On the $T\bar T$ deformation of the compactified boson and its interpretation in Lattice Gauge Theory

We study the effective string description of space-like Polyakov loop correlators at finite temperature with the goal of describing the behaviour of the space--like string tension in the vicinity of the deconfinement transition. To this end we construct the partition function of the Nambu-Goto effective string theory in presence of a compact transverse direction of length $L$ equal to the inverse temperature. We then show that, under particular conditions, our result can be interpreted as the partition function of the $T\bar T$ deformation of the 2d quantum field theory describing a compactified bosonic field and that this mapping allows a deeper insight on the behaviour of the space-like observables of the theory. In particular we show, by imposing that the spectrum of the model obeys the inviscid Burgers equation, that the $T\bar T$ deformations follow well defined trajectories in the parameter space $(σ,L)$ of the model, where $σ$ is the string tension, which are characterized by a constant value of the dimensionless compactification radius $ρ=L\sqrt{σ/2π}$. We discuss the potential usefulness of these results for studying the space--like string tension of the underlying Lattice Gauge Theory and its behaviour across the deconfinement transition.