Minimal Models

We go over some examples of commonly encountered minimal models in 2D CFT as well as common calculations for them. These are taken from Schottenloher and Di Francesco.

Introduction

As we have seen in here, a representation $\rho :\text{Vir}\to \text{End}(V)$ of the highest weight representation of a Virasoro Algebra is called a Virasoro module. If, in addition, the elements of the form $\rho (L_{-n_1})\rho (L_{-n_2})\cdots \rho (L_{-n_k})v$ form a basis, where $v$ is the vacuum vector of $V$, then $V$ is called a Verma module. What is interesting is that Verma modules are indecomposable, but not (in general) irreducible!

Remember indecomposable is being able to find two proper subreprsentations whose direct sum is isomorphic to the full Verma module not just as a vector space, but also as a representation. While irreducible, is simply being able to find a proper subrepresentation. One can take the vector space quotient but it isn't necessary that the resulting thing will be naturally isomorphic as a module to the vector space decomposition by the quotient.

Useful Tools

Singular Vectors

Let's be more specific. As we have already seen here there is a natural bilinear Hermitian (possibly degenerate) form $H:M^2\to \mathbb{C}$ on any Verma module $M$ that takes advantage of the grading in the level subspaces.

Definition: Elements of the kernel of $H$ are known as singular vectors, or null vectors, and $\text{ker}H$ is known as the singular, maximal, or null subspace. A representation $M(c,h)$ with $c<1$ such that there exist singular vectors is called vanishing.

Corollary: $M/\text{ker}H$ is an irreducible Virasoro module.

We can show their existence in certain Verma Modules.

Theorem: (Existence of Singular Vectors) Consider a Verma module $M(h_{pq}(c),c)$ with

$$h_{pq}(c)=\frac{1}{48}[(13-c)(p^2+q^2)+\sqrt{(c-1)(c-25)}(p^2-q^2)-24pq+2(c-1)],$$

for some natural numbers $p,q\in \mathbb{N}$, then there exists a singular vector at level $pq$ in the level decomposition, i.e. $L_0\chi =h_{pq}+pq$ and $⟨u,\chi ⟩=0$ for any $u\in M$.

Note: All minimal models are built from Verma modules of this form.

Proof: It is enough to show that the Gram determinant vanishes at level $pq$. By Kac's theorem, we have that

$$\det A_{pq}\propto (h_{pq}-h_{pq})=0.$$ $$\begin{array}{}◻& \end{array}$$

Classifying Vanishing Representations

Ok so we have shown some elementary results for singular vectors, it is time to find properties.

A vanishing representation with central charge $c$ has highest weight $h_{pq}(c)$ for some $p,q\in \mathbb{Z}$. What is interesting about these is that depending on the value of the central charge, some of them show some pretty cool recursions.

Lemma: (Classification of Rational Modules) Let $r,s\in \mathbb{Z}$ be coprime integers and $0\le c\le 1$, such that

$$r{\alpha }_{+}+s{\alpha }_{-}=0,$$

where ${\alpha }_{\pm }$ are defined by Kac's theorem as

$$\begin{array}{rl}{h}_{p,q}& =h_0+\frac{1}{4}(p{\alpha }_{+}+q{\alpha }_{-}{)}^2\\ h_0& =\frac{c-1}{24}\\ {\alpha }_{\pm }& =\frac{\sqrt{1-c}\pm \sqrt{25-c}}{\sqrt{24}}.\end{array}$$

Then, the following identity holds

$$h_{p,q}=h_{p+r,q+s},$$

And any highest weight representation with this central charge is called rational.

If no such pair of integers exists (i.e. $\frac{{\alpha }_{+}}{{\alpha }_{-}}$ is irrational), the associated highest weight representations are called irrational.

Proof: Check this out! Here is a diagram of ${\mathbb{Z}}^2$ along with lines passing through the origin with slope $-\frac{{\alpha }_{+}}{{\alpha }_{-}}$ one where the slope is rational and another where the slope is irrational. Notice that this slope is completely determined by the central charge $c$ that we picked.

You can also notice that $\delta$ is the distance between one of the points (marked my $(p,q)$) and the lines. In fact, by using this $\delta$ we can write that for that central charge, and any $p,q\in \mathbb{N}$

$$h_{p,q}=h_0+\frac{{\delta }^2}{4}({\alpha }_{+}^2+{\alpha }_{-}^2),$$

where again $\delta$ is the distance of closest approach of the corresponding line through the origin to the lattice point $(p,q)\in {\mathbb{Z}}^2$. It is clear that now if the slope $-\frac{{\alpha }_{+}}{{\alpha }_{-}}$ is irrational, it will not pass through any points in the lattice. On the other hand, if it is rational then it can be written in irreducible form as

$$-\frac{{\alpha }_{+}}{{\alpha }_{-}}=\frac{s}{r},$$

for coprime integers $s,r\in \mathbb{Z}$ then we see that for that particular conformal weight there is a periodicity in the vanishing Virasoro representations, since the line will cross infinitely many of these points, proving the lemma.

$$\begin{array}{}◻& \end{array}$$

This diagram is usually called the diagram of dimensions, and it is a nice visualization for the periodicity of certain vanishing representations.

Note: Since the sign of ${\alpha }_{\pm }$ doesn't affect the value of $h_{p,q}$ we can pick $s,r\in \mathbb{N}$ without loss of generality. This will simplify some of the calculations in what follows.

Corollary: A Virasoro highest weight representation is rational iff there exist coprime natural numbers $r,s\in \mathbb{N}$ and integers numbers $p,q\in \mathbb{Z}$ such that the conformal weight and central charge are given by

$$\begin{array}{rl}c& =1-6\frac{(r-s{)}^2}{rs}\\ h& =\frac{(sp-rq{)}^2-(r-s{)}^2}{4rs}.\end{array}$$

Here are some properties of such representations

Proposition: Consider a rational representation as defined in the corollary above. Then the following identities hold

1. $h_{p,q}+pq=h_{r+p,s-q}=h_{r-p,s+q}$
2. $h_{p,q}+(r-p)(s-q)=h_{p,2s-q}=h_{2r-p,q}$

Proof: Rearrange the formula in the corollary a couple times.

$$\begin{array}{}◻& \end{array}$$

When it comes to studying singular vectors the proposition above, has the direct corollary:

Proposition: In a rational Verma module, there are infinitely many singular Virasoro submodules.

We have already shown the existence of a singular vector at level $pq$ and as a result, the existence of a singular Virasoro submodule which consists of the descendants of the singular vector at $pq$. The interesting fact about rational Verma modules is that there are infinitely many such submodules.

Proof: First notice that there exists a singular vector at level $(r-p)(s-q)$. We can show this by plugging it in Kac's formula for the Gram Matrix determinant at that level, and using the previous proposition. Now, consider the singular vector at level $pq$, This has eigenvalue $h_{p,q}+pq$ for $L_0$. For the proposition above we can see that that vector is the cyclic vector for the Verma module $M(c,h_{r+p,s-q})$, and since all of its descendants are by definition contained in $M(c,h_{p,q})$, then the module homomorphism $M(c,h_{r+p,s-q})\to M(c,h_{p,q})$ is an inclusion!

This is a pretentious way of saying that such a singular vector forms a singular Verma submodule that must contain at least one singular Verma submodule and so on, leading to infinite singular vectors!

$$\begin{array}{}◻& \end{array}$$

Irreducible Rational Highest Weight Representations

We can be more precise about this classification. Since we can classify the singular submodules, we can classify how to remove them in order to obtain a description of the irreducible rational highest weight representations.

Note: Since we are interested in studying minimal models, what we will see is that for a given pair of coprime naturals $r,s\in \mathbb{N}$ the only distinct conformal weights $h_{p,q}$ that will appear are the ones where $1\le p<r$ and $1\le q<s$. That restriction comes from fusion which we haven't seen yet, but we will prove later.

Proposition: Given a rational Verma module $M(c,h_{p,q})$ with $c,h_{p,q}$ given by $r,s\in \mathbb{N}$ and $1\le p<r$, $1\le q<s$, where $H:M^2\to \mathbb{C}$ is the canonical Hermitian form, we have that

$$\text{ker}H=\text{span}[M(c,h_{r+p,s-q})\cup M(c,h_{p,2s-q})],$$

where we identify $M(c,h_{r+p,s-q})$ and $M(c,h_{p,2s-q})$ with their canonical inclusions in $M(c,h_{p,q})$.

Proof: We have already shown that the two modules above are singular submodules of $M(c,h_{p,q})$. We just need to show that any other submodules are contained in them. But I don't want to write this, this can definitely be shown by induction on Kac's formula, or maybe a nifty trick that I am unaware of.

$$\begin{array}{}◻& \end{array}$$

Unfortunately, the quotient is not entirely straightforward to calculate because the intersection between the singular submodules is not empty! Schematically, if we quotient by one, we have inadvertently quotiented by part of the other.

Characters for Rational Modules

One useful tool in characterizing a Virasoro module is it's character. This is simply a generating function for the dimension each level, i.e. taking the appropriate number of derivatives gives us the number of independent vectors at that level.

Definition: Given a Virasoro module $V$ and its level decomposition

$$V=\underset{n\in \mathbb{N}}{⨁}{V}_n,$$

a Virasoro character for that module is a smooth function $\chi \in C^{\mathrm{\infty }}(\mathbb{R})$ at such that

$$\text{dim}{V}_n=\frac{1}{n!}{\frac{{\partial }^n}{\partial t^n}|}_0\chi (t)t^{c/24-h}.$$

Note: The reason for the factor of $t^{c/24-h}$ is because it helps us simplify some calculations that involve modular invariance. The information of the character is still the same.

Proposition: Given a Virasoro module $V$ its character is uniquely given by

$$\chi (t)=\text{Tr}{t}^{L_0-\frac{c}{24}}.$$

Proof: Since $\chi$ is smooth, it has a Taylor series, each term of which is constrained by the definition.

$$\begin{array}{}◻& \end{array}$$

[tbc, but honestly Di Francesco has a table]

Fusion

What is interesting about singular vectors? Primarily that they constrain the correlation functions further than conformal symmetry. Constraining correlation functions implies the famous fusion rules as we will see in this section.

Let's first discuss 3 point correlation functions. Since we live in a virasoro module we know that the following set

$$v_{N,h}=\rho (L_{-n_1})\rho (L_{-n_2})\cdots \rho (L_{-n_k})v_h,$$

where $\rho$ is the Virasoro representation, $N=\{n_i{\}}_{i=1}^k$ with $n_1+n_2+\cdots +n_k\in \mathbb{N}$ and $n_1\le n_2\le \cdots \le n_k$, and $v$ a cyclic vector, spans the entire module! Such states correspond to quasi-primary fields in a CFT. We can therefore denote their corresponding fields as ${\varphi }_h^N(z)$ defined by

$$\underset{z\to 0}{lim}{\varphi }_h^N(z)\mathrm{\Omega }=v_{N,h},$$

where $\mathrm{\Omega }$ is the global, unique, conformal vacuum in the Hilbert space. By the way this can be defined similarly including the antiholomorphic component. By conformal covariance of correlation functions, we have that if

$$⟨{\varphi }_{h,\overline{h}}^{N,\overline{N}}(w,\overline{w}){\varphi }_{h^{\prime },{\overline{h}}^{\prime }}^{N^{\prime },{\overline{N}}^{\prime }}(z,\overline{z})⟩=\{\begin{array}{ll}\frac{C}{(w-z{)}^{2H}(\overline{w}-\overline{z}{)}^{2\overline{H}}}& h+N=h^{\prime }+N^{\prime }=H,\overline{h}+\overline{N}={\overline{h}}^{\prime }+{\overline{N}}^{\prime }=\overline{H}\\ 0& \text{otherwise}\end{array},$$

where $h+N$ is shortcut for $h+\sum _{n\in N}n$. This would benefit from some compact notation. One will notice that, in fact, we have normalized our choices for ${\varphi }_{h,\overline{h}}^{N,\overline{N}}$ such that $C=1$. From now we will make the conformal weight implicit when it gets cumbersome. The interesting thing is what results this has in the operator product expansions.

Proposition: The product of two fields $\varphi ,\psi$ with respective conformal weights $\alpha ,\beta$ can be expressed in the basis of quasi-primary fields as

$$\varphi (z,\overline{z})\varphi (w,\overline{w})=\sum _{h,N}{C}_{\alpha \beta }^{h,N}(z-w{)}^{h+N-\alpha -\beta }(\overline{z}-\overline{w}{)}^{\overline{h}+\overline{N}-\overline{\alpha }-\overline{\beta }}{\varphi }_h^N(w,\overline{w}),$$

here we have made the dependence on $\overline{h},\overline{N}$ implicit in our labeling of the basis. Each term here has conformal weight $\alpha +\beta$.

Now we can use the 3 point correlation function. To simplify notation, let's keep the antiholomorphic part implicit. Given two quasiprimary fields ${\varphi }_{\alpha },{\varphi }_{\beta }$ and a primary field $\psi$ with conformal weights $\alpha ,\beta ,\gamma$ respectively, we have that

$$⟨\psi (z){\varphi }_{\alpha }(v){\varphi }_{\beta }(w)⟩=\sum _{h,N}{C}_{\alpha \beta }^{h,N}(v-w{)}^{h+N-\alpha -\beta }⟨\psi (z){\varphi }_h^N(w)⟩.$$

However, we know that $⟨\psi (z){\varphi }_h^N(w)⟩=0$ unless $\gamma =h=h+N$, that is because Verma modules are orthogonal. Therefore, not only they must share the same conformal weight, but they must also be in the same irreducible representation for the correlation function to not vanish. As a result, we have that

$$⟨\psi (z){\varphi }_{\alpha }(v){\varphi }_{\beta }(w)⟩=C_{\alpha \beta }^{\gamma }(v-w{)}^{\gamma -\alpha -\beta }(z-w{)}^{-2\gamma }.$$

Therefore, in a sense, the 3 point function, extracts the coefficient of the OPE $C_{\alpha \beta }^{\gamma }$!

Note: This is a really important result in CFT. Knowing the 3 point functions as well as the primaries, completely defines the operator algebra.

All this was context for the next theorem. It basically says that the existence of null vectors constrains the 3 point correlation functions.

Theorem: Let $\chi \in M(c,h)$ be a singular vector. Then there exists a differential opeator ${\mathcal{L}}_{\chi }$ such that any 3-point correlation function $G$ must satisfy

$${\mathcal{L}}_{\chi }G=0.$$

The operator is hard to calculate in general, but becomes possible when studying minimal models! The far more interesting corollary is that the OPE defines a product in the set of Virasoro representations that form our model. We will see this in the following section.

Definition

We are finally ready to define a Minimal model!

Definition: A conformal field theory is a minimal model iff its associated Hilbert space can be written as a direct sum of finitely many irreducible tensor products of holomorphic and antiholomorphic Virasoro modules that is closed under fusion.

This has a very interesting property.

Proposition: In a minimal model the following are true

1. There are finitely many conformal primaries.
2. All the Virasoro modules that appear in its decomposition are rational.

Corollary: For every minimal model there exist a unique pair of coprime naturals $r,s\in \mathbb{N}$ such that

$$\begin{array}{rl}c& =1-6\frac{(r-s{)}^2}{rs},r>s\end{array}$$

and every Virasoro module in the decomposition has conformal weight

$$h=\frac{(sp-rq{)}^2-(r-s{)}^2}{4rs},$$

for some integers $p,q\in \mathbb{Z}$ such that $0<p<r$ and $0<q<s$.

Note: For that reason we use the notation $\mathcal{M}(r,s)$ to refer to the corresponding minimal model.

Proof: (2) Every CFT must be closed under fusion. This puts constraints into which highest weight representations can appear. If the representation is rational then there are finitely many conformal weights available through fusion, since at higher and higher Kac indices the conformal weights repeat. If the representation is irrational, highest weights do not repeat, and therefore closure under fusion would include infinitely many of them.

(1) Via the state-field correspondence, a conformal primary is associated to a cyclic vector in a Virasoro submodule of the theory. Since there are finitely many of them, there are finitely many cyclic vectors. Additionally, since we have the description in terms of irreducible representations we can't find a different decomposition into Virasoro modules with more terms.

(Corollary) By (2) the representations are rational. Therefore the central charge (which is shared by all of them otherwise it wouldn't be a cft) defines the two coprime indices. Representations of conformal weights with Kac indices greater than those are either prohibited by fusion or by periodicity.

$$\begin{array}{}◻& \end{array}$$

Corollary: All minimal models have $c<1$.

Direct Sum Decomposition into Primaries

Another way of expressing Minimal models is through the decomposition into irreducible highest weight Virasoro modules $W(c,h)$ (that we define here).

Proposition: The Hilbert space of a minimal model with central charge $c$ can be written as

$$\mathcal{H}=\underset{h,\overline{h}}{⨁}{M}_{h,\overline{h}}W(c,h)\otimes W(c,\overline{h}),$$

where $M_{h,\overline{h}}\in \mathbb{N}$ is the number of copies of the module $W(c,h)\otimes W(c,\overline{h})$, i.e.

$$2W(c,h)\otimes W(c,\overline{h}):=[W(c,h)\otimes W(c,\overline{h})]\oplus [W(c,h)\otimes W(c,\overline{h})],$$

and $W(c,h)$ is the reduced Verma module $M(c,h)$ given by quotienting the singular subrepresentations.

Note: The multiplicities $M_{h,h}$ of each module are given by fusion. However, as we will see soon, we can calculate them (or at least constrain them) from modular invariance by constraining the partition function.

Minimal Models as Fusion Rings

We have seen that the existence of singular vectors constrains the possible conformal weights that can appear in an OPE and therefore in our theory. However, the OPE itself, can help us define a product called fusion on the level of Virasoro representations that form our theory.

Definition: Let $\mathcal{M}$ be a set of Virasoro modules with a basis of irreducible representations. A fusion on $\mathcal{M}$ is a map $\odot :\mathcal{M}\times \mathcal{M}\to \mathcal{M}$ such that for every irreducible representations $V,W\in \mathcal{M}$ and every $v\in V,w\in W$ then $v\times w\in V\odot W$ where $v\times w$ is understood as the operator product expansion of $v,w$.

Corollary: A minimal model is a fusion ring of Virasoro modules under fusion and direct sum with a basis of irreducible representations and involution the identity.

I am using the term fusion ring out of nowhere, but a full definition is found here.

Example: (2D Ising) Let's try and bootstrap an example of a CFT. We will impose unitarity as well as $c=\frac{1}{2}$ in our theory. The question is which Virasoro modules do we need to close under fusion.

From Kac's theorem, the only highest weight unitary representations of the Virasoro algebra at $c=\frac{1}{2}$ occur at $h=0,h=\frac{1}{2},h=\frac{1}{16}$. Therefore the only characters that can appear in our Modular partition function are these. As a result, the Hilbert space can be written as

$$\mathcal{H}=\underset{h,\overline{h}}{⨁}{M}_{h,\overline{h}}W(c,h)\otimes W(c,\overline{h}),$$

which implies that the torus partition function is written as

$$Z(q)=\sum _{h,\overline{h}}{M}_{h,\overline{h}}{\chi }_h(q){\overline{\chi }}_{\overline{h}}(\overline{q}).$$

By defining a vector $\chi (q)={\chi }_0(q)e_1+{\chi }_{\frac{1}{2}}(q)e_2+{\chi }_{\frac{1}{16}}(q)e_3$ and a 3x3 matrix $M$ we can write

$$Z(q)=\chi (q{)}^{†}M\chi (q).$$

So now, if we find the matrix $M$ we have found our model! Modular invariance tells us that $Z(\tau )=Z(-\frac{1}{\tau })$ by invariance under the $S$ transformation. However, the $S$ transformation is a conformal transformation on the plane, therefore we know how our fields with the corresponding conformal weights would change.

However, if we know the characters, we don't have to transform the fields, we can just plug in $-1/\tau$. Either way, we obtain the following $S$ matrix such that $\chi (-1/\tau )=S\chi (\tau )$

$$S=\frac{1}{2}\left(\begin{array}{ccc}1& 1& \sqrt{2}\\ 1& 1& -\sqrt{2}\\ \sqrt{2}& -\sqrt{2}& 0\end{array}\right).$$

In addition to this, we have $T$ invariance which is $Z(\tau +1)=Z(\tau )$. By doing the necessary grunt-work we find that

$$h-\overline{h}=0\text{mod}1.$$

In our model we notice that there isn't a combination of weights such that $h-\overline{h}\ge 1$ therefore we have that $h=\overline{h}$ which implies $M$ is diagonal.

Additionally, we take that the identity operator is nondegenrate, i.e. $M_{11}=1$. Therefore, we are finally ready to put all these together, and figure out what are the possible minimal models for $c=\frac{1}{2}$. They should satisfy

$$S^{†}MS=M⟹MS=SM$$

Taking all of them together we find that $M=\mathbb{I}$.

This is fantastic! The only thing that is left is to find out the fusion rules. To do so, we will use singular vectors.

As we have explored before the Verma module $M(\frac{1}{2},0)$ has a null vector at level $1$. There is only one linearly independent vector there, which is $\rho (L_{-1})v_0$. For $M(\frac{1}{2},\frac{1}{16})$ and $M(\frac{1}{2},\frac{1}{2})$ there is a singular vector at level $2$. Using these to constrain the 3-point correlation functions we obtain the following Fusion rules.

$$\begin{array}{rl}{W}_0\odot W& =W\\ W_{\frac{1}{16}}\odot W_{\frac{1}{16}}& =W_0\oplus W_{\frac{1}{2}}\\ W_{\frac{1}{16}}\odot W_{\frac{1}{2}}& =W_{\frac{1}{16}}\\ W_{\frac{1}{2}}\odot W_{\frac{1}{2}}& =W_0.\end{array}$$

Where $W_{\alpha }=W(\frac{1}{2},\alpha )$. This creates a fusion ring with a basis $B=\{W_0,W_{\frac{1}{16}},W_{\frac{1}{2}}\}$ under fusion and direct sums. By the way, this is called the Ising fusion ring. We will see that the defects of the Ising model also form the same fusion ring.

$$\begin{array}{}◻& \end{array}$$

Verlinde's Formula

We have a way, using modular invariance, to over-constrain the multiplicity of the highest weight modules that can appear in our theory. We also have a way, using unitarity and Kac's theorem, to find out which irreducible highest weight modules can appear. What we are left to constrain is how to obtain fusion rules.

We already have a way to do so using highest weight vectors, as we saw in the previous example. But what I skipped there is 2 days of tedious calculations. It is not simple to obtain them using singular vectors directly, especially for more complicated models. So let's try to develop a new methodology.