Nongeometric $F$-theory–heterotic duality

In this work we study the duality between $F$-theory and the heterotic string beyond the stable degeneration limit in $F$-theory and large fiber limit in the heterotic theory. Building upon a recent proposal by Clingher and Doran and by Malmendier and Morrison—which phrases the duality on the heterotic side for a particular class of models in terms of (fibered) genus-two curves as nongeometric heterotic compactifications—we establish the precise limit to the semiclassical heterotic string in both eight and lower space-time dimensions. In particular for six-dimensional theories, we argue that this class of nongeometric heterotic compactifications capture ${\alpha }^{\prime }$ quantum corrections to the semiclassical heterotic supergravity compactifications on elliptically fibered $K3$ surfaces. From the nongeometric heterotic theory, the semiclassical phase on the $K3$ surface is recovered from a remarkable limit of genus-two Siegel modular forms combined with a geometric surgery operation. Finally, in four dimensions we analyze another limit deep in the quantum regime of the nongeometric heterotic string, which we refer to as the heterotic Sen limit. In this limit we can explicitly argue that the semiclassical two-staged fibrational structure of the heterotic hypermultiplet moduli space—recently established by Alexandrov, Louis, Pioline, and Valandro—gets corrected by quantum effects.

Figure 1

Depicted is the topological surgery operation performed in the limit $\sigma \to 0$. It maps the singular genus-two fibers of type $I_{1-0-0}$ (left side) to smooth elliptic fibers with the spectral cover data of an $SU(2)$ bundle (right side).