Geometric U-folds in four dimensions

TL;DR

This paper constructs and analyzes geometric U-folds in four-dimensional supergravity, revealing their topological conditions, relation to string theory, and providing explicit examples in a generalized axion-dilaton model.

Contribution

It introduces a global construction of geometric U-folds in 4D supergravity, linking their topology to flat fiber bundles and discrete U-duality groups, with explicit examples in a non-compact Riemann surface setting.

Findings

Non-trivial U-folds require non-trivial fundamental groups of scalar and space-time manifolds.

Smooth geometric U-folds are compatible with string theory if glued via subgroups of the discrete U-duality group.

Explicit examples are constructed in a generalized axion-dilaton model with a high-genus scalar manifold.

Abstract

We describe a construction of geometric U-folds compatible with a non-trivial extension of the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. The topology of geometric U-folds depends on certain flat fiber bundles which encode how supergravity fields are globally glued together. We show that smooth non-trivial U-folds of this type can exist only in theories where both the scalar and space-time manifolds have non-trivial fundamental group and in addition, the scalar map of the solution is homotopically non-trivial. Consistency with string theory requires smooth geometric U-folds to be glued using subgroups of the effective discrete U-duality group, implying that the fundamental group of the scalar manifold of such solutions must be a subgroup of the latter. We construct simple examples of geometric U-folds in a generalization of the axion-dilaton model of $\mathcal{N}=2$ supergravity coupled to a single vector multiplet, whose scalar manifold is a generally non-compact Riemann surface of genus at least two endowed with its uniformizing metric. We also discuss the relation between geometric U-folds and a moduli space of flat connections defined on the scalar manifold, which involves certain character varieties not studied in the literature.