Correlation Functions in TT-deformed Theories on the Torus

We study the correlation functions of local operators in unitary TT-deformed field theories defined on a torus, using their formulation in terms of Jackiw-Teitelboim gravity. We focus on the two-point correlation function in momentum space when the undeformed theory is a conformal field theory. The large momentum behavior of the correlation function is computed and compared to that of TT-deformed field theories defined on a plane. For the latter, the behavior found was |q|^-tq²{}, where q is the momentum and t is the deformation parameter. For a torus, the same behavior is found for |q|>L/t, a different behavior is found: (4t⁵q² e L³|T|²) ^tq²{}, where T is the modular parameter of the torus. Hence, at large momentum, the correlator decays and then grows. This behavior suggests that operators carrying momentum q are smeared on a distance scale t|q|. The difference from the plane's result illustrates the non-locality of the theory and the UV-IR mixing.