Knizhnik-Zamolodchikov equations

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In a conformal field theory with an additional affine symmetry the Knizhnik-Zamolodchikov equations are a set of additional constraints satisfied by the correlation functions of the theory.

[edit] Definition

Let $\hat{\mathfrak{g}}_k$ denote the affine Lie algebra with level $k$ and dual Coxeter number $h$ . Let $v$ be a vector from a zero mode representation of $\hat{\mathfrak{g}}_k$ and $Φ(v | z)$ the primary field associated with it. Let $t a$ be the generators of the Lie algebra $\mathfrak{g}$ , $t^a_i$ their representation on the primary field $Φ(v i | z)$ and $η$ the Killing form. Then for $i,j=1,2,\ldots,N$ the Knizhnik-Zamolodchikov equations read

$\left( (k-h)\partial_{z_i} + \sum_{j \neq i} \frac{\sum_{a,b} \eta_{ab} t^a_i \otimes t^b_j}{z_i-z_j} \right) \langle \Phi(v_N|z_N)\dots\Phi(v_1|z_1) \rangle = 0$ .

[edit] Derivation

The Knizhnik-Zamolodchikov equations result from the existence of null vectors in the $\hat{\mathfrak{g}}_k$ module. This is quite similar to the case in minimal models, where the existence of null vectors result in additional constraints on the correlation functions.

The null vectors of a $\hat{\mathfrak{g}}_k$ module are of the form

$(L_{-1} - \frac{1}{2(k-h)} \sum_{k \in \mathbf{Z}} \sum_{a,b} \eta_{ab} J^a_{-k}J^b_{k-1})v = 0$ ,

where $v$ is a highest weight vector and $J^a_k$ the conserved current associated with the affine generator $t a$ . Since $v$ is of highest weight, the action of most $J^a_k$ on it vanish and only $J^a_{-1}J^b_{0}$ remain. The operator-state correspondence then leads directly to the Knizhnik-Zamolodchikov equations as given above.