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In string theory models there is known to be a duality between heterotic string theory and F-theory. In particular, a heterotic model (on an elliptically fibered Calabi-Yau three-fold) can have an F-theory dual (on a K3 fibered Calabi-Yau fourfold), where bundle data on the heterotic side is dual to geometric data on the F-theory side.

I'm familiar with the realisation of the duality in terms of the spectral cover construction for the vector bundle on the heterotic side. (This is developed by Friedman, Morgan, and Witten in https://arxiv.org/abs/hep-th/9701162 and https://arxiv.org/abs/alg-geom/9709029 .) However, not every vector bundle can be given a spectral cover description (in particular, I believe the vector bundle must be semi-stable and 'regular' on every fibre, with 'regularity' defined in the papers by Friedman, Morgan, and Witten, and both of these conditions can separately fail for a general vector bundle).

My question then is whether there is an F-theory dual for a heterotic model in the absence of a spectral cover description for the vector bundle on the heterotic side. A related question is whether it is known whether there always exists an F-theory dual for a heterotic model (on an elliptic fibration), but I believe there is no definitive answer to this in the literature.

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