On possible Chern Classes of stable Bundles on Calabi-Yau threefolds

TL;DR

This paper proves that on elliptically fibered Calabi-Yau threefolds, infinitely many second Chern classes can be realized by stable vector bundles, addressing a conjecture related to heterotic string models.

Contribution

It demonstrates the existence of infinitely many stable vector bundles with specific Chern classes on elliptically fibered Calabi-Yau threefolds, supporting a conjecture about their realizability.

Findings

Existence of infinitely many cohomology classes of a certain form.

Construction of stable SU(n) vector bundles for these classes.

Relevance to heterotic string anomaly cancellation.

Abstract

Supersymmetric heterotic string models, built from a Calabi-Yau threefold $X$ endowed with a stable vector bundle $V$, usually lead to an anomaly mismatch between $c_2(V)$ and $c_2(X)$; this leads to the question whether the difference can be realized by a further bundle in the hidden sector. In math.AG/0604597 a conjecture is stated which gives sufficient conditions on cohomology classes on $X$ to be realized as the Chern classes of a stable reflexive sheaf $V$; a weak version of this conjecture predicts the existence of such a $V$ if $c_2(V)$ is of a certain form. In this note we prove that on elliptically fibered $X$ infinitely many cohomology classes $c\in H^4(X, {\bf Z})$ exist which are of this form and for each of them a stable SU(n) vector bundle with $c=c_2(V)$ exists.