Correction to: Local Energy Bounds and Strong Locality in Chiral CFT

Commun. Math. Phys. 390, 169–192 (2022)

Digital Object Identifier https://doi.org/10.1007/s00220-021-04291-9

We correct some errors in [CTW22].

The first one is concerned with Proposition 3.3. The constant $C'$ should actually be $C'' = (1+|k|)^\frac{\beta +1}{2}\sqrt{\frac{C}{|k|(\beta +1)}}$ instead of $C' = \sqrt{\frac{C}{k(\beta +1)}}$. This does not affect the main results of the paper, because we only need that there is such a constant, and we do not need the specific dependence on k. On the other hand, this specific dependence could be useful for other purposes.

The previous value of $C'$ works for $\phi _k$ with $k>0$, but for $\phi _{-k}$ there is a sign mistake at the commutation relation between $\phi _{-k}$ and $L_0$ in the last displayed formula of the proof.

To see that the corrected $C''$ work for $-k$ (in the assumption k and $-k$ needs to be exchanged but it results in the same inequality) for $\Psi _n$ a generic vector in $V_n$, we calculate

$$\begin{aligned} \Vert \phi _{-k}(L_0 + \mathbbm {1})^{-\frac{\beta +1}{2}}\Psi _n\Vert ^2&=\Vert \phi _{-k}(n + \mathbbm {1})^{-\frac{\beta +1}{2}}\Psi _n\Vert ^2 \\&\le (1 + k)^{\beta +1}\Vert \phi _{-k}(n + \mathbbm {1} + k)^{-\frac{\beta +1}{2}}\Psi _n\Vert ^2 \\&= (1 + k)^{\beta +1}\Vert (L_0 + \mathbbm {1})^{-\frac{\beta +1}{2}}\phi _{-k}\Psi _n\Vert ^2 \\&\le (1 + k)^{\beta +1}\Vert (L_0 + \mathbbm {1})^{-\frac{\beta +1}{2}}\phi _{-k}\Vert ^2 \cdot \Vert \Psi _n\Vert ^2 \\&\le (1 + k)^{\beta +1}\Vert \phi _{k}(L_0 + \mathbbm {1})^{-\frac{\beta +1}{2}}\Vert ^2 \cdot \Vert \Psi _n\Vert ^2 \\&\le (1 + k)^{\beta +1}(C')^2 \Vert \Psi _n\Vert ^2. \end{aligned}$$

As $\phi _{-k}(L_0 + \mathbbm {1})^{-\frac{\beta +1}{2}}\Psi _n$ are mutually orthogonal for different n, for a general $\Psi$ we obtain $\Vert \phi _{-k}(L_0 + \mathbbm {1})^{-\frac{\beta +1}{2}}\Psi \Vert ^2 \le (1 + k)^{\beta +1}(C')^2 \Vert \Psi \Vert ^2$, which is what we had to prove with $C'' = (1+k)^{\frac{\beta +1}{2}}\sqrt{\frac{C}{k(\beta + 1)}}$.

In addition, in Proposition 3.2, Theorem 3.5, Lemma 3.6, Theorem 3.7, the fields $\phi$ etc. should be assumed to be hermitian. Note however that in a unitary vertex operator algebra every primary field $\phi$ with conformal dimension d is equal to $\phi _1 + i \phi _2$ with $\phi _1$ and $\phi _2$ hermitian primary fields of the same dimension d.