Lie algebra lattices and strings on T-folds

TL;DR

This paper systematically constructs conformal field theories for T-folds using Lie algebra lattices, identifying consistent cases under T-duality and deriving modular invariant partition functions for bosonic and heterotic strings.

Contribution

It introduces a Lie algebra lattice-based framework for T-fold CFTs, classifies possible algebras under T-duality constraints, and constructs new modular invariant partition functions including non-free fermion cases.

Findings

Identified four Lie algebras compatible with T-duality fixed points.

Constructed modular invariant partition functions for T-folds in bosonic string theory.

Developed a class of partition functions for heterotic T-folds parametrized by three integers.

Abstract

We study the world-sheet conformal field theories for T-folds systematically based on the Lie algebra lattices representing the momenta of strings. The fixed point condition required for the T-duality twist restricts the possible Lie algebras. When the T-duality acts as a simple chiral reflection, one is left with the four cases, $A_1, D_{2r}, E_7, E_8$, among the simple simply-laced algebras. From the corresponding Englert-Neveu lattices, we construct the modular invariant partition functions for the T-fold CFTs in bosonic string theory. Similar construction is possible also by using Euclidean even self-dual lattices. We then apply our formulation to the T-folds in the $E_8 \times E_8$ heterotic string theory. Incorporating non-trivial phases for the T-duality twist, we obtain, as simple examples, a class of modular invariant partition functions parametrized by three integers. Our construction includes the cases which are not reduced to the free fermion construction.