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

Cylinder quantum field theories at small coupling