Cylinder quantum field theories at small coupling

TL;DR

This paper demonstrates that 2D scalar field theories on a cylinder with Fourier-expandable potentials simplify to 1D theories with KK modes at small coupling, revealing universal T-duality invariance in their partition functions.

Contribution

It establishes a universal equivalence between 2D cylinder QFTs and 1D theories with KK modes in the small coupling limit, and computes the torus partition function for Liouville theory.

Findings

Partition function invariant under T-duality at leading order

Interactions between zero mode and KK modes are suppressed by coupling

Universal behavior observed across different cylinder QFTs

Abstract

We show that any 2D scalar field theory compactified on a cylinder and with a Fourier expandable potential $V$ is equivalent, in the small coupling limit, to a 1D theory involving a massless particle in a potential $V$ and an infinite tower of free massive Kaluza-Klein (KK) modes. Moving slightly away from the deep IR region has the effect of switching on interactions between the zero mode and the KK modes, whose strength is controlled by powers of the coupling, hence making the interactions increasingly suppressed. We take the notable example of Liouville field theory and, starting from its worldline version, we compute the torus (one-loop) partition function perturbatively in the coupling constant. The partition function at leading order is invariant under a T-duality transformation that maps the radius of the cylinder to its inverse and rescales it by the square of the Schwinger parameter of the cylinder. We show that this behavior is a universal feature of cylinder QFTs.