Modular invariance and uniqueness of $$ T\overline{T} $$ deformed CFT

Abstract

Any two dimensional quantum field theory that can be consistently defined on a torus is invariant under modular transformations. In this paper we study families of quantum field theories labeled by a dimensionful parameter t, that have the additional property that the energy of a state at finite t is a function only of t and of the energy and momentum of the corresponding state at t = 0, where the theory becomes conformal. We show that under this requirement, the partition sum of the theory at t = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in t, to be that of a TT deformed CFT. Non-perturbatively, we find that for one sign of t (for which the energies are real) the partition sum is uniquely determined, while for the other sign we find non-perturbative ambiguities. We characterize these ambiguities and comment on their possible relations to holography.

TT¯ deformations as TsT transformations

“…The index i denotes the flavour of the particle. 8 Non-diagonal S matrices can diagonalised by nested Bethe ansatz, leading to the same conclusion. Note also that the exact spectrum will also have to account for finite size effects of the type of refs.…”

“…In a similar way, the classical Hamiltonian and Lagrangian obey an ODE in the space of fields, which can be often solved in closed form [3,4]. Over the last three years, T T deformations of a number of integrable [3,5,6], as well as of more general [7][8][9][10] theories have been considered. 1 A particularly striking link emerged between string theory and T T deformations, fueled by the initial observation that the T T deformation of a theory of free bosons is related to strings in flat space [3], see also refs.…”

Correlation functions, entanglement and chaos in the $$ T\overline{T}/J\overline{T} $$-deformed CFTs

“…Recently many attentions have been paid to the deformations of two-dimensional quantum field theory by irrelevant operators constructed from bilinears of conserved currents [1], of which the so-called TT /JT -deformation [2,3] has been extensively investigated [4][5][6][7][8][9][10][11][12][13][14][15][16]. Although these deformations are irrelevant in the renormalization group sense, the deformed theory appears to be more predictive than the generic nonrenormalizable QFT.…”

Current-current deformations, conformal integrals and correlation functions

“…The worldsheet theory obtained in this way admits the interpretation of string theory on a background that interpolates between AdS 3 and a linear dilaton background, resulting in a solvable deformation of AdS 3 /CFT 2 that exhibits Hagedorn spectrum at high energy and Cardy spectrum at low energy. This theory has been studied in [6][7][8][9][10][11][12][13][14][15][16][17][18] and in references thereof. In particular, correlation functions were studied in [8,11,12], where it was shown that the insertion of the operator (1.1) in the correlation functions produces a logarithmic divergence that leads to the renormalization of the primary operators, which consequently acquire an anomalous dimension.…”