Correlation functions in \( \textrm{T}\overline{\textrm{T}} \)-deformed theories on the torus

We study the correlation functions of local operators in unitary \( \textrm{T}\overline{\textrm{T}} \)-deformed field theories defined on a torus, using their formulation in terms of Jackiw-Teitelboim gravity. We focus on the two-point correlation functions in momentum space when the undeformed theory is a conformal field theory. The large momentum behavior of the correlation functions is computed and compared to that of \( \textrm{T}\overline{\textrm{T}} \)-deformed field theories defined on a plane. For the latter, the behavior found was \( {\left(\frac{\sqrt{t}\left|q\right|}{\pi e}\right)}^{-\frac{tq^2}{\pi }} \), 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: \( {\left(\frac{2{\sqrt{t}}^5{q}^2}{\pi e{L}^3{\left|T\right|}^2}\right)}^{\frac{tq^2}{\pi }} \), where T is the complex structure 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.