1. Relevant Representations of the Virasoro Algebra

1.1. Representations

In minimal models, the central charge of the Virasoro algebra takes values of the type

[math]\displaystyle{ c_{p,q} = 1 - 6 {(p-q)^2 \over pq}\ . }[/math]

where [math]\displaystyle{ p, q }[/math] are coprime integers such that [math]\displaystyle{ p,q \geq 2 }[/math]. Then the conformal dimensions of degenerate representations are

[math]\displaystyle{ h_{r,s} = \frac{(pr-qs)^2-(p-q)^2}{4pq}\ , \quad \text{with}\ r,s\in\mathbb{N}^*\ , }[/math]

and they obey the identities

[math]\displaystyle{ h_{r,s} = h_{q-r,p-s} = h_{r+q,s+p}\ . }[/math]

The spectrums of minimal models are made of irreducible, degenerate lowest-weight representations of the Virasoro algebra, whose conformal dimensions are of the type [math]\displaystyle{ h_{r,s} }[/math] with

[math]\displaystyle{ 1\leq r \leq q-1 \quad , \quad 1\leq s \leq p-1\ . }[/math]

Such a representation [math]\displaystyle{ \mathcal{R}_{r,s} }[/math] is a coset of a Verma module by its infinitely many nontrivial submodules. It is unitary if and only if [math]\displaystyle{ |p-q|=1 }[/math]. At a given central charge, there are [math]\displaystyle{ \frac12(p-1)(q-1) }[/math] distinct representations of this type. The set of these representations, or of their conformal dimensions, is called the Kac table with parameters [math]\displaystyle{ (p, q) }[/math]. The Kac table is usually drawn as a rectangle of size [math]\displaystyle{ (q-1)\times (p-1) }[/math], where each representation appears twice due to the relation

[math]\displaystyle{ \mathcal{R}_{r,s} = \mathcal{R}_{q-r,p-s}\ . }[/math]

1.2. Fusion Rules

The fusion rules of the multiply degenerate representations [math]\displaystyle{ \mathcal{R}_{r,s} }[/math] encode constraints from all their null vectors. They can therefore be deduced from the fusion rules of simply degenerate representations, which encode constraints from individual null vectors.^[1] Explicitly, the fusion rules are

[math]\displaystyle{ \mathcal{R}_{r_1,s_1} \times \mathcal{R}_{r_2,s_2} = \sum_{r_3\overset{2}{=}|r_1-r_2|+1}^{\min(r_1+r_2,2q-r_1-r_2)-1}\ \sum_{s_3\overset{2}{=}|s_1-s_2|+1}^{\min(s_1+s_2,2p-s_1-s_2)-1} \mathcal{R}_{r_3,s_3}\ , }[/math]

where the sums run by increments of two.

2. Classification

2.1. A-Series Minimal Models: the Diagonal Case

For any coprime integers [math]\displaystyle{ p,q }[/math] such that [math]\displaystyle{ p,q\geq 2 }[/math], there exists a diagonal minimal model whose spectrum contains one copy of each distinct representation in the Kac table:

[math]\displaystyle{ \mathcal{S}_{p,q}^\text{A-series} = \frac12 \bigoplus_{r=1}^{q-1}\bigoplus_{s=1}^{p-1} \mathcal{R}_{r,s}\otimes \bar{\mathcal{R}}_{r,s}\ . }[/math]

The [math]\displaystyle{ (p,q) }[/math] and [math]\displaystyle{ (q,p) }[/math] models are the same.

The OPE of two fields involves all the fields that are allowed by the fusion rules of the corresponding representations.

2.2. D-Series Minimal Models

A D-series minimal model with the central charge [math]\displaystyle{ c_{p,q} }[/math] exists if [math]\displaystyle{ p }[/math] or [math]\displaystyle{ q }[/math] is even and at least [math]\displaystyle{ 6 }[/math]. Using the symmetry [math]\displaystyle{ p\leftrightarrow q }[/math] we assume that [math]\displaystyle{ q }[/math] is even, then [math]\displaystyle{ p }[/math] is odd. The spectrum is

[math]\displaystyle{ \mathcal{S}_{p,q}^{\text{D-series}} \ \ \underset{q\equiv 0\operatorname{mod} 4,\ q\geq 8}{=}\ \ \frac12 \bigoplus_{r\overset{2}{=}1}^{q-1} \bigoplus_{s=1}^{p-1} \mathcal{R}_{ r,s} \otimes \bar{\mathcal{R}}_{r,s}\oplus \frac12\bigoplus_{r\overset{2}{=}2}^{q-2} \bigoplus_{s=1}^{p-1} \mathcal{R}_{r,s} \otimes \bar{\mathcal{R}}_{q-r,s}\ , }[/math]

[math]\displaystyle{ \mathcal{S}_{p,q}^{\text{D-series}} \ \ \underset{q\equiv 2\operatorname{mod} 4,\ q\geq 6}{=}\ \ \frac12 \bigoplus_{r\overset{2}{=}1}^{q-1} \bigoplus_{s=1}^{p-1} \mathcal{R}_{ r,s} \otimes \bar{\mathcal{R}}_{r,s}\oplus \frac12\bigoplus_{r\overset{2}{=}1}^{q-1} \bigoplus_{s=1}^{p-1} \mathcal{R}_{r,s} \otimes \bar{\mathcal{R}}_{q-r,s}\ , }[/math]

where the sums over [math]\displaystyle{ r }[/math] run by increments of two. In any given spectrum, each representation has multiplicity one, except the representations of the type [math]\displaystyle{ \mathcal{R}_{\frac{q}{2},s}\otimes \bar{\mathcal{R}}_{\frac{q}{2},s} }[/math] if [math]\displaystyle{ q\equiv 2\ \mathrm{mod}\ 4 }[/math], which have multiplicity two. These representations indeed appear in both terms in our formula for the spectrum.

The OPE of two fields involves all the fields that are allowed by the fusion rules of the corresponding representations, and that respect the conservation of diagonality: the OPE of one diagonal and one non-diagonal field yields only non-diagonal fields, and the OPE of two fields of the same type yields only diagonal fields. ^[2] For this rule, one copy of the representation [math]\displaystyle{ \mathcal{R}_{\frac{q}{2},s}\otimes \bar{\mathcal{R}}_{\frac{q}{2},s} }[/math] counts as diagonal, and the other copy as non-diagonal.

2.3. E-Series Minimal Models

There are three series of E-series minimal models. Each series exists for a given value of [math]\displaystyle{ q\in\{12,18,30\}, }[/math] for any [math]\displaystyle{ p\geq 2 }[/math] that is coprime with [math]\displaystyle{ q }[/math]. (This actually implies [math]\displaystyle{ p\geq 5 }[/math].) Using the notation [math]\displaystyle{ |\mathcal{R}|^2 = \mathcal{R}\otimes \bar{\mathcal{R}} }[/math], the spectrums read:

[math]\displaystyle{ \mathcal{S}^\text{E-series}_{p,12} = \frac12 \bigoplus_{s=1}^{p-1} \left\{ \left| \mathcal{R}_{1,s}\oplus \mathcal{R}_{7,s}\right|^2 \oplus \left| \mathcal{R}_{4,s} \oplus \mathcal{R}_{8,s}\right|^2 \oplus \left| \mathcal{R}_{5,s} \oplus \mathcal{R}_{11,s} \right|^2 \right\}\ , }[/math]

[math]\displaystyle{ \mathcal{S}^\text{E-series}_{p,18} = \frac12 \bigoplus_{s=1}^{p-1} \left\{ \left|\mathcal{R}_{9,s}\oplus 2\mathcal{R}_{3,s}\right|^2 \ominus 4\left|\mathcal{R}_{3,s}\right|^2 \oplus \bigoplus_{r\in\{1, 5, 7\}} \left|\mathcal{R}_{r,s}\oplus \mathcal{R}_{18-r,s}\right|^2 \right\}\ , }[/math]

[math]\displaystyle{ \mathcal{S}^\text{E-series}_{p,30} = \frac12 \bigoplus_{s=1}^{p-1} \left\{ \left|\bigoplus_{r\in\{1,11,19,29\}} \mathcal{R}_{r,s}\right|^2 \oplus \left|\bigoplus_{r\in\{7,13,17,23\}} \mathcal{R}_{r,s}\right|^2 \right\}\ . }[/math]

3. Examples

The following A-series minimal models are related to well-known physical systems:^[1]

• [math]\displaystyle{ (p,q)=(3,2) }[/math] : trivial CFT,
• [math]\displaystyle{ (p,q)=(5,2) }[/math] : Yang-Lee edge singularity,
• [math]\displaystyle{ (p,q)=(4,3) }[/math] : critical Ising model,
• [math]\displaystyle{ (p,q)=(5,4) }[/math] : tricritical Ising model,
• [math]\displaystyle{ (p,q)=(6,5) }[/math] : tetracritical Ising model.

The following D-series minimal models are related to well-known physical systems:

• [math]\displaystyle{ (p,q)=(6,5) }[/math] : 3-state Potts model at criticality,
• [math]\displaystyle{ (p,q)=(7,6) }[/math] : tricritical 3-state Potts model.

The Kac tables of these models, together with a few other Kac tables with [math]\displaystyle{ 2\leq q \leq 6 }[/math], are:

[math]\displaystyle{ \begin{array}{c}\begin{array}{c|cc} 1 & 0 & 0 \\ \hline & 1 & 2 \end{array}\\ c_{3,2}=0 \end{array} \qquad \begin{array}{c}\begin{array}{c|cccc} 1 & 0 & - \frac{1}{5} & - \frac{1}{5} & 0 \\ \hline & 1 & 2 & 3 & 4 \end{array}\\ c_{5,2}=- \frac{22}{5} \end{array} }[/math]

[math]\displaystyle{ \begin{array}{c}\begin{array}{c|ccc} 2 & \frac{1}{2} & \frac{1}{16} & 0 \\ 1 & 0 & \frac{1}{16} & \frac{1}{2} \\ \hline & 1 & 2 & 3 \end{array}\\ c_{4,3}=\frac{1}{2} \end{array} \qquad \begin{array}{c}\begin{array}{c|cccc} 2 & \frac{3}{4} & \frac{1}{5} & - \frac{1}{20} & 0 \\ 1 & 0 & - \frac{1}{20} & \frac{1}{5} & \frac{3}{4} \\ \hline & 1 & 2 & 3 & 4 \end{array}\\ c_{5,3}=- \frac{3}{5} \end{array} }[/math]

[math]\displaystyle{ \begin{array}{c}\begin{array}{c|cccc} 3 & \frac{3}{2} & \frac{3}{5} & \frac{1}{10} & 0 \\ 2 & \frac{7}{16} & \frac{3}{80} & \frac{3}{80} & \frac{7}{16} \\ 1 & 0 & \frac{1}{10} & \frac{3}{5} & \frac{3}{2} \\ \hline & 1 & 2 & 3 & 4 \end{array}\\ c_{5,4}=\frac{7}{10} \end{array} \qquad \begin{array}{c}\begin{array}{c|cccccc} 3 & \frac{5}{2} & \frac{10}{7} & \frac{9}{14} & \frac{1}{7} & - \frac{1}{14} & 0 \\ 2 & \frac{13}{16} & \frac{27}{112} & - \frac{5}{112} & - \frac{5}{112} & \frac{27}{112} & \frac{13}{16} \\ 1 & 0 & - \frac{1}{14} & \frac{1}{7} & \frac{9}{14} & \frac{10}{7} & \frac{5}{2} \\ \hline & 1 & 2 & 3 & 4 & 5 & 6 \end{array}\\ c_{7,4}=- \frac{13}{14} \end{array} }[/math]

[math]\displaystyle{ \begin{array}{c}\begin{array}{c|ccccc} 4 & 3 & \frac{13}{8} & \frac{2}{3} & \frac{1}{8} & 0 \\ 3 & \frac{7}{5} & \frac{21}{40} & \frac{1}{15} & \frac{1}{40} & \frac{2}{5} \\ 2 & \frac{2}{5} & \frac{1}{40} & \frac{1}{15} & \frac{21}{40} & \frac{7}{5} \\ 1 & 0 & \frac{1}{8} & \frac{2}{3} & \frac{13}{8} & 3 \\ \hline & 1 & 2 & 3 & 4 & 5 \end{array}\\ c_{6,5}=\frac{4}{5} \end{array} \qquad \begin{array}{c}\begin{array}{c|cccccc} 4 & \frac{15}{4} & \frac{16}{7} & \frac{33}{28} & \frac{3}{7} & \frac{1}{28} & 0 \\ 3 & \frac{9}{5} & \frac{117}{140} & \frac{8}{35} & - \frac{3}{140} & \frac{3}{35} & \frac{11}{20} \\ 2 & \frac{11}{20} & \frac{3}{35} & - \frac{3}{140} & \frac{8}{35} & \frac{117}{140} & \frac{9}{5} \\ 1 & 0 & \frac{1}{28} & \frac{3}{7} & \frac{33}{28} & \frac{16}{7} & \frac{15}{4} \\ \hline & 1 & 2 & 3 & 4 & 5 & 6 \end{array}\\ c_{7,5}=\frac{11}{35} \end{array} }[/math]

[math]\displaystyle{ \begin{array}{c}\begin{array}{c|cccccc} 5 & 5 & \frac{22}{7} & \frac{12}{7} & \frac{5}{7} & \frac{1}{7} & 0 \\ 4 & \frac{23}{8} & \frac{85}{56} & \frac{33}{56} & \frac{5}{56} & \frac{1}{56} & \frac{3}{8} \\ 3 & \frac{4}{3} & \frac{10}{21} & \frac{1}{21} & \frac{1}{21} & \frac{10}{21} & \frac{4}{3} \\ 2 & \frac{3}{8} & \frac{1}{56} & \frac{5}{56} & \frac{33}{56} & \frac{85}{56} & \frac{23}{8} \\ 1 & 0 & \frac{1}{7} & \frac{5}{7} & \frac{12}{7} & \frac{22}{7} & 5 \\ \hline & 1 & 2 & 3 & 4 & 5 & 6 \end{array}\\ c_{7,6}=\frac{6}{7} \end{array} }[/math]

4. Related Conformal Field Theories

4.1. Coset Realizations

The A-series minimal model with indices [math]\displaystyle{ (p,q) }[/math] coincides with the following coset of WZW models:^[1]

[math]\displaystyle{ \frac{SU(2)_k\times SU(2)_1}{SU(2)_{k+1}}\ , \quad \text{where} \quad k = \frac{q}{p-q}-2\ . }[/math]

Assuming [math]\displaystyle{ p\gt q }[/math], the level [math]\displaystyle{ k }[/math] is integer if and only if [math]\displaystyle{ p=q+1 }[/math] i.e. if and only if the minimal model is unitary.

There exist other realizations of certain minimal models, diagonal or not, as cosets of WZW models, not necessarily based on the group [math]\displaystyle{ SU(2) }[/math].^[1]

4.2. Generalized Minimal Models

For any central charge [math]\displaystyle{ c\in\mathbb{C} }[/math], there is a diagonal CFT whose spectrum is made of all degenerate representations,

[math]\displaystyle{ \mathcal{S}=\bigoplus_{r,s=1}^\infty \mathcal{R}_{r,s}\otimes \bar{\mathcal{R}}_{r,s} \ . }[/math]

When the central charge tends to [math]\displaystyle{ c_{p,q} }[/math], the generalized minimal models tend to the corresponding A-series minimal model.^[3] This means in particular that the degenerate representations that are not in the Kac table decouple.

4.3. Liouville Theory

Since Liouville theory reduces to a generalized minimal model when the fields are taken to be degenerate,^[3] it further reduces to an A-series minimal model when the central charge is then sent to [math]\displaystyle{ c_{p,q} }[/math].

Moreover, A-series minimal models have a well-defined limit as [math]\displaystyle{ c\to 1 }[/math]: a diagonal CFT with a continuous spectrum called Runkel-Watts theory,^[4] which coincides with the limit of Liouville theory when [math]\displaystyle{ c\to 1^+ }[/math].^[5]

4.4. Products of Minimal Models

There are three cases of minimal models that are products of two minimal models.^[6] At the level of their spectrums, the relations are:

[math]\displaystyle{ \mathcal{S}^\text{A-series}_{2,5}\otimes \mathcal{S}^\text{A-series}_{2,5} = \mathcal{S}^\text{D-series}_{3,10}\ , }[/math]

[math]\displaystyle{ \mathcal{S}^\text{A-series}_{2,5}\otimes \mathcal{S}^\text{A-series}_{3,4} = \mathcal{S}^\text{E-series}_{5,12} \ , }[/math]

[math]\displaystyle{ \mathcal{S}^\text{A-series}_{2,5}\otimes \mathcal{S}^\text{A-series}_{2,7} = \mathcal{S}^\text{E-series}_{7,30} \ . }[/math]

4.5. Fermionic Extensions of Minimal Models

If [math]\displaystyle{ q\equiv 0\bmod 4 }[/math], the A-series and the D-series [math]\displaystyle{ (p,q) }[/math] minimal models each have a fermionic extension. These two fermionic extensions involve fields with half-integer spins, and they are related to one another by a parity-shift operation.^[7]