Partition function (quantum field theory)

Quantum mechanics

Modular invariance

Theoretical physics

Mathematical physics

Quantum field theory

Topological quantum field theory

Field theory (psychology)

Operating system

Computer science

Modular design

Physics

Mathematics

Invariant (physics)

Mathematical analysis

Geometry

Conformal field theory

Conformal map

Uniqueness

Torus

Journal of High Energy Physics

A bstract 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 $$ T\overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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.

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