======Virasoro Algebra======

The Virasoro algebra $\mathcal{V}$ is defined by the commutation relation

$$ [ L_n, L_m] = (n - m) L_{n + m} + \frac{c}{12} (n - 1)n(n + 1) \delta_{n, -m}$$

the number $c$ is called the central charge. 2d conformal field theories transform under two copies of the Virasoro algebra, $\mathcal{V} \,   \times \, \bar{ \mathcal{V}} $. [[Chiral CFTs]] transform under a single copy of the Virasoro algebra.

====Construction from Conformal Transformations====

Conformal transformations are those which preserve angles. In two dimensions, this includes any holomorphic function $f(z)$. Infinitesimally, any such transformation takes the form

$$ z \to z + \epsilon z^{n + 1}$$

for integers $n$. 

The action of these transformations on functions of $z$ is given by the differential operators

$$ \ell_n = - z^{n + 1} \frac{\partial}{\partial z} . $$

These operators satisfy the Witt algebra

$$[ \ell_n, \ell_m ] = (n - m) \ell_{m + n} .  $$

$\ell_{-1}$, $\ell_{0}$, and $\ell_{1}$ generate a subalgebra which is often called the //global// part of the conformal algebra.

Now we would like to determine how conformal transformations act on the states in the theory. First, the algebra needs to be complexified, because the spectrum of a CFT is a complex vector space. This requires that we take complex linear combinations of the generators. The basis of the complex Witt algebra is denoted by $(\ell_n, \bar \ell_n)$. $\ell_n$ and $\bar \ell_n$ commute with each other, so this algebra acts on $C^2$ and we must take $z$ and $\bar z$ to be independent.

Second, we need to consider possible central extensions of the Witt algebra.  This is because quantum states are only defined up to a total phase, so symmetry algebras need only act projectively on the states. A projective action of a symmetry alegebra is equivalent to the action of a centrally-extended algebra. The result is that the conformal transformations act on the states via the central extension of the Witt algebra, which is called the Virasoro algebra:

$$ [ L_n, L_m] = (n - m) L_{n + m} + \frac{c}{12} (n - 1)n(n + 1) \delta_{n, -m}.$$

Here $c$ is a generator which commutes with every other generator of the algebra, which means it is equal to a complex number times the identity. By abuse of notation, we also refer to this complex number as $c$-- this is the central charge of the theory. It must be real for the theory to be unitary.

====Representations====

One of the axioms of 2d CFTs is that the spectrum breaks into irreps of the Virasoro algebra where $L_0$ is bounded from below. Consider such an irrep $\mathcal{R}$, whose lowest eigenvalue is $\Delta$, corresponding to an eigenstate $|\Delta\rangle$. It is easy to see from the structure of the algebra that $L_n$ decrease the $L_0$ eigenvalue for $n>0$. Therefore this state must be a **primary** state, which is defined to satisfy

\begin{align}
L_0 | \Delta \rangle \, &= \, \Delta | \Delta \rangle,  \\
L_n | \Delta \rangle \, &= \, 0 , \qquad \quad  n > 0. 
\end{align}

This allows us to define $\mathcal{R}$ by the $| \Delta \rangle$, plus every element of $\mathcal{V}$ applied to $| \Delta \rangle$. Schematically, these might include things like $ L_{-1}| \Delta \rangle$, $ L^2_{-1}| \Delta \rangle$, $ L_{-1} L_{-2} L_{-1}| \Delta \rangle$, $ L_{-1} L_{2} | \Delta \rangle$, etc. However we only need to consider $L_n$ with $n$ negative because the positive ones annihilate $| \Delta \rangle$. Furthermore, the commutation relations can be used to order the operators $L_n$ (meaning that unordered operators may be rewritten in terms of ordered ones). Therefore the representation consists of the states

\begin{align}
& | \Delta \rangle,  \\
& L_{-1} | \Delta \rangle,  \\
& L_{-1}^2 | \Delta \rangle, \ \ L_{-2} | \Delta \rangle \\
& L_{-1}^3 | \Delta \rangle, \ \ L_{-1} L_{-2} | \Delta \rangle, \ \ L_{-3} | \Delta \rangle \\
&...
\end{align}

In this case, $\mathcal{R}$ is called a **highest-weight representation**. The collection $U(\mathcal{V})$ of all of the operators in the algebra is called the universal enveloping algebra of $\mathcal{V}$. Furthermore, we denote the set of all operators generated by $L_n$, $n<0$, by $U(\mathcal{V^+})$ As a result, we have $ \mathcal{R} =  U(\mathcal{V}) \,   | \Delta \rangle =  U(\mathcal{V^+}) \,   | \Delta \rangle $.

The lines in the above equation are organized by **level**, defined by 

\begin{align}
N = |L_{-n_1} ... L_{-n_p}| = \sum_{i = 1}^p n_i \, .
\end{align}

The number of basis states at level $N$ is the number of partitions of $N$. Any basis state with level $N>0$ is called a **descendent** state.

The construction above allows us to define two types of representations: Verma modules, and degenerate representations.

===Verma Modules===

The first are essentially those where every element of the ordered basis $\left\{ L_{-n_1} ... L_{-n_p} \right\}_{n_1 \leq n_2 ... \leq n_p >0}$ creates a linearly independent element when acting on $|\Delta \rangle$. Such representations are called **Verma modules**, denoted by $\mathcal{V}_\Delta$. Verma modules are infinite-dimensional representations, and are isomorphic to $U(\mathcal{V^+})$.

===Null vectors===

Another possibility is to create a finite-dimensional representation by quotienting a Verma module by one of its subrepresentations.  Consider a subrepresentation of $\mathcal{V}_\Delta$. It must also have $L_0$ eigenvalues bounded from below, and have a highest-weight vector which we denote $|\chi \rangle$. For a non-trivial subrepresentation, $|\chi \rangle \neq |\Delta \rangle$, so $|\chi \rangle$ must be a descendent in the representation $\mathcal{V}_\Delta$. We conclude that $\mathcal{V}_\Delta$ has subrepresentations when there exists a descendent $|\chi \rangle$ which is also annihilated by all $L_n$ for $n>0$. The state $|\chi \rangle$ is called a **null vector** or a **singular vector**.

==Example: null vectors at level 1==

The only descendent at level 1 is $|\chi \rangle = L_{-1} | \Delta \rangle$. We now need to check that $L_n$ annihilates $|\chi \rangle $ for all $n \geq 1$. This is automatic for  $n \geq 2$ because $|\Delta \rangle$ is a highest-weight state. 

So we need only check $L_1$. 

$$ L_1 |\chi \rangle = L_1 L_{-1} |\Delta \rangle = 2 \Delta |\Delta \rangle \, . $$

Therefore $\mathcal{V}_\Delta$ has a null vector if and only if $\Delta = 0$.

==Example: null vectors at level 2==

We can do the same thing at level two. Now there is a family of possible null vectors

$$| \chi \rangle = \left( a L_{-1}^2 + b L_{-2} \right) |\Delta \rangle \, . $$

We need to check that $L_1 |\chi \rangle = 0$ and $L_2 |\chi \rangle = 0$. These equations can only be solved when 

$$ \Delta = \frac{5 - c \pm \sqrt{(c - 1)(c - 25)}}{16} \, . $$

In particular, we only find level 2 null vectors when $c \leq 1$ or $c \geq 25$.

===Denegerate representations===

If a Verma module $\mathcal{V}_\Delta$ has null vectors, then we can construct a new highest-weight representation by quotienting it by the subrepresentation constructed from its null vectors. in other words,

$$ \mathcal R = \frac{\mathcal{V}_\Delta}{U(\mathcal{V}^+) |\chi \rangle} \, . $$

This forms a **degenerate representation**.  All representations are either Verma modules or degenerate representations. The latter are important in the study of [[minimal models]]. 



====Sources====

Like most of the 2d CFT content, this page follows the excellent notes of Sylvain Ribault. The most complete form of those notes is given [[ https://arxiv.org/abs/1406.4290 | here]]