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virasoro_symmetries [2021/09/14 21:11]
Brian McPeak [Representations] 		virasoro_symmetries [2021/09/15 17:52] (current)
Brian McPeak
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 ====Representations==== ====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 satisfy+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} \begin{align}
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 \end{align} \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 $\mathcal{V}$. As a result, we have $ R =  U(\mathcal{V}) \,   | \Delta \rangle $.+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  The lines in the above equation are organized by **level**, defined by 
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 \end{align} \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]]. 
  
  
Last modified: 2021/09/14 21:11
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