In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, a minimal model or Virasoro minimal model is a
two-dimensional conformal field theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations.
In contrast to other types of conformal field theories, two-dimensional conformal fi ...
whose spectrum is built from finitely many
irreducible representations of the
Virasoro algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
.
Minimal models have been classified and solved, and found to obey an
ADE classification
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
.
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
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
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 Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
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 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
In mathematics and theoretical physics, fusion rules are rules that determine the exact decomposition of the tensor product of two representations of a group into a direct sum of irreducible representations. The term is often used in the context of ...
of simply degenerate representations, which encode constraints from individual null vectors.
Explicitly, the fusion rules are
:
where the sums run by increments of two.
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
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 models
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.
For this rule, one copy of the representation
counts as diagonal, and the other copy as non-diagonal.
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:
[P. Di Francesco, P. Mathieu, and D. Sénéchal, ''Conformal Field Theory'', 1997, ]
*
: 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 theories
Coset realizations
The A-series minimal model with indices
coincides with the following coset of
WZW models:
:
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
.
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.
[S. Ribault, "Conformal field theory on the plane"]
arXiv:1406.4290
/ref> 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, 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, which coincides with the limit of Liouville theory when .
Products of minimal models
There are three cases of minimal models that are products of two minimal models.[T. Quella, I. Runkel, G. Watts, "Reflection and Transmission for Conformal Defects"]
arxiv:hep-th/0611296
/ref>
At the level of their spectrums, the relations are:
:
:
:
Fermionic extensions of minimal models
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.
References
{{DEFAULTSORT:Minimal Models
Conformal field theory
Exactly solvable models