# 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 ${\displaystyle Q}$-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 CFTs

### Ising model

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 parameters

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 ${\displaystyle c}$, but is unitary only if ${\displaystyle c>1}$. 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 models

In two dimensions, a diagonal minimal model exists for any two coprime integers ${\displaystyle 2\leq p, but it is unitary only if ${\displaystyle q=p+1}$. It is possible to normalize fields such that all three-point structure constants are real, even in the non-unitary cases. Non-unitarity manifests itself by the negativity of some two-point structure constants,[5]

${\displaystyle B(P)={\frac {1}{\prod _{\pm }\Upsilon _{\beta }(\beta \pm 2P)}}}$

where ${\displaystyle \beta }$ is related to the central charge by ${\displaystyle c=13-6\beta ^{2}-6\beta ^{-2}}$ and ${\displaystyle P}$ to the conformal dimension by ${\displaystyle \Delta ={\frac {c-1}{24}}+P^{2}}$. The relevant values of the momentum are ${\displaystyle 2P_{(r,s)}=r\beta -s\beta ^{-1}}$ where ${\displaystyle 1\leq r\leq q-1}$ and ${\displaystyle 1\leq s\leq p-1}$ are Kac table indices, with ${\displaystyle \beta ^{2}={\frac {p}{q}}}$. The function ${\displaystyle \Upsilon _{\beta }(x)}$ is a type of w:multiple Gamma function.

Using the known behaviour of ${\displaystyle \Upsilon _{\beta }(x)}$ under shifts of its argument by ${\displaystyle \beta ,\beta ^{-1}}$, we find

${\displaystyle \operatorname {sign} B(P_{(r,s)})=(-1)^{\left\lfloor (q-p){\frac {r}{q}}\right\rfloor +\left\lfloor (q-p){\frac {s}{p}}\right\rfloor }}$

In particular, this is always positive if ${\displaystyle q=p+1}$. In the case ${\displaystyle \beta ^{2}={\frac {2}{5}}}$ of the Yang-Lee minimal model, ${\displaystyle B(P_{(1,1)})<0}$. More generally, in any non-unitary minimal model, some primary fields have negative conformal dimensions i.e. violate the unitarity bound, and some primary fields have negative two-point structure constants, while all irreducible representations are non-unitary. For example, here is the Kac table of the model with ${\displaystyle \beta ^{2}={\frac {3}{5}}}$, where the dimensions in red are for primary fields with negative two-point structure constants:

${\displaystyle {\begin{array}{c|cccc}2&\color {red}{\frac {3}{4}}&\color {red}{\frac {1}{5}}&-{\frac {1}{20}}&0\\1&0&-{\frac {1}{20}}&\color {red}{\frac {1}{5}}&\color {red}{\frac {3}{4}}\\\hline &1&2&3&4\end{array}}}$

## Methods for solving non-unitary CFTs

### Semi-definite programming

The unitarity-based numerical methods that rely on positivity of squared structure constants can surely be applied to some non-unitary CFTs. What matters is indeed not that squared structure constants are positive, but that we know their signs, whatever these signs are. There are non-unitary CFTs such as non-unitary minimal models, where squared three-point structure constants are positive, while two-point functions of primary fields can be either ${\displaystyle +1}$ or ${\displaystyle -1}$ (instead of always ${\displaystyle +1}$ in unitary CFTs).[6] Semi-definite programming could be adapted to allow for these discrete variables: signs of two-point functions.

## References

1. Gliozzi, Ferdinando (2013-07-11). "More constraining conformal bootstrap". arXiv.org. 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". arXiv.org. 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". arXiv.org. 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". arXiv.org. doi:10.1007/JHEP01(2018)095. Retrieved 2020-12-22.
5. Migliaccio, Santiago; Ribault, Sylvain (2017-11-24). "The analytic bootstrap equations of non-diagonal two-dimensional CFT". arXiv.org. doi:10.1007/JHEP05(2018)169. Retrieved 2021-02-21.
6. "Reality of three-point structure constants in CFT, unitary or not". Research Practices and Tools. 2014-10-08. Retrieved 2020-12-22.