Moduli in N=1 heterotic/F-theory duality

TL;DR

The paper analyzes moduli in 4D N=1 heterotic/F-theory duality, showing a decomposition into base (even) and fiber/twisting data (continuous odd plus discrete). By relating the heterotic moduli to cohomology of spectral covers and matching them to F-theory moduli in a stable degeneration, it identifies the continuous twisting data with the F-theory intermediate Jacobian $J^3(X^4)$ and the discrete twisting data with components of Deligne cohomology, up to finite-index corrections. It then proves that the three geometric descriptions—spectral covers, cameral covers, and del Pezzo fibrations—encode the same base data and that the continuous twisting data correspond to the relative intermediate Jacobian $J^3(U/B)$, with the discrete part captured by flux-related cohomology. The analysis culminates in a duality dictionary linking heterotic bundle moduli to F-theory geometric data, including stability under degeneration and flux quantization constraints, with implications for brane impurities and four-flux consistency.

Abstract

The moduli in a 4D N=1 heterotic compactification on an elliptic CY, as well as in the dual F-theoretic compactification, break into "base" parameters which are even (under the natural involution of the elliptic curves), and "fiber" or twisting parameters; the latter include a continuous part which is odd, as well as a discrete part. We interpret all the heterotic moduli in terms of cohomology groups of the spectral covers, and identify them with the corresponding F-theoretic moduli in a certain stable degeneration. The argument is based on the comparison of three geometric objects: the spectral and cameral covers and the ADE del Pezzo fibrations. For the continuous part of the twisting moduli, this amounts to an isomorphism between certain abelian varieties: the connected component of the heterotic Prym variety (a modified Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the discrete part generalizes the matching of heterotic 5brane / F-theoretic 3brane impurities.