Quantum Mechanics of the Doubled Torus

TL;DR

The paper analyzes the quantum mechanics of Hull's doubled torus by treating it as a constrained Hamiltonian system with second-class constraints and Dirac-bracket quantization, and demonstrates quantum equivalence to the conventional non-doubled torus for a simple T-fold. It derives the spectrum, Virasoro operators, and a modular-invariant partition function, incorporating both untwisted and twisted sectors and addressing zero-mode quantization through a T-duality phase. The work also extends the formalism to a consistent supersymmetric doubled torus with supersymmetric constraints, preserving the constrained-geometry approach. The results support the idea that the doubled torus provides a polarization-invariant, geometrically natural description of T-folds, yielding the same quantum physics as the standard formulation and offering a pathway to more general supersymmetric backgrounds.

Abstract

We investigate the quantum mechanics of the doubled torus system, introduced by Hull [1] to describe T-folds in a more geometric way. Classically, this system consists of a world-sheet Lagrangian together with some constraints, which reduce the number of degrees of freedom to the correct physical number. We consider this system from the point of view of constrained Hamiltonian dynamics. In this case the constraints are second class, and we can quantize on the constrained surface using Dirac brackets. We perform the quantization for a simple T-fold background and compare to results for the conventional non-doubled torus system. Finally, we formulate a consistent supersymmetric version of the doubled torus system, including supersymmetric constraints.