Modular invariance and uniqueness of $T\bar{T}$ 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 $T\bar{T}$ 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.

20 pages; v2 : minor improvements in presentation

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Cited by in corpus (102)

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• $T\bar{T},J\bar{T}$, $T\bar{J}$ Partition Sums From String Theory
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• $T\overline{T}$ Deformations of Supersymmetric Quantum Mechanics
• The Most Irrational Rational Theories
• Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy
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• Exact $T\bar{T}$-deformation of two-dimensional Yang-Mills theory on the sphere
• Quantization of the ModMax Oscillator
• A Path Integral Realization of Joint $J\bar{T}$, $T\bar{J}$ and $T\bar{T}$ Flows
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• Composite operators in $T\bar{T}$-deformed free QFTs
• Characters of irrelevant deformations
• Root-$T\overline{T}$ deformed CFT partition functions at large central charge
• Soliton, breather and shockwave solutions of the Heisenberg and the $T\bar{T}$ deformations of scalar field theories in 1+1 dimensions
• Comments on D3-Brane Holography
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• Complete Minimal Form Factors for Irrelevant Deformations of Integrable Quantum Field Theory