“Minimal models”的意思、由来-开放百科全书

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 algebra

Representations

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

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

Fusion rules

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

Classification

A-series minimal models: the diagonal case

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 OPE of two fields involves all the fields that are allowed by the fusion rules of the corresponding representations.

D-series minimal models

A D-series minimal model with the central charge

exists if

is even and at least

. Using the symmetry

In any given spectrum, each representation has multiplicity one, except the representations of the type

, 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.

E-series minimal models

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:

Examples

The following A-series minimal models are related to well-known physical systems:^[3]

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:

Related conformal field theories

Coset realizations

The A-series minimal model with indices

coincides with the following coset of WZW models:^[3]

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

.^[3]

Generalized minimal models

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 theory

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 models

There are three cases of minimal models that are products of two minimal models.^[7]

References

1. ^A. Cappelli, J-B. Zuber, "A-D-E Classification of Conformal Field Theories", Scholarpedia
2. ^ I. Runkel, "Structure constants for the D series Virasoro minimal models", [https://arxiv.org/abs/hep-th/9908046 hep-th/9908046]
3. ^¹²³P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, {{ISBN|0-387-94785-X}}
4. ^¹S. Ribault, "Conformal field theory on the plane", [https://arxiv.org/abs/1406.4290 arXiv:1406.4290]
5. ^I. Runkel, G. Watts, "A Nonrational CFT with c = 1 as a limit of minimal models", [https://arxiv.org/abs/hep-th/0107118 arXiv:hep-th/0107118]
6. ^V. Schomerus, "Rolling tachyons from Liouville theory",[https://arxiv.org/abs/hep-th/0306026 arXiv:hep-th/0306026]
7. ^T. Quella, I. Runkel, G. Watts, "Reflection and Transmission for Conformal Defects", [https://arxiv.org/abs/hep-th/0611296 arxiv:hep-th/0611296]