Liouville field theory

Liouville theory is defined for all complex values of the central charge

, and two eigenvectors are linearly dependent if their momenta are related by the reflection where the background charge is While the exponential potential breaks momentum conservation, it does not break conformal symmetry, and Liouville theory is a conformal field theory with the central charge Under conformal transformations, an energy eigenvector with momentum

These quantum symmetries of Liouville theory are however not manifest in the Lagrangian formulation, in particular the exponential potential is not invariant under the duality.

of Liouville theory is a diagonal combination of Verma modules of the Virasoro algebra, where

denote the same Verma module, viewed as a representation of the left- and right-moving Virasoro algebra respectively.

In terms of momenta, corresponds to The reflection relation is responsible for the momentum taking values on a half-line, instead of a full line for a free theory.

A vacuum state can be defined, but it does not contribute to operator product expansions.

In Liouville theory, primary fields are usually parametrized by their momentum rather than their conformal dimension, and denoted

, the three-point structure constant is given by the DOZZ formula (for Dorn–Otto[2] and Zamolodchikov–Zamolodchikov[3]), where the special function

-point functions on the sphere can be expressed in terms of three-point structure constants, and conformal blocks.

[5][6] Liouville theory exists not only on the sphere, but also on any Riemann surface of genus

Technically, this is equivalent to the modular invariance of the torus one-point function.

Due to remarkable identities of conformal blocks and structure constants, this modular invariance property can be deduced from crossing symmetry of the sphere four-point function.

that appears in the action must be marginal, i.e. have the conformal dimension This leads to the relation between the background charge and the coupling constant.

-point correlation function of primary fields is It has been difficult to define and to compute this path integral.

It is only in the 2010s that a rigorous probabilistic construction of the path integral was found, which led to a proof of the DOZZ formula[9] and the conformal bootstrap.

[6][10] When the central charge and conformal dimensions are sent to the relevant discrete values, correlation functions of Liouville theory reduce to correlation functions of diagonal (A-series) Virasoro minimal models.

WZW model) can be expressed in terms of correlation functions of Liouville theory.

In two dimensions, the Einstein-Hilbert action is topological, i.e. it is proportional to the Euler characteristic.

Nevertheless, after quantization, general relativity is no longer topological, because of the Weyl anomaly: under a rescaling of the metric

This leads to the construction of Liouville gravity as a product of three CFTs: Liouville theory for the gravitational sector, Faddeev-Popov ghosts for Weyl invariance (viewed as a gauge symmetry), and an arbitrary CFT that describes matter.

The central charges of these CFTs must sum to zero in order to cancel the Weyl anomaly, and ensure that the quantum theory is topological.

Correlation numbers can be computed explicitly in some examples, such as the Virasoro minimal string.

The tachyon field equation of motion in the linear dilaton background requires it to take an exponential solution.

The Polyakov action in this background is then identical to Liouville field theory, with the linear dilaton being responsible for the background charge term while the tachyon contributing the exponential potential.

[22] These models describe a thermal particle in a random potential that is logarithmically correlated.

This has applications to extreme value statistics of the two-dimensional Gaussian free field, and allows to predict certain universal properties of the log-correlated random energy models (in two dimensions and beyond).

Liouville theory is related to other subjects in physics and mathematics, such as three-dimensional general relativity in negatively curved spaces, the uniformization problem of Riemann surfaces, and other problems in conformal mapping.

It is also related to instanton partition functions in a certain four-dimensional superconformal gauge theories by the AGT correspondence.

[24] It was first called Liouville theory when it was found to actually exist, and to be spacelike rather than timelike.