Virasoro Algebra

We discuss abstract properties of the Virasoro Algebra and how it applies to conformal field theory in particular. That includes representation theory for Virasoro and so on.

The path we will follow is first construct the Witt algebra, a Lie algebra formed by transformations on the Euclidean plane. And then obtain the Virasoro Algebra as a central extension of it. Then we will study projective representation of the Virasoro algebra.

Witt Algebra

We have shown here that the conformal transformations in the eucledian plane are the holomorphic and anti-holomorphic functions, wherever the derivative does not vanish. Identifying the eucledian plane with $\mathbb{C}$ we can use taylor’s theorem to express these functions. To be more formal we have

Proposition: For some (anti) holomorphic map $f:\mathbb{C}\to \mathbb{C}$ we can write

$$z\mapsto f(z)=z+\sum _{n\in \mathbb{Z}}{a}_n{z}^n$$

for an appropriately defined laurent series. Therefore there exists a vector field $X\in \mathfrak{X}(\mathbb{C})$ given by

$$X=a_n{z}^n\frac{d}{dz}$$

such that

$$z\mapsto z+Xz$$

By that property we can call such an $X$ an infinitesimal holomorphic transformation. It is clear how the infinitesimal conformal transformations can be spanned by the following basis $\{L_n{\}}_{n\in \mathbb{Z}}\subset \mathfrak{X}(\mathbb{C})$ given by

$$L_n:=-z^{n+1}\frac{\partial }{\partial z}$$

Similarly, the antiholomorphic transformations would be spanned by a basis

$${\overline{L}}_n:=-{\overline{z}}^{n+1}\frac{\partial }{\partial \overline{z}}$$

We picked the $n+1$ with a bit of hindsight so that using the commutator of vector fields both of them have the property that

$$[L_n,L_m]=(n-m)L_{n+m}$$

With this nice property we can now proceed to the following proposition

Proposition: The linear span $W:=\mathbb{C}\{L_n\mid n\in \mathbb{Z}\}$ with the commutator of vector fields forms a Lie algebra called the Witt Algebra.

We can see from here that the Witt algebra is the algebra that contains all the holomorphic (1/2 of the conformal transformations). We can also see this, by the way, by plotting some of the vector fields.

Here they are, from left to right they go as $L_{-1}\cdots L_3$! First of all, they look so cool! Secondly, we notice that each of the vector fields is like a flower with exactly $n$ leafs. Therefore these are the flows in ${\mathbb{R}}^2$ that have to be linearly combined in order to create an infinitesimal holomorphic, conformal, transformation.

Sidenote on global and local conformal transformations

A priori there is no reason that the Laurent expansion of an arbitrary function is well defined everywhere. In fact, for the holomorphic functions to be globally defined we want it to not blow up at any point on the Riemann sphere.

Proposition: The conformal transformations generated by $L_n\in W$ is gobally defined iff $-1\le n\le 1$.

Proof: Requiring the laurent expansion of the transformation to be finite at $z\to 0$ implies that $n\ge -1$. Requiring that it must be finite for $\frac{1}{z}\to 0$ implies that $n\le 1$. The re are no other singularities in the laurent expansion. A similar result holds for the anti-holomorphic transformations.

Virasoro Algebra

In here we have shown how the quantization of a Symmetry, really is the central extension of the classical symmetry by $U(1)$. The Virasoro algebra is the central extension of the Witt Algebra by $\mathbb{C}$. The proof is in A Mathematical Introduction to Conformal Field Theory, but the theorem is

Theorem: The central extension of the Witt algebra $W$ by $\mathbb{C}$ exists and is unique up to group isomorphism.

Definition: The Virasoro Algebra $\text{Vir}$ is the central extension of the Witt algebra $W$ by $\mathbb{C}$, given by

$$\text{Vir}:=W\oplus \mathbb{C}Z$$

as a complex vector space, where for any $n,m\in \mathbb{Z}$

1. $[L_n,L_m]=(n-m)L_{n+m}+{\delta }_{n,-m}\frac{1}{12}n(n^2-1)Z$
2. $[L_n,Z]=0$

fun fact, the $\frac{1}{12}$ factor comes from the analytic continuation of the $\zeta$ function. The extra generator $Z\in \text{Vir}$ is related to the central charge.

Representations of Virasoro Algebra

In CFT we will always encounter representations of the Virasoro Algebra in different spaces. This section will study cool representations as well as properties of the vector spaces the representations are on. Let’s start with unitary representations.

Definition: A lie algebra representation $\rho :\text{Vir}\to {\text{End}}_{\mathbb{C}}V$ to a complex vector space $V$ is called unitary if there exists a positive sem-definite Hermitian form $H=⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{C}$ such that for any $v,w\in V$

$$\begin{gathered} ⟨\rho (L_n)v,w⟩=⟨v,\rho (L_{-n})w⟩ \\ ⟨\rho (Z)v,w⟩=⟨v,\rho (Z)w⟩ \end{gathered}$$

(A cool way to think of unitarity of a representation at a higher level of abstraction is one that preserves an involution for a $\ast$-algebra.) Ok cool, the thing we notice is that under a unitary representation the adjoint of $L_n$ is $L_{-n}$. Let’s see some more useful concepts. One of the most useful concepts is cyclic vectors.

Definition: Given a lie algebra representation $\rho :\text{Vir}\to {\text{End}}_{\mathbb{C}}V$ to a complex vector space $V$, a vector $v\in V$ is called cyclic if

$$V=\text{span}\{\rho (X_1)\rho (X_2)\cdots \rho (X_m)v\mid X_i\in \text{Vir},m\in \mathbb{N}\}$$

This is conducive to talking about the vaccuum state of the Hilbert space. That is because we can use the Virasoro algebra to get to any state of our conformal theory.

Definition: A lie algebra representation $\rho :\text{Vir}\to {\text{End}}_{\mathbb{C}}V$ to a complex vector space $V$ is called a highest weight representation if there are complex numbers $h,c\in \mathbb{C}$ and a cyclic vector $v\in V$ such that

$$\begin{array}{rl}\rho (Z)v& =cv\\ \rho (L_0)v& =hv\\ \rho (L_n)v& =0\mathrm{\forall }n\ge 1\end{array}$$

If further $h>0$ the representation is called a positive energy representation. The vector space $V$ equipped with a highest weight representation is called a Virasoro Module. Additionally, $c$​ is referred to as the central charge.

Here is a nice proposition that explains why highest weight representations are something that we care about.

Proposition: Given a representation of $\rho :\text{Vir}\to {\text{End}}_{\mathbb{C}}V$ to a complex vector space $V$, if $\rho (L_0)$ is bounded from below and diagonalizable then

$$\rho (L_n)v=0\mathrm{\forall }n\ge 1$$

So as we can see, the highest weight representation is the one that keeps the transformation generated by $L_0$ invertible, which is why we often associate it to time translation.

Let’s now take a look at the objects that generalize those vector spaces. These are verma modules.

Definition: A Verma module is a vector space $M(h,c)$ equipped with a highest weight representation of the Virasoro algebra with conformal parameters $h,c\in \mathbb{C}$ and highest weight vector $v\in M(h,c)$​ such that

$$\{\rho (L_{-n_1})\rho (L_{-n_2})\cdots \rho (L_{-n_k})v\mid 0<n_1\le n_2\le \cdots \le n_k,k\in \mathbb{N}\}\cup \{v\}$$

is a basis for $M(h,c)$.

An interesting consequence is that a Verma module for fixed $c,h\in \mathbb{C}$ always exists! In fact it works the other way around. We can build a Verma module using one vector and a representation of the Virasoro algebra.

Corollary: For every Virasoro module $V$ with highest weight representation parameters $c,h$, there is a surjective homomorphism $M(c,h)\to V$ that respects the representation.

Another important property of virasoro modules is the following decomposition.

Proposition: Let $V$ be a virasoro module for $h,c\in \mathbb{C}$. Then there exists a direct sum decomposition

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

where $V_0=\mathbb{C}v$, and $V_n$ for $n>0$ is the complex vector space

$$V_n:=\text{span}\{\rho (L_{-n_1})\rho (L_{-n_2})\cdots \rho (L_{-n_k})v|0<n_1\le n_2\le \cdots \le n_k,k\in \mathbb{N},\sum _{i=1}^k{n}_i=n\}$$

i.e. $V_n$ is the eigenspace of $\rho (L_0)$ with eigenvalue $n+h$.

Another interesting object are submodules that are defined below.

Definition: Given a Verma module $V$, a Verma submodule $A$ of $V$ is a vector subspace space $A\subset V$ such that $A$ is invariant under the representation of the Virasoro algebra.

Theorem: The kernel of $H$ is a maximal proper Verma submodule of $M$. In other words that kernel is the biggest possible Verma module with that $h,c$ as highest weight and central charge that is strictly included in $M$. Elements of $\text{ker}H$ are called singular, or null vectors

Corollary: $M/\text{ker}H$ is an irreducible highest weight representation of the Virasoro algebra.

Reducing Representations

In CFTs we will obtain a symmetry of the form of a representation of the Virasoro algebra in some Hilbert space. It would be nice to decompose the Hilbert space into highest weight representations. Let’s see how to do that

Definition: A vector space $M$ with a lie algebra representation of the Virasoro Algebra is indecomposable if there exist no proper subspaces $V,W$ invariant by the rerpesentation such that $M=V\oplus W$. Otherwise it is called decomposable.

Definition: M is called irreducible if there is no invariant proper subspace of $M$ under the representation.

Theorem: For each conformal weights $(c,h)\in {\mathbb{C}}^2$ we have that

1. The verma module $M(c,h)$ is indecomposable.
2. If $M(c,h)$ is reducible then there is a maximal invariant subspace $I(c,h)$ such that $M(c,h)/I(c,h)$ is an irreducible, heighest weight representation.
3. There is at most one, up to isomorphism, positive definite unitary highest weight representation of the virasoro algebra. We call the corresponding verma module $W(c,h)$
4. Any positive definite highest weight unitary representation $W(c,h)$ is irreducible.

Kac Determinant

We have been studying positive highest weight unitary representations. The Kac determinant is a a tool to help us decide if these representations have these properties.

We have seen that every Virasoro module (and by extension every Verma module) can be decomposed into a direct sum of $V_n$ subspaces. We can use this decomposition to assign a number to each state similar to a norm.

Definition: Given a Verma module $M(c,h)$ we define the expectation value of a vector $v\in M(c,h)$ using a linear map $⟨\cdot ⟩:M(c,h)\to \mathbb{C}$ such that

$$⟨v⟩={\pi }_0(v),$$

where ${\pi }_0:M(c,h)=\underset{n\in \mathbb{N}}{⨁}{V}_n\to V_0=\mathbb{C}{v}_0$ is the canonical projection map with respect the direct sum decomposition of $M(c,h)$, and $v_0$ is the highest weight vector.

Now we can use this expectation value concept to define a Hermitian norm on the Verma module

Definition: Given a Verma Module $M(c,h)$ its canonical hermitian form $H:M(c,h{)}^2\to \mathbb{C}$ is given by

$$H(v_{n_1\cdots n_k},v_{m_1\cdots m_l})=⟨\rho (L_{n_1})\cdots \rho (L_{n_k})v_{m_1\cdots m_l}⟩$$

where $v_{n_1\cdots n_k}=L_{-n_1}\cdots L_{-n_k}{v}_0$ is the basis of $M(c,h)$ with respect to the decomposition $M(c,h)=\underset{n\in \mathbb{N}}{⨁}{V}_n$.

Why this form? The reason is that this is a particularly nice form where $L_{-n}$ is the adjoint of $L_n$. In particular check out this theorem

Theorem: Consider the Verma modlue $M(c,h)$ where $c,h\in \mathbb{R}$. Then the following statements are true

1. $H:M(c,h{)}^2\to \mathbb{C}$ is the unique Hermitian form such that $H(v_0,v_0)=1$, and for any $v,w\in M(c,h)$ $H(L_{n}v,w)=H(v,L_{-n}w)$ and $H(Zv,w)=H(v,Zw)$.
2. $H(v,w)=0$ if $v\in V_n,w\in V_m$ for $n\ne m$. Or in other words the eigenspaces of $L_0$ are orthogonal.
3. $\ker H\subset M(c,h)$ is the maximal proper submodule of $M(c,h)$.

Corollary: $M(c,h)/\ker H$ is a Virasoro module where $H$ is nondegenerate.

Corollary: If $H$ is positive semi definite then $h\ge 0$ and $c\ge 0$.

The proof of this uses the following proposition that is going to be useful

Proposition: $H$ has the following property

$$H(v_n,v_n)=2nh+\frac{c}{12}n(n^2-1)$$

Now we are ready to define the the Kac determinant.

Proof: We use the commutation relation and notice that

$$H(v_n,v_n):=⟨L_n{L}_{-n}{v}_0⟩=⟨[L_n,L_{-n}]v_0⟩=2nh+\frac{c}{12}n(n^2-1).$$ $$\begin{array}{}◻& \end{array}$$

Proof of Corollary: Using the above proposition we see that if $c<0$ or there is a high enough $n$ to make $H(v_n,v_{-n})<0$, and if $h<0$ then $H(v_1,v_{-1})<0$.

$$\begin{array}{}◻& \end{array}$$

Definition: Let $\mathcal{B}=\{b_1,\cdots ,b_k\}$ be a basis for $V_n$. Then the Kac determinant is the determinant of the Gram Matrix $A_n$ of the basis $\mathcal{B}$ given by

$$\det A_n=\det {(H(b_i,b_j))}_{ij}$$

What we can see is that if all the Gram matrices are positive definite, then so is $H$. The highest weight representation associated with the Verma module $M(c,h)$ will be unitary if $H$ is in addition positive definite or positive semidefinite. So if we could calculate the determinant as a function of $c,h$ then everything would be awesome! As luck would have it, someone did!

Theorem: (Kac's Theorem) Let $c,h\in \mathbb{R}$ the determinant of the Gram Matrix at level $n$ is given by

$$\det A_n=K_n\prod _{\begin{array}{c}p,q\in \mathbb{N}\\ pq\le n\end{array}}(h-h_{p,q}(c){)}^{\dim V_{n-pq}},$$

where $K_n>0$ is the number given by

$$K_n=\prod _{\begin{array}{c}p,q\in \mathbb{N}\\ pq\le n\end{array}}{[{\left(2p\right)}^{q}q!]}^{P(n-pq)-P(n-pq-p)},$$

where $P(k)$ is the number of integer partitions of $k$ and $h_{p,q}(c)$ is a number given by

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

What we see is that if $h$ is greater or equal to $h_{p,q}$ for all such $p,q$ then the Kac determinant is going to be positive semidefinite. As a result, requiring unitarity (almost) fixes the values of $h$ as a function of $c$! This is a really cool result in the classification of CFTs.

Proposition: Another way to write $h_{p,q}$ is by introducing the following quantities

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

This description appears (in a slightly more refined form) in the derivation of fusion rules.

Unitary Representations

As we have already seen, the existence of unitary representations is identical to the existence of a positive semi-definite Hermitian form. Now with Kac's theorem we can find when our Hermitian form follows such rules.

Essentially we want no negative eigenvalues on $H$. Therefore, we can use the following theorem.

Theorem: (Classification of Unitary representations) Let $M(c,h)$ be a Verma module. Then the representation is unitary iff

1. $c\ge 1$ and $h\ge 0$ or

2. For some integer $m\ge 2$ there exist $p,q\in \mathbb{N}$ where $1\le q<p<m$ such that

   $$\begin{array}{rl}c& =1-\frac{6}{m(m+1)}\\ h& =\frac{{[(m+1)q-mp]}^2-1}{4m(m+1)}.\end{array}$$

This condition might seem terribly arbitrary, but it has its origin in enumerating a discrete set of intersections for when plotting $h_{p,q}(c)$. The intersections of these diagrams host unitary representations.

Proof: We have shown (1) in a previous corollary. For (2) we can prove this using cosets, but I haven't learned that yet so I will fill it in soon.

$$\begin{array}{}◻& \end{array}$$