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.[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.

Relevant representations of the Virasoro algebraEdit


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


Fusion rulesEdit

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.


A-series minimal models: the diagonal caseEdit

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.

D-series minimal modelsEdit

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. The spectrum is


where the sums over   run by increments of two. In any given spectrum, each representation has multiplicity one, except the representations of the type   if  , 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. [3] For this rule, one copy of the representation   counts as diagonal, and the other copy as non-diagonal.

E-series minimal modelsEdit

There are three series of E-series minimal models. Each series exists for a given value of   for any   that is coprime with  . (This actually implies  .) Using the notation  , the spectrums read:



The following A-series minimal models are related to well-known physical systems:[2]

  •   : trivial CFT,
  •   : Yang-Lee edge singularity,
  •   : critical Ising model,
  •   : tricritical Ising model,
  •   : tetracritical Ising model.

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

  •   : 3-state Potts model at criticality,
  •   : tricritical 3-state Potts model.

The Kac tables of these models, together with a few other Kac tables with  , are:


Related conformal field theoriesEdit

Coset realizationsEdit

The A-series minimal model with indices   coincides with the following coset of WZW models:[2]


Assuming  , the level   is integer if and only if   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  .[2]

Generalized minimal modelsEdit

For any central charge  , there is a diagonal CFT whose spectrum is made of all degenerate representations,


When the central charge tends to  , the generalized minimal models tend to the corresponding A-series minimal model.[4] This means in particular that the degenerate representations that are not in the Kac table decouple.

Liouville theoryEdit

Since Liouville theory reduces to a generalized minimal model when the fields are taken to be degenerate,[4] 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,[5] which coincides with the limit of Liouville theory when  .[6]

Products of minimal modelsEdit

There are three cases of minimal models that are products of two minimal models.[7] At the level of their spectrums, the relations are:


Fermionic extensions of minimal modelsEdit

If  , the A-series and the D-series   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.[8]


  1. ^ A. Cappelli, J-B. Zuber, "A-D-E Classification of Conformal Field Theories", Scholarpedia
  2. ^ a b c d P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X
  3. ^ I. Runkel, "Structure constants for the D series Virasoro minimal models", hep-th/9908046
  4. ^ a b S. Ribault, "Conformal field theory on the plane", arXiv:1406.4290
  5. ^ I. Runkel, G. Watts, "A Nonrational CFT with c = 1 as a limit of minimal models", arXiv:hep-th/0107118
  6. ^ V. Schomerus, "Rolling tachyons from Liouville theory",arXiv:hep-th/0306026
  7. ^ T. Quella, I. Runkel, G. Watts, "Reflection and Transmission for Conformal Defects", arxiv:hep-th/0611296
  8. ^ Runkel, Ingo; Watts, Gerard (2020). "Fermionic CFTs and classifying algebras". Journal of High Energy Physics. 2020 (6): 25. arXiv:2001.05055. Bibcode:2020JHEP...06..025R. doi:10.1007/JHEP06(2020)025. S2CID 210718696.