1) symmetries present at the Gepner point.