Correlation Functions in TT̄-deformed Theories on the Torus

We study the correlation functions of local operators in unitary TT̄-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 TT̄-deformed field theories defined on a plane. For the latter, the behavior found was |q|-tq²/π, where q is the momentum and t is the deformation parameter. For a torus, the same behavior is found for $|q|<<L/t$, where L is the torus' length scale. However, for $|q|>>L/t$, a different behavior is found: (4√t⁵q²π e L³|T|²)tq²/π, 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.