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, giving rise to an ADE classification. Most minimal models have been solved, i.e. their 3-point structure constants have been computed analytically. 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.
Relevant representations of the Virasoro algebra
Representations In minimal models, the central charge of the Virasoro algebra takes values of the type
c p , q = 1 − 6 ( p − q ) 2 p q . {\displaystyle c_{p,q}=1-6{(p-q)^{2} \over pq}\ .}
where p , q {\displaystyle p,q} are coprime integers such that p , q ≥ 2 {\displaystyle p,q\geq 2} . Then the conformal dimensions of degenerate representations are
h r , s = ( p r − q s ) 2 − ( p − q ) 2 4 p q , with r , s ∈ N ∗ , {\displaystyle 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 . {\displaystyle h_{r,s}=h_{q-r,p-s}=h_{r+q,s+p}\ .}
The spectra 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 {\displaystyle h_{r,s}} with
1 ≤ r ≤ q − 1 , 1 ≤ s ≤ p − 1 . {\displaystyle 1\leq r\leq q-1\quad ,\quad 1\leq s\leq p-1\ .}
Such a representation R r , s {\displaystyle {\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 {\displaystyle |p-q|=1} . At a given central charge, there are 1 2 ( p − 1 ) ( q − 1 ) {\displaystyle {\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 ) {\displaystyle (p,q)} . The Kac table is usually drawn as a rectangle of size ( q − 1 ) × ( p − 1 ) {\displaystyle (q-1)\times (p-1)} , where each representation appears twice due to the relation
R r , s = R q − r , p − s . {\displaystyle {\mathcal {R}}_{r,s}={\mathcal {R}}_{q-r,p-s}\ .}
Fusion rules The fusion rules of the multiply degenerate representations R r , s {\displaystyle {\mathcal {R}}_{r,s}} 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. Explicitly, the fusion rules are
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