Motivated by considerations coming from string theory compactifications, the authors study extremal vector bundles on Calabi-Yau threefolds, i.e. stable \(\mathrm{SU}(n)\) bundles which satisfy the cancellation of the Green-Schwarz anomaly. More explicitly, a stable vector bundle \(V\) on a Calabi-Yau threefold \(X\) is called extremal if (1) \(c_1(V)=0\) and (2) \(c_2(TX)=c_2(V)\).
Most of the article takes an experimental point of view, using various databases of stable bundles on standard constructions of Calabi-Yau threefolds to determine the rarity of extremal bundles as well as bounds on their \(c_3(V)\). Section 3, however, is the theoretical core of the paper, where the authors first show that the spectral cover construction cannot produce extremal bundles on elliptic Calabi-Yau threefolds, which serve as the main testing ground for the authors’ investigation. The authors are thus lead to consider a different construction that generalizes the classical Hartshorne-Serre correspondence and gives locally free sheaves of any rank \(n\). The essential idea is to use semi-continuity and openness of stability to find stable bundles in the same deformation space as \(\mathcal O_X^{n-1}\oplus I_C\) for some smooth curve \(C\), but not obtained simply as a non-trivial extension, as these are not always even reflexive sheaves. In rank two, for example, they first consider a coherent sheaf obtained as an extension \[ 0\to\mathcal O_X\to\mathcal E_0\to I_C\to 0, \] where \(C\) is a smooth curve in \(X\) with \([C]=c_2(TX)\), just as in the classical Serre construction. They consider a seemingly unrelated second extension \[ 0\to\mathcal O_X(-D)\to\mathcal E_1\to \tilde{\mathcal F}\to 0, \] where \(D\) is an ample divisor not containing \(C\) and \(\tilde{\mathcal F}\) is a torsion-free extension in \(\mathrm{Ext}^1(\mathcal O_D,I_C)\). The crucial observation is that \(\mathcal E_0\) and \(\mathcal E_1\) can also be seen as representing linearly independent classes \(v_0,v_1\) in some other extension space, and by using semi-continuity, openness of stability, and the construction of \(\mathcal E_0,\mathcal E_1\), the authors are able to show that the generic member of the family of sheaves represented by \((1-t)v_0+tv_1,t\in\mathbb A^1\) is a stable bundle. The authors also consider which bundles of monad type on CICY’s (complete intersection Calabi-Yau’s) satisfy the conditions of being extremal.
In Section 4 of the article, the authors use the extremality conditions to translate the DRY conjecture, which gives sufficient conditions on topological invariants for the existence of a stable vector bundle with those invariants on a Calabi-Yau threefold \(X\), into a purely topological condition on \(X\). They then determine, for example, that all of the 7890 CICY’s satisfy this topological condition upon using extremal bundles of monad type, except for 37 self-mirror manifolds. The authors proceed similarly in Section 5 by studying the statistics of the bundles they consider with regard to extremality and the DRY conjecture. For extremal bundles, as an example, they plot the “difference” between \(\mathrm{ch}_2(V)\) and \(\mathrm{ch}_2(TX)\), appropriately defined as these represent 1-dimensional cycles, versus \(\mathrm{ch}_3(V)\) as \(X\) and \(V\) range over all possibilities in a given construction and rank. In this way, they obtain some interesting general trends for characteristic classes of known bundles.