Correlation Functions in $\textrm{T}\bar{\textrm{T}}$-deformed Theories on the Torus

We study the correlation functions of local operators in unitary $\textrm{T}\bar{\textrm{T}}$-deformed field theories defined on a torus, using their formulation in terms of Jackiw-Teitelboim gravity. We focus on the two-point correlation 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 $\textrm{T}\bar{\textrm{T}}$-deformed field theories defined on a plane. For the latter, the behavior found was $\left(\frac{\sqrt{t}|q|}{\pi e}\right)^{-\frac{tq^2}{\pi}}$, 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: $\left(\frac{2\sqrt{t}^5q^2}{\pi e L^3|T|^2}\right)^{\frac{tq^2}{\pi}}$, 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.