Partition function (quantum field theory)

Quantum mechanics

Pure mathematics

Mathematical physics

Quantum field theory

Field (mathematics)

Moduli

Physics

Mathematics

Geometry

Cauchy stress tensor

Moduli space

Torus

Journal of High Energy Physics

A bstract We revisit the results of Zamolodchikov and others on the deformation of two-dimensional quantum field theory by the determinant det T of the stress tensor, commonly referred to as $$ T\overline{T} $$ <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> . Infinitesimally this is equivalent to a random coordinate transformation, with a local action which is, however, a total derivative and therefore gives a contribution only from boundaries or nontrivial topology. We discuss in detail the examples of a torus, a finite cylinder, a disk and a more general simply connected domain. In all cases the partition function evolves according to a linear diffusion-type equation, and the deformation may be viewed as a kind of random walk in moduli space. We also discuss possible generalizations to higher dimensions.

The $$ T\overline{T} $$ deformation of quantum field theory as random geometry