#### Quick reminders on Virasoro null vectors.

#### A null vector of the Virasoro algebra is labelled by two integers \(r,s\geq 1\), whose product is the level of the null vector. This null vector occurs in the Verma module with a specific conformal dimension \(\Delta_{r,s}\), and it can be written as

\[|\chi_{r,s}\rangle = L_{r,s} |\Delta_{r,s}\rangle\] where \(|\Delta_{r,s}\rangle\) is the primary state of our Verma module, and \(L_{r,s}\) is a level \(rs\) creation operator.

Let \(c=1+6Q^2\) with \(Q=b+\frac{1}{b}\) be the central charge, and let us write conformal dimensions \(\Delta = \alpha(Q-\alpha)\) in terms of momentums \(\alpha\). The momentum that corresponds to \(\Delta_{r,s}\) is

\[\alpha_{r,s} = \frac12\left[Q-rb-\frac{s}{b}\right]\] In particular the degenerate momentums up to the level \(5\) are

\[\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \alpha_{1,1} & \alpha_{1,2} & \alpha_{2,1} & \alpha_{1,3} & \alpha_{3,1} & \alpha_{1,4} &\alpha_{2,2} &\alpha_{4,1} & \alpha_{1,5} & \alpha_{5,1} \\ \hline 0 & -\frac{b}{2} & -\frac{1}{2b} & -b & -\frac{1}{b} & -\frac{3b}{2} & -Q & -\frac{3}{2b} & -2b & -\frac{2}{b} \\ \hline \end{array}\]

#### Null vectors from deformed matrix integrals.

For any given level \(N\), Manabe and Sulkowski find a level \(N\) creation operator \(A_N^\alpha\) that depends on a momentum \(\alpha\), such that

\[rs\leq N \quad \Rightarrow \quad A^{\alpha_{r,s}}_N \propto L_{r,s}\] where the proportionality coefficient is itself a creation operator if \(rs<N\). In particular,

\[A_1^\alpha = L_{-1}\] \[A_2^\alpha = L_{-1}^2 +4(\alpha-\alpha_{1,1})^2 L_{-2}\] \[A_3^\alpha = L_{-1}A_2^\alpha +8(\alpha-\alpha_{1,1})^2(\alpha-\alpha_{1,2})(\alpha-\alpha_{2,1}) L_{-3}\] \[\begin{array}{rl} A_4^\alpha = L_{-1}A_3^\alpha + \frac{4}{5\alpha+3Q} (\alpha-\alpha_{1,1})(\alpha-\alpha_{1,3})(\alpha-\alpha_{3,1}) L_{-2}A^\alpha_2 \\ -\frac{8}{5\alpha+3Q}(\alpha-\alpha_{1,1})(\alpha-\alpha_{1,2}) (\alpha-\alpha_{2,1})(\alpha-\alpha_{1,3}) (\alpha-\alpha_{3,1}) L_{-1}L_{-3} \\ +\frac{64}{5\alpha+3Q} (\alpha-\alpha_{1,1})(\alpha-\alpha_{1,2})^2(\alpha-\alpha_{2,1})^2 (\alpha-\alpha_{1,3}) (\alpha-\alpha_{3,1}) L_{-4} \end{array}\]

Let us look for a conformal field theory interpretation of \(A^\alpha_N\), irrespective of its derivation from matrix integrals. These quantities do not depend on the conformal dimension \(\Delta\), but rather on the corresponding momentum \(\alpha\). This suggest that they should not be understood in terms of the Virasoro algebra, but rather in terms of the corresponding affine Lie algebra \(\hat u_1\).

#### Quick reminders on the affine Lie algebra \(\hat u_1\).

#### The generators \(L_n\) of the Virasoro algebra can be rewritten in terms of generators \(J_n\) of the affine Lie algebra \(\hat u_1\) as

\[L_{n\neq 0} = -\sum_{m=-\infty}^\infty J_{n-m}J_m + Q(n+1)J_n\] \[L_0 =-2\sum_{m=1}^\infty J_{-m}J_m -J_0^2+QJ_0\] where \([J_m,J_n ] = \frac12 n \delta_{m+n,0}\). The Verma module with conformal dimension \(\Delta =\alpha(Q-\alpha)\) corresponds to the affine highest-weight representation with momentum \(\alpha\) and a primary state \(|\alpha\rangle\) such that \(J_{n>0}|\alpha\rangle =0\) and \(J_0|\alpha\rangle = \alpha |\alpha\rangle\). The primary state \(|\alpha\rangle\) is identical to \(|\Delta\rangle\) as far as the Virasoro algebra is concerned, but it can be distinguished from \(|Q-\alpha\rangle\) by the action of the \(\hat u_1\) algebra.

#### The creation operators in terms of the affine Lie algebra.

#### In the expression for \(A_N^\alpha\), let us replace the Virasoro generators with \(\hat u_1\) generators , and then replace the central generator \(J_0\) with its eigenvalue \(\alpha\). We first find

\[A_1^\alpha = -2\alpha J_{-1}\] For \(\alpha=\alpha_{1,1}=0\), the operator \(A_1^\alpha\) does not correspond to a \(\hat u_1\) null vector: rather, its action in our \(\hat u_1\) representation vanishes identically. It turns out that the other operators \(A_N^\alpha\) have the same behaviour:

\[A_2^\alpha = -8\prod_{rs\leq 2} (\alpha-\alpha_{r,s}) \ J_{-2}\] \[A_3^\alpha = -8\prod_{rs\leq 3}(\alpha -\alpha_{r,s})\ J_{-3}\] \[A_4^\alpha = -\frac{128}{5\alpha+3Q} \prod_{rs\leq 4}(\alpha -\alpha_{r,s})\ J_{-4}\] Assuming this pattern goes on at higher levels, this suggests that \(A_N^\alpha\) could be constructed without resorting to matrix models, by rewriting \(J_{-N}\) in terms of Virasoro generators.

#### Conclusion.

#### It seems that the construction of Manabe and Sulkowski becomes simpler when formulated in terms of the affine Lie algebra \(\hat u_1\). This might lead to an efficient algorithm for computing Virasoro null vectors. The role of \(\hat u_1\) reminds me of Alba, Fateev, Litvinov and Tarnopolsky’s CFT interpretation of the AGT correspondence, with their formulas for Virasoro conformal blocks as sums over states that are better expressed in terms of \(\hat u_1\) rather than Virasoro.

#### Generalization.

#### I had an email exchange with David Ridout and Simon Wood, and this helped clarify the algebraic structure behind the operators \(A_N^\alpha\). I have already written the expression for Virasoro generators in terms of \(\hat u_1\) generators. The map from Virasoro to \(\hat u_1\) cannot be inverted, but given a momentum \(\alpha\), we can write a unique map from \(\hat u_1\) creation operators to Virasoro creation operators, for example

\[\varphi_\alpha(J_{-1}) = -\frac{1}{2\alpha} L_{-1}\] \[\varphi_\alpha(J_{-2}) = -\frac{1}{8\prod_{rs\leq 2} (\alpha-\alpha_{r,s})} \left(L_{-1}^2 +4\alpha^2 L_{-2}\right)\] \[\varphi_\alpha(J_{-1}^2) = \frac{1}{8\prod_{rs\leq 2} (\alpha-\alpha_{r,s})}\left((2\alpha+Q)L_{-1}^2-2\alpha L_{-2}\right)\] For \(\alpha = \alpha_{r,s}\), the Verma module becomes reducible, but nothing special happens on the \(\hat u_1\) side. The map \(\varphi_\alpha\) becomes singular, and more specifically it has a simple pole, with a residue that is proportional to the Virasoro null vector.

So there is nothing special with the creation operator \(J_{-N}\) that appears in the work of Manabe and Sulkowski, and with the corresponding operator \(A_N^\alpha \propto \varphi_\alpha(J_{-N})\). Applying the map \(\varphi_\alpha\) to any \(\hat u_1\) creation operator produces similar results.

According to Ridout and Wood, this type of map between a Verma module and a Fock space is related to an object called a Jantzen filtration, see Chapter 3 of the book by Iohara and Koga.

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