Worldsheet CFTs for Flat Monodrofolds

TL;DR

Hellerman and Walcher resolve a central puzzle in string propagation on locally flat spacetimes with discrete ``gauge'' Wilson lines by showing that twisted sectors can be made level-matched not by oscillator energy alone but by fractionating base momentum in units of $\frac{1}{N^2R}$. They formalize a general Wilson line CFT (the CCC principle) that encompasses tame and wild cases, and demonstrate how to realize these backgrounds as refined orbifolds with winding-dependent phases, enabling consistent modular invariance and OPE closure. Their construction includes explicit bosonic-string examples with T-duality Wilson lines (T-folds) and SU(2) current-algebra fibers, and extends to a type II monodrofold in seven dimensions preserving ${\cal N}=1$ SUSY with fixed fiber moduli and enhanced gauge symmetry at small base radius, including self-T-duality on the base. The work provides a comprehensive, solvable framework for nongeometric string backgrounds, highlighting the crucial role of long winding strings and fractional base momenta in achieving consistent, physically meaningful theories. Overall, the results offer new avenues for nongeometric compactifications and reinforce the CCC principle as a guiding tool in constructing consistent worldsheet CFTs for stringy Wilson lines.

Abstract

We resolve a puzzle in the theory of strings propagating on locally flat spacetimes with nontrivial Wilson lines for stringy Z_N gauge symmetries. We find that strings probing such backgrounds are described by consistent worldsheet CFTs. The level mismatch in the twisted sectors is compensated by adjusting the quantization of momentum of strings winding around the Wilson line direction in units of 1/(N^2 R) rather than 1/(N R), as might have been classically expected. We demonstrate in various examples how this improvement of the naive orbifold prescription leads to satisfaction of general physical principles such as level matching and closure of the OPE. Applying our techniques to construct a Wilson line for T-duality of a torus in the type II string (``T-fold''), we find a new 7D solution with N=1 SUSY where the moduli of the fiber torus are fixed. When the size of the base becomes small this simple monodrofold exhibits enhanced gauge symmetry and a self-T-duality on the S^1 base.