Knizhnik-Zamolodchikov equations

In mathematical physics the Knizhnik-Zamolodchikov equations are a set of additional constraints satisfied by the correlation functions of the conformal field theory associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the "N"-point functions of primary fields and can be derived using either the formalism of Lie algebras or that of vertex algebras. The structure of the genus zero part of the conformal field theory is encoded in the monodromy properties of these equations. In particular the braiding and fusion of the primary fields (or their associated representations) can be deduced from the properties of the four-point functions, for which the equations reduce to a single matrix-valued first order complex ordinary differential equation of Fuchsian type. Originally the Russian physicists Vadim Knizhnik and Alexander Zamolodchikov deduced the theory for SU(2) using the classical formulas of Gauss for the connection coefficients of the hypergeometric differential equation.

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 $Phi(v,z)$ the primary field associated with it. Let $t^a$ be a basis of the underlying Lie algebra $mathfrak\{g\}$, $t^a\_i$ their representation on the primary field $Phi(v\_i,z)$ and $eta$ the Killing form. Then for $i,j=1,2,ldots,N$ the Knizhnik-Zamolodchikov equations read

: $left(\; (k+h)partial\_\{z\_i\}\; +\; sum\_\{j\; eq\; i\}\; frac\{sum\_\{a,b\}\; eta\_\{ab\}\; t^a\_i\; otimes\; t^b\_j\}\{z\_i-z\_j\}\; ight)\; langle\; Phi(v\_N,z\_N)dotsPhi(v\_1,z\_1)\; angle\; =\; 0.$

Informal 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

: $left(\; L\_\{-1\}\; -\; frac\{1\}\{2(k+h)\}\; sum\_\{k\; in\; mathbf\{Z\; sum\_\{a,b\}\; eta\_\{ab\}\; J^a\_\{-k\}J^b\_\{k-1\}\; ight)v\; =\; 0,$

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.

Mathematical formulation

Since the treatment in
*citation|first= Edward|last= Frenkel|first2= David|last2= Ben-Zvi|title=Vertex algebras and Algebraic Curves|series= Mathematical Surveys and Monographs|volume= 88|publisher=American Mathematical Society|year= 2001|id= ISBN 0-8218-2894-0