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Concept
Minimal model (physics)
Summary
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.
In minimal models, the central charge of the Virasoro algebra takes values of the type
where are coprime integers such that .
Then the conformal dimensions of degenerate representations are
and they obey the identities
The spectrums of minimal models are made of irreducible, degenerate lowest-weight representations of the Virasoro algebra, whose conformal dimensions are of the type with
Such a representation is a coset of a Verma module by its infinitely many nontrivial submodules. It is unitary if and only if . At a given central charge, there are distinct representations of this type. The set of these representations, or of their conformal dimensions, is called the Kac table with parameters . The Kac table is usually drawn as a rectangle of size , where each representation appears twice
due to the relation
The fusion rules of the multiply degenerate representations 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
where the sums run by increments of two.
For any coprime integers such that , there exists a diagonal minimal model whose spectrum contains one copy of each distinct representation in the Kac table:
The and models are the same.
The OPE of two fields involves all the fields that are allowed by the fusion rules of the corresponding representations.
A D-series minimal model with the central charge exists if or is even and at least . Using the symmetry
we assume that is even, then is odd.
Official source
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