Non-unitary conformal field theory

Unitary conformal field theories are the subject of many numerical bootstrap studies. These studies rely on the positivity of squared three-point structure constants in unitary theories.

However, there are many interesting non-unitary CFTs from statistical physics, such as the -state Potts model. Numerical methods for solving such CFTs, such as Gliozzi's method,[1] are less developed and less powerful.

In the solvable examples of two-dimensional w:minimal models and w:Liouville theory, unitarity does not play an important role in the structure and solvability of the CFTs. The widespread use of unitarity-based numerical bootstrap techniques could lead to an overestimation of unitarity's role. In this article we introduce some non-unitary CFTs and non-unitary bootstrap methods.

Examples of non-unitary CFTsEdit

Ising modelEdit

The Ising model can be defined in terms of spins on a lattice, whose dynamics is determined by a self-adjoint Hamiltonian. The conformal limit of the model is a unitary CFT, whose observables are correlation functions of spins and other local observables. However, non-local observables such as connectivities of clusters can in general not be expressed in terms of spins.[2]

In two dimensions, it is possible to decompose four-point cluster connectivities in terms of Virasoro conformal blocks, at the price of introducing many more representations than exist in the local CFT. These representations are not unitary. Therefore, the CFT that would describe cluster connectivites is not unitary either.[3] There is no reason for the situation to be more favourable in higher dimensions, and it is tempting to speculate that cluster connectivities again belong to a non-unitary CFT.

CFTs with continuous parametersEdit

A CFT that has a continuous parameter can be made non-unitary by giving a complex value to that parameter. This is the case with w:Liouville theory, which depends analytically on the central charge  , but is unitary only if  . Another type of continuous parameter is the coupling constant of w:N = 4 supersymmetric Yang–Mills theory, which is unitary if the coupling constant is real. Analytically continuing to complex coupling constants seems possible, and surely destroys unitarity. A toy model of w:N = 4 supersymmetric Yang–Mills theory is the conformal fishnet CFT, which also depends on a coupling constant, but is non-unitary for any value of that constant.[4]

Generalized free fields provide other toy examples of CFTs with continuous parameters.

Minimal modelsEdit

Isolated correlatorsEdit

Solutions of crossing symmetry can provide examples of correlation functions, which may or may not belong to a known CFT.[5]

Methods for solving non-unitary CFTsEdit

Gliozzi's methodEdit

Analytic bootstrapEdit


  1. Gliozzi, Ferdinando (2013-07-11). "More constraining conformal bootstrap". doi:10.1103/PhysRevLett.111.161602. Retrieved 2020-12-22.
  2. Delfino, Gesualdo; Viti, Jacopo (2011-04-21). "Potts q-color field theory and scaling random cluster model". doi:10.1016/j.nuclphysb.2011.06.012. Retrieved 2021-02-21.
  3. He, Yifei; Jacobsen, Jesper Lykke; Saleur, Hubert (2020-05-14). "Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap". doi:10.1007/JHEP12(2020)019. Retrieved 2021-02-21.
  4. Gromov, Nikolay; Kazakov, Vladimir; Korchemsky, Gregory; Negro, Stefano; Sizov, Grigory (2017-06-13). "Integrability of Conformal Fishnet Theory". doi:10.1007/JHEP01(2018)095. Retrieved 2020-12-22.
  5. Esterlis, Ilya; Fitzpatrick, A. Liam; Ramirez, David (2016-06-23). "Closure of the Operator Product Expansion in the Non-Unitary Bootstrap". doi:10.1007/JHEP11(2016)030. Retrieved 2021-05-27.