Non-Geometric F-Theory-Heterotic Duality

Abstract

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-Doran and Malmendier-Morrison -which phrases the duality on the heterotic side for a particular class of models in terms of (fibered) genus two curves as non-geometric heterotic compactifications -we establish the precise limit to the semi-classical heterotic string in both eight and lower space-time dimensions. In particular for six dimensional theories, we argue that this class of non-geometric heterotic compactifications capture α ′ quantum corrections to the semi-classical heterotic supergravity compactifications on elliptically fibered K3 surfaces. From the non-geometric heterotic theory, the semi-classical 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 non-geometric heterotic string, which we refer to as the heterotic Sen limit. In this limit we can explicitly argue that the semi-classical two-staged fibrational structure of the heterotic hypermultiplet moduli spacerecently established by Alexandrov, Louis, Pioline and Valandro -gets corrected by quantum effects.

Unbroken $E_7\times E_7$ nongeometric heterotic strings, stable degenerations and enhanced gauge groups in F-theory duals

“…Comparing the Weierstrass equation (32) with equation ( 19), we find that the following substitutions:…”

“…Now we demonstrate that F-theory on an extremal K3 surface (32) can be seen as a deformation of stable degeneration, owing to an effect of coincident 7-branes. As deduced in [23], the K3 extremal fibration (32) is obtained as the quadratic base change of the extremal rational elliptic surface X [III * , 2,1] in which two type I 2 fibers and two type I 1 fibers collide. Whereas the quadratic base change of a rational elliptic surface generally yields an elliptic K3 surface, with twice as many singular fibers as the original rational elliptic surface, at the special limits at which singular fibers collide, the singularity type of the resulting K3 surface is enhanced.…”

“…Because a K3 surface with two type III * fibers, a type I 4 fiber, and a type I 2 fiber has the singularity type E 2 7 A 3 A 1 , the K3 surface S 1 deforms and it becomes an extremal K3 surface (32) in this limit. In short, F-theory on the extremal K3 surface (32) can be seen as deformation of the stable degeneration because of the coincident 7-branes.…”

“…Two Wilson line expectation values are non-trivial in non-geometric heterotic strings with unbroken e 7 e 7 algebra. See also [32,33,34,35,36,37,38,39,40] for recent progress on non-geometric heterotic strings.…”

“…We also discuss applications to SO (32) heterotic strings in this study. We deduce the Weierstrass equations of elliptic K3 surfaces appearing as the compactification spaces of the F-theory duals of some SO(32) heterotic strings, which are obtained as the transformations of e 7 e 7 non-geometric heterotic strings.…”