An Uplifting Discussion of T-Duality

TL;DR

This paper explores the mathematical structure of T-duality in string theory, revealing it acts as an order four symmetry on the state space and discussing its implications for orbifold constructions and discrete gauge symmetries.

Contribution

It generalizes the understanding of T-duality as an order four symmetry on the conformal field theory state space and connects it to mathematical concepts in Lie groups.

Findings

T-duality acts as order four on the state space.

Reevaluation of mod two conditions in orbifold constructions.

Implications for T-duality as a discrete gauge symmetry.

Abstract

It is well known that string theory has a T-duality symmetry relating circle compactifications of large and small radius. This symmetry plays a foundational role in string theory. We note here that while T-duality is order two acting on the moduli space of compactifications, it is order four in its action on the conformal field theory state space. More generally, involutions in the Weyl group $W(G)$ which act at points of enhanced $G$ symmetry have canonical lifts to order four elements of $G$, a phenomenon first investigated by J. Tits in the mathematical literature on Lie groups and generalized here to conformal field theory. This simple fact has a number of interesting consequences. One consequence is a reevaluation of a mod two condition appearing in asymmetric orbifold constructions. We also briefly discuss the implications for the idea that T-duality and its generalizations should be thought of as discrete gauge symmetries in spacetime.