# Analytic bootstrap approach to 2d CFT: from BPZ equations to the DOZZ formula

Sylvain Ribault

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Sylvain Ribault. Analytic bootstrap approach to 2d CFT: from BPZ equations to the DOZZ formula. Master. France. 2025, pp.9. hal-05394676

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Submitted on 3 Dec 2025

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# Analytic bootstrap approach to 2d CFT: from BPZ equations to the DOZZ formula

Sylvain Ribault, December 2025

sylvain.ribault@ipht.fr

**Abstract:** In 2d conformal field theories that have degenerate fields, analytic results can be derived in the bootstrap approach. Degenerate fields indeed imply Belavin-Polyakov-Zamolodchikov equations for 4-point functions. Together with crossing symmetry and single-valuedness, this leads to shift equations for structure constants. Such shift equations are solved in terms of Barnes' double Gamma function. In the case of Liouville theory, the solution is called the DOZZ formula. The same formula also provides the solution of diagonal minimal models. Similar techniques can be applied to other CFTs such as critical loop models.

**Based on:** [1, 2]

## Contents

<table><tbody><tr><td><strong>1 Principles of 2d CFT</strong></td><td><strong>2</strong></td></tr><tr><td>1.1 Introduction</td><td>2</td></tr><tr><td>1.2 Degenerate fields</td><td>2</td></tr><tr><td>1.3 Operator produce expansions</td><td>3</td></tr><tr><td><strong>2 Analytic formulas for structure constants</strong></td><td><strong>4</strong></td></tr><tr><td>2.1 BPZ equations and their solutions</td><td>4</td></tr><tr><td>2.2 Single-valued 4-point functions</td><td>5</td></tr><tr><td>2.3 Shift equations for 3-point structure constants</td><td>5</td></tr><tr><td>2.4 Barnes double Gamma function</td><td>6</td></tr><tr><td><strong>3 Application to particular CFTs</strong></td><td><strong>7</strong></td></tr><tr><td>3.1 A-series minimal models</td><td>7</td></tr><tr><td>3.2 Liouville theory</td><td>8</td></tr><tr><td>3.3 Critical loop models</td><td>8</td></tr><tr><td><strong>Acknowledgements</strong></td><td><strong>9</strong></td></tr></tbody></table>
# 1 Principles of 2d CFT

## 1.1 Introduction

In 2d, the conformal algebra is infinite-dimensional: a product of two Virasoro algebras. This makes conformal symmetry much more constraining than in $d \ge 3$ dimensions. But this is not enough for solving any nontrivial theory. To solve CFTs such as minimal models and Liouville theory, we also need the existence of degenerate fields.

Now, when a theory is solvable, there are usually several ways to solve it. In the case of minimal models and Liouville theory, there is also the Coulomb gas method, based on path integrals. However, the analytic bootstrap method, based on degenerate fields, is simpler and more general.

Solving a CFT using degenerate fields is to some extent analogous to deriving conformal symmetry constraints. In both cases, if we want to constrain a 3-point function $\langle VVV \rangle$, we consider an auxiliary 4-point function with an extra field $\langle TVVV \rangle$ or $\langle V_{\langle 1,2 \rangle}^d VVV \rangle$, where $V_{\langle 1,2 \rangle}^d$ is a degenerate field. In the case of conformal symmetry, the extra field is the energy-momentum tensor $T$, whose OPE with a primary field reads

$$ T(y)V_{\Delta}(z) = \frac{\Delta}{(y-z)^2}V_{\Delta}(z) + \frac{1}{y-z}\frac{\partial}{\partial z}V_{\Delta}(z) + \sum_{n=2}^{\infty}(y-z)^{n-2}L_{-n}V_{\Delta}(z) \quad (1) $$

This includes infinitely many descendant fields $L_{-n}V_{\Delta}(z)$, including $L_{-1}V_{\Delta}(z) = \partial_z V_{\Delta}(z)$. Using the properties $\bar{\partial}T = 0$ and $T(y) = O(y^{-4})$, we obtain Ward identities that allow us to deduce any correlation function of descendant fields, from the corresponding correlation function of primary fields. For example,

$$ \left\langle L_{-2}V_{\Delta_1}(z_1) \prod_{i=2}^{N} V_{\Delta_i}(z_i) \right\rangle = \sum_{i=2}^{N} \left( \frac{1}{z_1 - z_i} \frac{\partial}{\partial z_i} + \frac{\Delta_i}{(z_i - z_1)^2} \right) \left\langle \prod_{i=1}^{N} V_{\Delta_i}(z_i) \right\rangle \quad (2) $$

(Exercise: derive this formula. Same question for later formulas in this lecture.)

## 1.2 Degenerate fields

Now, what about degenerate fields? A degenerate field may be defined as a primary field that has a vanishing descendant field. A generic primary field belongs to an irreducible Verma module, which implies that we cannot set a descendant field to zero without setting the primary field to zero. The descendant fields that can be set to zero are called singular vectors, they only exist in reducible Verma modules, with particular values of the conformal dimension. A singular vectors obeys $L_{n>0}V = 0$.

$V_{\Delta}$ has a singular vector $L_{-1}V_{\Delta}$ at level 1 if $L_1L_{-1}V_{\Delta} = 0$ which implies $\Delta = 0$. This implies that there are 2 types of primary fields of dimension $\Delta = 0$: the non-degenerate field $V_0$, which generates a Verma module where $L_{-1}V_0 \neq 0$, and the degenerate field $V_{\langle 1,1 \rangle}^d$, which obeys $\partial_z V_{\langle 1,1 \rangle}^d(z) = L_{-1}V_{\langle 1,1 \rangle}^d(z) = 0$. This field is $z$-independent, it is also called the identity field.

To get nontrivial constraints on correlation functions, we need to consider the next simplest degenerate fields: $V_{\langle 2,1 \rangle}^d$ and $V_{\langle 1,2 \rangle}^d$, which have vanishing descendants at level 2. To write their conformal
dimensions, let us introduce the notations

$$c = 1 - 6 (\beta - \beta^{-1})^2, \quad \Delta = \frac{c-1}{24} + P^2, \quad P_{(r,s)} = \frac{1}{2} (\beta r - \beta^{-1} s) \quad (3)$$

A primary field $V_P$ with the momentum $P = P_{(r,s)}$ with $r, s \in \mathbb{N}^*$ has a singular vector at level $rs$. If we set that singular vector to zero, we call this field degenerate and write it $V_{\langle r,s \rangle}^d$. In the case of level 2, the vanishing descendants are

$$(L_{-1}^2 - \beta^2 L_{-2}) V_{\langle 2,1 \rangle}^d = 0, \quad (L_{-1}^2 - \beta^{-2} L_{-2}) V_{\langle 1,2 \rangle}^d = 0 \quad (4)$$

## 1.3 Operator produce expansions

The existence of OPEs is the fundamental axiom of the bootstrap approach to CFT. The idea is that a product of any two fields is a linear combination of fields:

$$V_1(z_1)V_2(z_2) = \sum_k C_{12}^k(z_1, z_2)V_k(z_2) \quad (5)$$

If we know the functions $C_{12}^k(z_1, z_2)$, called OPE coefficients, we can compute any N-point function by reducing it to simpler correlation functions. In the bootstrap approach we determine OPE coefficients by solving crossing symmetry equations, which are obtained by performing different OPEs in the same 4-point function $\langle V_1V_2V_3V_4 \rangle$.

To have a hope of solving crossing symmetry equations, we should first simplify them using conformal symmetry. Conformal symmetry implies that OPEs of primary fields are of the form

$$V_1(z_1)V_2(z_2) = \sum_{k \text{ primary}} C_{12k} \left| z_{12}^{\Delta_k - \Delta_1 - \Delta_2} \sum_{L \in \mathcal{L}} z_{12}^{|L|} f_{\Delta_1, \Delta_2}^{\Delta_k, L} \right|^2 L \bar{L} V_k(z_2) \quad (6)$$

Here the sum is over primary fields, $C_{12k}$ is a 3-point structure constant, $\mathcal{L} = \{1, L_{-1}, L_{-1}^2, L_{-2}, \dots\}$ is the set of creation operators, and the coefficients $f_{\Delta_1, \Delta_2}^{\Delta_k, L}$ are determined by conformal symmetry, with $f_{\Delta_1, \Delta_2}^{\Delta_k, 1} = 1$. The modulus squared notation means

$$|f(c, \Delta, z)|^2 = f(c, \Delta, z) f(c, \bar{\Delta}, \bar{z}) \quad (7)$$

where $z, \bar{z}$ are complex conjugates, while $\Delta, \bar{\Delta}$ are generally not. The OPE can be summarized as

$$V_1(z_1)V_2(z_2) = \sum_{k \text{ primary}} C_{12k} \left| z_{12}^{\Delta_k - \Delta_1 - \Delta_2} \right|^2 (V_k(z_2) + \text{descendants}) \quad (8)$$

If $V_1 = V_{\langle 1,1 \rangle}^d$, since $\partial z_1 V_{\langle 1,1 \rangle}^d(z_1) = 0$, we find $\Delta_k = \Delta_2$ so that

$$V_{\langle 1,1 \rangle}^d(z_1)V_2(z_2) = C_{122}V_2(z_2) \quad (9)$$

with no descendants, and we can set $C_{122} = 1$ by a choice of normalization. This is why $V_{\langle 1,1 \rangle}^d$ is called the identity field. In the case of level 2 degenerate fields, a more complicated calculation yields

$$V_{\langle 2,1 \rangle}^d(z_1)V_P(z_2) = \sum_{\pm} C_{\pm}(P) \left| z_{12}^{\pm \beta P + \frac{1-\beta^2}{2}} \right|^2 \left( V_{P \pm \frac{\beta}{2}}(z_2) + \text{descendants} \right) \quad (10)$$

$$V_{\langle 1,2 \rangle}^d(z_1)V_P(z_2) = \sum_{\pm} \tilde{C}_{\pm}(P) \left| z_{12}^{\pm \beta^{-1}P + \frac{1-\beta^{-2}}{2}} \right|^2 \left( V_{P \pm \frac{1}{2\beta}}(z_2) + \text{descendants} \right) \quad (11)$$
More generally, in the OPE $V_{\langle r,s \rangle}^d V_P$ of a level $rs$ degenerate field with a generic primary field, $rs$ primary fields appear. In fact, we could define a degenerate field as a field whose OPE with a primary field yields finitely many primary fields.

# 2 Analytic formulas for structure constants

## 2.1 BPZ equations and their solutions

We can now compute 4-point functions of one degenerate and 3 primary fields,

$$Z(z_i) = \langle V_{\langle 2,1 \rangle}^d(z_0) V_{P_1}(z_1) V_{P_2}(z_2) V_{P_3}(z_3) \rangle \quad (12)$$

The vanishing of a level 2 descendant of the degenerate field (4) leads to a second-order differential equation for $Z$, called a BPZ equation. This equation is derived with the help of Eq. (2):

$$\left( \frac{1}{\beta^2} \frac{\partial^2}{\partial z_0^2} - \sum_{i=1}^{3} \left( \frac{1}{z_0 - z_i} \frac{\partial}{\partial z_i} + \frac{\Delta_i}{(z_i - z_0)^2} \right) \right) Z(z_i) = 0 \quad (13)$$

This looks like a complicated PDE in 4 variables, but global conformal symmetry reduces $Z$ to a function of one variable, the cross-ratio $z = \frac{z_{01}z_{23}}{z_{02}z_{13}}$. Equivalently, we may fix $(z_1, z_2, z_3) = (0, \infty, 1)$, and obtain an ODE for $Z(z) = \langle V_{\langle 2,1 \rangle}^d(z) V_{P_1}(0) V_{P_2}(\infty) V_{P_3}(1) \rangle$:

$$\left( \frac{z(z-1)}{\beta^2} \frac{\partial^2}{\partial z^2} + (2z-1) \frac{\partial}{\partial z} + \frac{\Delta_1}{z} + \Delta_{(2,1)} - \Delta_2 + \frac{\Delta_3}{1-z} \right) Z(z) = 0. \quad (14)$$

To solve this equation, we may look for solutions of the type

$$\mathcal{F}(z) = z^{\gamma} [1 + a_1 z + a_2 z^2 + \dots] \quad \text{or} \quad \mathcal{F}(z) = (1-z)^{\delta} [1 + b_1(1-z) + b_2(1-z)^2 + \dots] \quad (15)$$

The characteristic exponents $\gamma, \delta$ are constrained by the differential equation, and may only take the values

$$\gamma_{\pm} = \pm \beta P_1 + \frac{1-\beta^2}{2}, \quad \delta_{\pm} = \pm \beta P_3 + \frac{1-\beta^2}{2} \quad (16)$$

These agree with the exponents in Eq. (10). This tells us that the solution $\mathcal{F}_{\pm}^{(s)}$ corresponding to $\gamma_{\pm}$ is the sum of the contributions of $V_{P_1 \pm \frac{\beta}{2}}$ and its left-moving descendants in the OPE $V_{\langle 2,1 \rangle}^d(z) V_{P_1}(0)$. The solution $\mathcal{F}_{\pm}^{(t)}$ corresponding to $\delta_{\pm}$ similarly sums up contributions to the OPE $V_{\langle 2,1 \rangle}^d(z) V_{P_3}(1)$. These solutions are called s- and t-channel conformal blocks. They may be written in terms of hypergeometric functions, for example

$$\mathcal{F}_{+}^{(s)}(z) = z^{\beta P_1 + \frac{1-\beta^2}{2}} (1-z)^{\beta P_3 + \frac{1-\beta^2}{2}} {}_2F_1\left(\frac{1}{2} + \beta P_{123}, \frac{1}{2} + \beta P_{13}^2, 1 + 2\beta P_1, z\right) \quad (17)$$

The second-order BPZ equation has a 2-dimensional space of solutions, with $\mathcal{F}_{\pm}^{(s)}$ and $\mathcal{F}_{\pm}^{(t)}$ providing two bases. We know everything about hypergeometric functions, and therefore we know the change of basis

$$\mathcal{F}_{\epsilon_1}^{(s)} = \sum_{\epsilon_3 \pm} F_{\epsilon_1, \epsilon_3} \mathcal{F}_{\epsilon_3}^{(t)} \quad \text{with} \quad F_{\epsilon_1, \epsilon_3} = \frac{\Gamma(1 + 2\beta\epsilon_1 P_1)\Gamma(-2\beta\epsilon_3 P_3)}{\prod_{\pm} \Gamma(\frac{1}{2} + \beta\epsilon_1 P_1 \pm \beta P_2 - \beta\epsilon_3 P_3)} \quad (18)$$
The degenerate fusing matrix $F$ is written in terms of Euler's Gamma function, a special function whose main properties are

$$ \Gamma(x+1) = x\Gamma(x) \quad , \quad \Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin(\pi x)} \qquad (19) $$

## 2.2 Single-valued 4-point functions

Which solution of the BPZ equation should we use to write our 4-point function $Z(z)$? BPZ equations are meromorphic in $z$, and do not depend on right-moving conformal dimensions. However, correlation functions are single-valued functions of $z$ like $|z^\delta|^2$ but unlike $z^\delta$, and also depend on right-moving dimensions. For simplicity we consider diagonal CFTs, i.e. CFTs such that all primary fields have zero conformal spin $\Delta - \bar{\Delta}$. From now on, $V_P$ therefore means a primary field with the same left and right momentum $P$. Then $Z(z)$ should be a linear combination of terms of the type $\mathcal{F}_\epsilon^{(x)}(z)\mathcal{F}_{\bar{\epsilon}}^{(x)}(\bar{z})$, with the same functions $\mathcal{F}_\pm$ of $z$ and $\bar{z}$.

Let us focus on $s$-channel blocks, which diagonalize the monodromy $z \to e^{2\pi i z}$ around $z=0$. For our 4-point function to have trivial monodromy around $z=0$ we need $Z(e^{2\pi i z}) = Z(z)$, and its $s$-channel decomposition should be of the type

$$ Z(z) = \sum_{\pm} d_{\pm}^{(s)} |\mathcal{F}_{\pm}^{(s)}(z)|^2 \qquad (20) $$

with no terms of the type $\mathcal{F}_\pm^{(x)}(z)\mathcal{F}_\mp^{(x)}(\bar{z})$. Similarly, trivial monodromy around $z=1$ implies a $t$-channel decomposition of the type

$$ Z(z) = \sum_{\pm} d_{\pm}^{(t)} |\mathcal{F}_{\pm}^{(t)}(z)|^2 \qquad (21) $$

And by crossing symmetry, both expressions should agree. Applying our change of basis, this implies

$$ \sum_{\pm} d_{\pm}^{(s)} F_{\pm, \epsilon} \bar{F}_{\pm, \bar{\epsilon}} = \delta_{\epsilon, \bar{\epsilon}} d_{\epsilon}^{(t)} \qquad (22) $$

These are 4 linear equations for the 4 unknowns $d_\pm^{(s)}, d_\pm^{(t)}$. It turns out that there are only 3 independent equations, so there exists a nonzero solution, with in particular

$$ \frac{d_{+}^{(s)}}{d_{-}^{(s)}} = - \frac{F_{+-}F_{--}}{F_{++}F_{+-}} \qquad (23) $$

This determines $Z(z)$ up to an overall, $z$-independent factor. But $Z(z)$ is only an auxiliary object: we are really interested in the ratio $\frac{d_+^{(s)}}{d_-^{(s)}}$, which will tell us about 3-point structure constants in our CFT.

## 2.3 Shift equations for 3-point structure constants

The coefficients $d_\pm^{(s)}$ of conformal blocks in the 4-point function $Z(z)$ can be expressed in terms of the degenerate OPE coefficients (10), and 3-point structure constants $C_{P_1,P_2,P_3} \propto \langle V_{P_1}V_{P_2}V_{P_3} \rangle$:

$$ d_{\pm}^{(s)} = C_{\pm}(P_1)C_{P_1 \pm \frac{\beta}{2}, P_2, P_3} \qquad (24) $$
Therefore, Eq. (23) determines how $C_{P_1,P_2,P_3}$ behaves under $P_i \to P_i + \beta$ – except for the unknown factors $C_\pm(P_1)$. However, these factors are not very important, as they depend on only 1 of the 3 fields. And they are subject to constraints from the 4-point function $\langle V_{\langle 2,1 \rangle}^d V_P V_{\langle 2,1 \rangle}^d V_P \rangle$. In the case of this 4-point function, we have

$$d_{\pm}^{(s)} = C_{\pm}(P)C_{\mp}(P \pm \frac{\beta}{2})B_P \quad \text{with} \quad B_P \propto \langle V_P V_P \rangle \qquad (25)$$

Therefore, we obtain

$$\frac{C_+(P)C_-(P + \frac{\beta}{2})}{C_-(P)C_+(P - \frac{\beta}{2})} = \text{known combination of } \Gamma \text{ functions} \qquad (26)$$

This equation does not completely determine $C_\pm(P)$, in particular because of the freedom to perform field renormalizations $V_P(z) \to \lambda_P V_P(z)$. A simple solution is

$$C_{\pm}(P) = \beta^{\pm 4\beta P} \frac{\Gamma(\mp 2\beta P)}{\Gamma(1 \pm 2\beta P)} \qquad (27)$$

Using this solution, the equation for the 3-point structure constant becomes

$$\frac{C_{P_1-\frac{\beta}{2},P_2,P_3}}{C_{P_1+\frac{\beta}{2},P_2,P_3}} = \beta^{8\beta P_1} \frac{\prod_{\pm,\pm} \Gamma(\frac{1}{2} - \beta P_1 \pm \beta P_2 \pm \beta P_3)}{\prod_{\pm,\pm} \Gamma(\frac{1}{2} + \beta P_1 \pm \beta P_2 \pm \beta P_3)} \qquad (28)$$

We want to solve this equation explicitly. We have to address 2 questions:

* Is the solution unique? Certainly not, as we only constrain the behaviour under $P \to P + \beta$. Any solution could be multiplied with a periodic function, i.e. any function of $\sin(\pi\beta^{-1}P_i)$. However, this changes if we also consider the analogous equation from the crossing symmetry of $\langle V_{\langle 1,2 \rangle}^d V_{P_1} V_{P_2} V_{P_3} \rangle$, which is obtained by $\beta \to \beta^{-1}$. This determines the behaviour under $P \to P + \beta^{-1}$. And for $\beta^2 \in \mathbb{R} \setminus \mathbb{Q}$, any continuous function that has the two periods $\beta, \beta^{-1}$ must be constant.

* Is the solution a combination of $\Gamma$ functions? Not really: on the right-hand side, the ratio does not involve $P_1 \to P_1 + \beta$, but rather $P_1 \to -P_1$. In fact, to write the solution, we need a function that produces $\Gamma$ functions when shifted by $\beta$ or $\beta^{-1}$. This will be the Barnes double Gamma function.

## 2.4 Barnes double Gamma function

We need a special function $\Gamma_\beta$ such that

$$\frac{\Gamma_{\beta}(x + \beta)}{\Gamma_{\beta}(x)} = \frac{1}{\Gamma(\beta x)} \quad , \quad \frac{\Gamma_{\beta}(x + \beta^{-1})}{\Gamma_{\beta}(x)} = \frac{1}{\Gamma(\beta^{-1} x)} \qquad (29)$$

In fact these equations are not quite compatible. There give us 2 ways to compute $\frac{\Gamma_\beta(x+\beta+\beta^{-1})}{\Gamma_\beta(x)}$, with slightly different results. Eq. (28) suggests the type of prefactors that are needed for making the equations compatible:

$$\frac{\Gamma_{\beta}(x + \beta)}{\Gamma_{\beta}(x)} = \frac{\beta^{\beta x - \frac{1}{2}}}{\Gamma(\beta x)} \quad , \quad \frac{\Gamma_{\beta}(x + \beta^{-1})}{\Gamma_{\beta}(x)} = \frac{\beta^{\frac{1}{2} - \beta^{-1}x}}{\Gamma(\beta^{-1} x)} \qquad (30)$$
Now it turns out that these equations have a solution $\Gamma_{\beta}$. And this solution is unique under reasonable analyticity assumption: that $\Gamma_{\beta}$ is meromorphic in $x \in \mathbb{C}$ and in $\beta \in \mathbb{C} \setminus i\mathbb{R}$. We know explicit integral or product formulas for $\Gamma_{\beta}$, we know its poles and residues, so we should consider it a known quantity, just like the $\Gamma$ function itself.

Using the double Gamma function, we can solve the shift equations for the 3-point structure constant, and we find

$$C_{P_1,P_2,P_3} = \prod_{\pm, \pm, \pm} \Gamma_{\beta}^{-1} \left( \frac{\beta+\beta^{-1}}{2} \pm P_1 \pm P_2 \pm P_3 \right) \quad (31)$$

This is a universal result, which relies on a few assumptions, in particular the existence of degenerate fields, and analyticity in $\beta$ and $P_i$. Now, to which CFTs is this result relevant?

# 3 Application to particular CFTs

## 3.1 A-series minimal models

A minimal model is a consistent 2d CFT with a finite number of primary fields. The corresponding torus partition functions have been classified by solving modular invariance equations – similar in spirit to crossing symmetry equations, but technically simpler. Modular invariance does not guarantee that a consistent CFT exists, but this is the case for minimal models, although not all of them have been fully solved.

The spectrum of a consistent CFT should be stable under OPEs. We want finitely many primary fields in OPEs, so only degenerate fields can appear in minimal models. But we could still generate infinitely many primary fields, by repeatedly doing OPEs. For example, starting from $V_{\langle 2,1 \rangle}^d$, we have the OPEs

$$V_{\langle 2,1 \rangle}^d V_{\langle 2,1 \rangle}^d \sim V_{\langle 1,1 \rangle}^d + V_{\langle 3,1 \rangle}^d, \quad V_{\langle 2,1 \rangle}^d V_{\langle r,1 \rangle}^d \sim V_{\langle r-1,1 \rangle}^d + V_{\langle r+1,1 \rangle}^d \quad (32)$$

so we get an infinite family of fields, with $r \in \mathbb{N}^*$. To really have finitely many, we need doubly degenerate fields of the type

$$V_{\langle r,s \rangle}^d = V_{\langle r',s' \rangle}^d \quad (33)$$

The equality $P_{(r,s)} = \pm P_{(r',s')}$ implies $\beta^2 \in \mathbb{Q}$, so minimal models exist for rational $\beta^2$, therefore rational central charge. In fact we even need $\beta^2 = \frac{q}{p} > 0$. Then we can show that the Kac table $\{V_{\langle r,s \rangle}^d\}_{\substack{r=1,\dots,p-1 \\ s=1,\dots,q-1}}$ is stable under OPEs. To build the spectrum of a minimal model, we use fields of the type $V_{\langle r,s \rangle \langle \bar{r}, \bar{s} \rangle}^d$ whose left and right dimensions are in the Kac table. There are 3 series called A, D, E, we will focus on the A-series which is diagonal i.e. $(r,s) = (\bar{r}, \bar{s})$.

For A-series minimal models, our 3-point structure constant applies. However, we usually do a bit of normalization. We want $V_{\langle 1,1 \rangle}^d$ to be a true identity field (not just up to a constant factor), and we want 2-point functions to be trivial $\langle V_{\langle r,s \rangle}^d V_{\langle r,s \rangle}^d \rangle = 1$. This implies that the 3-point structure constant $\tilde{C}$ should obey

$$\tilde{C}_{(1,1)(r,s)(r,s)} = 1 \quad (34)$$
To achieve this, we are free to modify $C_{P_1,P_2,P_3}$ by field renormalizations, and also by an overall constant factor:

$$ \tilde{C}_{(r_1,s_1)(r_2,s_2)(r_3,s_3)} = C_{P_{(r_1,s_1)},P_{(r_2,s_2)},P_{(r_3,s_3)}} \sqrt{\frac{C_{P_{(1,1)},P_{(1,1)},P_{(1,1)}}}{\prod_{i=1}^{3} C_{P_{(1,1)},P_{(r_i,s_i)},P_{(r_i,s_i)}}}} \quad (35) $$

This formula is valid for all 3-point functions in all A-series minimal models. Of course, simplifications can occur in examples. In the minimal model at $c = \frac{1}{2}$ i.e. $\beta^2 = \frac{3}{4}$, which describes a sector of the critical Ising model, there is only one nontrivial 3-point function $\langle \sigma\sigma\epsilon \rangle = \tilde{C}_{(2,1)(2,1)(3,1)} = \frac{1}{2}$.

## 3.2 Liouville theory

For any $c \in \mathbb{C}$, Liouville theory can be defined as a diagonal CFT whose primary fields are $\{V_P\}_{P \in \mathbb{R}_+}$. Degenerate fields are supposed to exist, without belonging to the CFT. This means that we can compute correlation functions that involve degenerate fields, but degenerate fields never appear in OPEs of the fields $V_P$.

The existence of degenerate fields implies that the 3-point structure constant of Liouville theory obeys shift equations, therefore it must coincide with the solution $C_{P_1,P_2,P_3}$ (31). This proves that Liouville theory is unique.

To prove that Liouville theory exists, we would have to prove crossing symmetry of arbitrary 4-point functions of the type $\langle \prod_{i=1}^4 V_{P_i} \rangle$. This is not easy at all, and has been done only for $c \ge 25$.

A big subtlety is that the solution $C_{P_1,P_2,P_3}$ actually works only for $c \le 1$ i.e. $\beta \in \mathbb{R}^*$. It can be analytically continued to complex $\beta$, but then crossing symmetry fails. However, we know another solution of the same equations, which is valid for $c \in \mathbb{C} \setminus (-\infty, 1)$ i.e. $\beta \in \mathbb{C} \setminus \mathbb{R}$:

$$ C_{P_1,P_2,P_3}^{\text{DOZZ}} = \frac{\prod_{\pm, \pm, \pm} \Gamma_b \left( \frac{b+b^{-1}}{2} \pm iP_1 \pm iP_2 \pm iP_3 \right)}{\prod_{k=1}^{3} \Gamma_b(2iP_k) \Gamma_b(b+b^{-1} - 2iP_k)} \quad \text{with} \quad b = i\beta \quad (36) $$

Compared with the original solution, the main double Gamma functions are now in the numerator, their arguments have picked $i$ factors, and a field renormalization has been performed. This particular field normalization comes from the original derivation by Dorn, Otto, Zamolodchikov and Zamolodchikov in the 1990s, who used a path integral approach to Liouville theory.

## 3.3 Critical loop models

The same formula for $C_{P_1,P_2,P_3}$ was found to describe 3-point functions of diagonal fields in critical loop models (including the Potts model). These models are non-diagonal CFTs, which include not only diagonal primary fields $V_P$, but also non-diagonal fields $V_{(r,s)}$ of momentums

$$ (P, \bar{P}) = (P_{(r,s)}, P_{(r,-s)}) \quad \text{with} \quad r \in \frac{1}{2}\mathbb{N}^*, s \in \frac{1}{r}\mathbb{Z} \quad (37) $$

The appearance of $C_{P_1,P_2,P_3}$ in loop models suggests that they might be closely related to Liouville theory. But this is wrong: while loop models and Liouville theory share some 3-point functions, their
4-point functions are very different, due to the very different OPEs.

$$V_{P_1} V_{P_2} \sim_{\text{Liouville}} \int_{\mathbb{R}_+} dP V_P \qquad (38)$$

$$V_{P_1} V_{P_2} \sim_{\text{loop models}} \sum_{k \in \mathbb{Z}} V_{P_0+k\beta-1} + \sum_{r \in \mathbb{N}^*} \sum_{s \in \frac{1}{r}\mathbb{Z}} V_{(r,s)} \qquad (39)$$

So, the same $C_{P_1,P_2,P_3}$ appears in both CFTs because it is a universal result, not because the CFTs are closely related.

Nevertheless, it is mysterious that $C_{P_1,P_2,P_3}$ appears in critical loop models, because while $V_{(1,2)}^d$ exists in these models, we do not have $V_{(2,1)}^d$. The spectrum of primary fields is indeed invariant under $s \to s+2$ but not $r \to r+2$. Solving this mystery is an open problem.

## Acknowledgements

This text is a preparatory note for a 2-hour lecture that was given in December 2025, in the context of a Physics Latam course on 2d CFT. I am grateful to the Physics Latam organizer Daniel Galviz, and to the 2d CFT lecturer Jiaxi Cao, for inviting me to give this lecture. I would also like to thank the students for their participation and questions.

## References

[1] S. Ribault (2016) [arXiv:1609.09523] [doi:10.21468/SciPostPhysLectNotes.1]

*Minimal lectures on two-dimensional conformal field theory*

[2] S. Ribault (2024) [arXiv:2411.17262]

*Exactly solvable conformal field theories*