Closed String Partition Functions in Toroidal Compactifications of Doubled Geometries

We revisit partition functions of closed strings on toroidal backgrounds, including their $mathbb{Z}_N$ shift orbifolds in the formalism where the dimension of the target space is doubled to make T-duality manifest. In such a T-duality covariant formalism, the constraint equation imposes a form of chiral factorization. Our computation furnishes a non-trivial consistency check for the quantum worldsheet theory of the doubled sigma model, when strings are placed on general toroidal backgrounds. The topological term that mixes the physical space and its T-dual is crucial in demonstrating that chiral factorization works, and that we obtain the correct partition function after imposing the constraints. Finally, we discuss how our results extend to $mathcal{N}=1$ worldsheet supersymmetry and string worldsheets of higher genus.