## 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.

Original language | American English |
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Article number | 86 |

Journal | Journal of High Energy Physics |

Volume | 2019 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2019 |

### Bibliographical note

Funding Information:Article funded by SCOAP3.

Publisher Copyright:

© 2019, The Author(s).

## Keywords

- Conformal Field Theory
- Effective Field Theories
- Field Theories in Lower Dimensions