# Critical 2d Potts model

In two dimensions, the critical Potts model is believed to have an infinite-dimensional symmetry algebra, which makes it accessible to analytic methods of two-dimensional conformal field theory. There may be a realistic prospect of solving it, in the sense of analytically (or semi-analytically) computing observables such as cluster connectivities.

## Motivations

The Potts model is a statistical model that generalizes the Ising model, and includes percolation as a special case. Therefore, it generalizes and unifies statistical models of fundamental interest.

In two dimensions and in the critical limit, the model may be solvable. The torus partition function is already known analytically. Powerful methods such as the conformal bootstrap can be used. In higher dimensions, the model's lack of unitarity would be an obstacle to using standard conformal bootstrap methods. But in two dimensions, conformal symmetry is more powerful and non-unitary models can be solved.

## Type of project

Tools: Given the known information on the spectrum that was obtained by other means, it may be possible to complete the solution of the model using the bootstrap method, analytic and/or semi-analytic. Interpreting the results in terms of statistical observables such as cluster connectivities and/or spin correlation functions may however require going back to the statistical description.

Chances of success: The known results on describing some observables exactly or approximately can surely be generalized to some extent. However, it is not known whether it is possible to build a consistent CFT from observables in the Potts model, and if so, whether that CFT includes all interesting observables. The scope of the project is therefore not precisely defined.

Length and difficulty: The known spectrum contains several qualitatively different infinite series of states. The statistical model contains various interesting observables. And even after solving the model for generic numbers of states, understanding special cases would require a lot of work. Solving the 2d critical Potts model is something in between a project and a field of research.

## Known results

Some aspects of the model are reviewed in the literature, in the loop model approach or in the conformal booststrap approach.

### Spectrum

The torus partition function is known exactly, see ref. or ref. for reviews. We write $\chi _{\langle r,s\rangle }$ the character of a diagonal degenerate representation with $r,s\in \mathbb {N} ^{*}$ . We write $\chi _{(r,s)}^{N}$ the character of the generally non-diagonal representation ${\mathcal {V}}_{\Delta _{(r,s)}}\otimes {\bar {\mathcal {V}}}_{\Delta _{(r,-s)}}$ . The torus partition function reads

$Z^{\text{Potts}}=\sum _{s=1}^{\infty }\chi _{\langle 1,s\rangle }+(Q-1)\sum _{s\in \mathbb {N} +{\frac {1}{2}}}\chi _{(0,s)}^{N}+\sum _{r=2}^{\infty }\sum _{s\in {\frac {1}{r}}\mathbb {Z} }D'_{r,s}\chi _{(r,s)}^{N}$ where we define

$D'_{r,s}={\frac {1}{r}}\sum _{r'=0}^{r-1}e^{2\pi ir's}w(r\wedge r')$ and

$w(d)=q^{2d}+q^{-2d}+(-1)^{d}(Q-1)\quad {\text{with}}\quad {\sqrt {Q}}=q+q^{-1}$ However, in contrast to what happens in CFTs such as minimal models, the partition function does not fully characterize the spectrum. The partition function characterizes the eigenvalues of the generator $L_{0}$ of the Virasoro algebra, but this is not enough for characterizing the action of the full algebra when that generator is not diagonalizable. And the spectrum of the Potts model is known to involve some representations where $L_{0}$ is not diagonalizable.

### Observables

In addition to the spectrum, the statistical model's observables include spin correlation functions and cluster connectivities. Spin correlation functions are a priori defined only when the number of states is an integer $Q=2,3,4,\dots$ . Spin clusters allow the model to be reformulated such that $Q$ is an arbitrary complex number.

It is natural to assume that in the critical limit, spin correlation functions tend to correlation functions of primary fields in a CFT. Somewhat less naturally, this assumption is also extended to cluster connectivities, at least for 2, 3 and 4-point connectivities.

### Fusion rules

In order to compute correlation functions, we should know not only the spectrum, but also which states contribute to which correlation functions: in other words, the fusion rules. A piece of information is already known: which states contribute to four-point connectivities in various channels. However, four-point connectivities are only a very special type of correlation functions.

### Correlation functions

Cluster connectivities are well approximated by correlation functions in an exactly solvable CFT, on the plane and on the torus. However, this approximation is not exact. Exact descriptions of correlation functions are available only for some particular correlation functions and some particular values of the number of states: $Q=0,2,3,4$ .

## Work to be done

### Determining the spectrum

To complete the determination of the spectrum, what is missing is the structure of the logarithmic representations, i.e. the representations where $L_{0}$ is not diagonalizable.

In principle this could first be determined in the lattice model (i.e. the non-critical model). However, it is not easy to take the critical limit of lattice results.

Alternatively, the needed structures could be determined from bootstrap considerations. The logarithmic representations are indeed needed for making conformal blocks finite, although this consideration does not fully determine their structure. Additional information may come from the existence of degenerate fields. In the boostrap approach, determining the structure of representations is closely related to computing the corresponding conformal blocks, which are needed for computing correlation functions.

### Bootstrapping connectivities

Four-point connectivities are already the subject of existing works, and can be compared to results from the lattice approach or from Monte-Carlo calculations. It is therefore natural to begin the study of four-point functions with four-point connectivities, i.e. four-point functions of fields with Kac table indices $(0,{\tfrac {1}{2}})$ .

Moreover, the lattice approach provides predictions for the necessary fusion rules. The knowledge of the fusion rules makes it possible to use the semi-analytic bootstrap method.

### Bootstrapping four-point functions

Connectivites are very special types of four-point functions. But in order to solve the model, we should in principle compute four-point functions of arbitrary fields. Once the spectrum is known and bootstrap techniques are available, the remaining missing ingredients are the fusion rules.

It is possible to adopt the null assumption that all states can appear in all channels of all four-point functions, but this may lead to unresolvable ambiguities. It would probably be better to guess the fusion rules, based on the known cases and on algebraic constraints from Virasoro symmetry, and to test the guesses by checking crossing symmetry.

### Interpreting correlation functions

Assuming that crossing-symmetric four-point functions are found, we have a consistent CFT. There remains the problem of interpreting its correlation functions in terms of the original Potts model.

The CFT will a priori include four-point functions that correspond to four-point connectivities. Linear combinations of four-point functions may correspond to spin four-point functions. Spin correlation functions are originally defined only if the number of states $Q$ is integer, but they may well make sense of any complex $Q$ by analytic continuation. The interpretation of higher correlation functions is however less clear.

### Limits and special cases

Cases of particular interest, including percolation, occur for rational values of the central charge, where the algebraic and analytic structures of CFT become more complicated. Understanding the behaviour of the spectrum in such cases is already challenging. Moreover, in correlation functions, we expect that the limit is finite due to cancellations of infinitely many singularities.

## Literature watch

### A 2019 article by Dotsenko

Main claims:

• For values of the Virasoro algebra's central charge $0\leq c\leq 1$ , the primary field $\sigma$ which describes spins in the q-state Potts model has nonvanishing couplings $\left\langle \sigma \sigma \Phi _{r,s}\right\rangle$ with degenerate fields whose first index $r$ is odd.
• These couplings can be computed with Coulomb gas techniques, in other words Dotsenko-Fateev integrals, and they are products of Gamma functions.
• A four-point function $\left\langle \sigma \sigma \sigma \sigma \right\rangle$ can be computed, whose s-channel decomposition is a sum over degenerate fields with one odd index.
• This four-point function belongs to the Potts model.
• In the limit $c\to 0$ , we obtain four-point functions in percolation.

Objections:

• The non-vanishing of $\left\langle \sigma \sigma \Phi _{r,s}\right\rangle$ for $r$ odd is a simple consequence of fusion rules. Computing it with Gamma functions (rather than double Gamma functions) obscures the fact that it depends on the conformal dimension $\Delta _{r,s}$ , and not on the indices $r,s$ separately.
• The dimensions $(\Delta _{r,s})_{(r,s)\in (2\mathbb {N} +1)\times \mathbb {N} ^{*}}$ are dense in the half-line $({\tfrac {c-1}{24}},\infty )$ . Therefore, in Dotsenko's expression for the four-point function, the sum over $r,s$ diverges.
• If we regularized the sum, we would obtain the continuous spectrum $\Delta \in ({\tfrac {c-1}{24}},\infty )$ of Liouville theory. Therefore, Dotsenko's four-point function belongs to Liouville theory, not the Potts model. These two CFTs are known to have different spectrums, in particular the Potts model's spectrum is discrete, with only positive conformal dimensions.