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

We study the correlation functions of local operators in unitary $\text{T}\bar{\text{T}}$-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 $\text{T}\bar{\text{T}}$-deformed field theories defined on a plane. For the latter, the behavior found was ${\left(\frac{\sqrt{t}\mathrm{\mid }q\mathrm{\mid }}{\pi e}\right)}^{-\frac{t{q}^2}{\pi }}$, where $q$ is the momentum and $t$ is the deformation parameter. For a torus, the same behavior is found for $\mathrm{\mid }q\mathrm{\mid }<<L\mathrm{/}t$, where $L$ is the torus’ length scale. However, for $\mathrm{\mid }q\mathrm{\mid }>>L\mathrm{/}t$, a different behavior is found: ${\left(\frac{2{\sqrt{t}}^5{q}^2}{\pi e{L}^3\mathrm{\mid }T{\mathrm{\mid }}^2}\right)}^{\frac{t{q}^2}{\pi }}$, 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 $t\mathrm{\mid }q\mathrm{\mid }$. The difference from the plane’s result illustrates the non-locality of the theory and the UV-IR mixing.