Minimal model (physics)

In theoretical physics, a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro algebra. Minimal models have been classified and solved, and found to obey an ADE classification. The term minimal model can also refer to a rational CFT based on an algebra that is larger than the Virasoro algebra, such as a W-algebra.

Classification

Relevant representations of the Virasoro algebra

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

$c_{p,q}=1-6{(p-q)^{2} \over pq}\ .$

where $p,q$ are coprime integers such that $p,q\geq 2$. Then the conformal dimensions of degenerate representations are

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

and they obey the identities

$h_{r,s}=h_{q-r,p-s}=h_{r+q,s+p}\ .$

The spectrums of minimal models are made of irreducible, degenerate lowest-weight representations of the Virasoro algebra, whose conformal dimensions are of the type $h_{r,s}$ with

$1\leq r\leq q-1\quad ,\quad 1\leq s\leq p-1\ .$

Such a representation ${\mathcal {R}}_{r,s}$ is a coset of a Verma module by its infinitely many nontrivial submodules. It is unitary if and only if $|p-q|=1$. At a given central charge, there are ${\frac {1}{2}}(p-1)(q-1)$ distinct representations of this type. The set of these representations, or of their conformal dimensions, is called the Kac table with parameters $(p,q)$. The Kac table is usually drawn as a rectangle of size $(q-1)\times (p-1)$, where each representation appears twice due to the relation $h_{r,s}=h_{q-r,p-s}$.

A-series minimal models: the diagonal case

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

${\mathcal {S}}_{p,q}^{\text{A-series}}={\frac {1}{2}}\bigoplus _{r=1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\ .$

The $(p,q)$ and $(q,p)$ models are the same.

D-series minimal models

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

${\mathcal {S}}_{p,q}^{\text{D-series}}\ \ {\underset {q\equiv 0\operatorname {mod} 4,\ q\geq 8}{=}}\ \ {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\oplus {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}2}^{q-2}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{q-r,s}\ ,$

${\mathcal {S}}_{p,q}^{\text{D-series}}\ \ {\underset {q\equiv 2\operatorname {mod} 4,\ q\geq 6}{=}}\ \ {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\oplus {\frac {1}{2}}\bigoplus _{r{\overset {2}{=}}1}^{q-1}\bigoplus _{s=1}^{p-1}{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{q-r,s}\ .$

Examples

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

• $(p,q)=(3,2)$ : trivial CFT,
• $(p,q)=(5,2)$ : Yang-Lee edge singularity,
• $(p,q)=(4,3)$ : Ising model at criticality,
• $(p,q)=(5,4)$ : tricritical Ising model.

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

• $(p,q)=(6,5)$ : 3-state Potts model,
• $(p,q)=(7,6)$ : tricritical 3-state Potts model.

The Kac tables of these models, together with a few other Kac tables with $2\leq q\leq 6$, are:

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

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

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

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

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

Related conformal field theories

Generalized minimal models

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

${\mathcal {S}}=\bigoplus _{r,s=1}^{\infty }{\mathcal {R}}_{r,s}\otimes {\bar {\mathcal {R}}}_{r,s}\ .$

When the central charge tends to $c_{p,q}$, the generalized minimal models tend to the corresponding A-series minimal model. This means in particular that the degenerate representations that are not in the Kac table decouple.

Liouville theory

Since Liouville theory reduces to a generalized minimal model when the fields are taken to be degenerate, it further reduces to an A-series minimal model when the central charge is then sent to $c_{p,q}$.

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

Products of minimal models

There are three cases of minimal models that are products of two minimal models.