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.
[1] 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
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
is a coset of a Verma module by its infinitely many nontrivial submodules.
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.
[2] Explicitly, the fusion rules are where the sums run by increments of two.
Minimal models are the only 2d CFTs that are consistent on any Riemann surface, and are built from finitely many representations of the Virasoro algebra.
[2] There are many more rational CFTs that are consistent on the sphere only: these CFTs are submodels of minimal models, built from subsets of the Kac table that are closed under fusion.
, there exists a diagonal minimal model whose spectrum contains one copy of each distinct representation in the Kac table: The
A D-series minimal model with the central charge
These representations indeed appear in both terms in our formula for the spectrum.
counts as diagonal, and the other copy as non-diagonal.
There are three series of E-series minimal models.
, the spectrums read: The following A-series minimal models are related to well-known physical systems:[2] The following D-series minimal models are related to well-known physical systems: The Kac tables of these models, together with a few other Kac tables with
, are: The A-series minimal model with indices
coincides with the following coset of WZW models:[2] Assuming
There exist other realizations of certain minimal models, diagonal or not, as cosets of WZW models, not necessarily based on the group
, there is a diagonal CFT whose spectrum is made of all degenerate representations, When the central charge tends to
[5] This means in particular that the degenerate representations that are not in the Kac table decouple.
Since Liouville theory reduces to a generalized minimal model when the fields are taken to be degenerate,[5] it further reduces to an A-series minimal model when the central charge is then sent to
Moreover, A-series minimal models have a well-defined limit as
: a diagonal CFT with a continuous spectrum called Runkel–Watts theory,[6] which coincides with the limit of Liouville theory when
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.