F-theory and linear sigma models

TL;DR

The paper develops an explicit dictionary between $(0,2)$ linear sigma models and spectral-cover data for $SU(r)$ bundles on elliptically fibered Calabi–Yau manifolds, enabling a computable map to the F-theory dual. It argues that much heterotic information is encoded in the spectral bundle and in its dual description as a gauge theory on multiple 7-branes, and provides a practical monad-based method to read off the spectral cover from $(0,2)$ data with constraints such as $c_1(V)=0$ and $c_2(V)=c_2(TZ)$. A key finding is that the simplest perturbative models often yield degenerate spectral covers, hinting at nonperturbative brane dynamics on the F-theory side that realize the missing data. These results illuminate how heterotic moduli map to F-theory data and point to deeper sub-duality structures involving degenerate spectral data and brane gauge configurations, with potential implications for constructing 4d $N=1$ vacua.

Abstract

We present an explicit method for translating between the linear sigma model and the spectral cover description of SU(r) stable bundles over an elliptically fibered Calabi-Yau manifold. We use this to investigate the 4-dimensional duality between (0,2) heterotic and F-theory compactifications. We indirectly find that much interesting heterotic information must be contained in the `spectral bundle' and in its dual description as a gauge theory on multiple F-theory 7-branes. A by-product of these efforts is a method for analyzing semistability and the splitting type of vector bundles over an elliptic curve given as the sheaf cohomology of a monad.