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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

## Informal derivation

## Mathematical formulation

Since the treatment in , the Knizhnik-Zamolodchikov equation has been formulated mathematically in the language of vertex algebras due to and . This approach was popularized amongst theoretical physicists by and amongst mathematicians by .#### Vertex algebra derivation

If (X_{s}) is an orthonormal basis of $mathfrak\{g\}$ for the Killing form, the Knizhnik-Zamolodchikov equations may be deduced by integrating the correlation function #### Lie algebra derivation

It is also possible to deduce the Knizhnik-Zamodchikov equations without explicit use of vertex algebras. The term Φ(v_{i},z_{i}) may be replaced in the correlation function by its commutator with L_{r} where r = 0 or ±1. The result can be expressed in terms of the derivative with respect to z_{i}. On the other hand L_{r} is also given by the Segal-Sugawara formula:#### Original derivation

The original proof of , reproduced in , uses a combination of both of the above methods. First note that for X in $mathfrak\{g\}$## References

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

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

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

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

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.

The vacuum representation H_{0} of an affine Kac-Moody algebra at a fixed level can be encoded in a vertex algebra.
The derivation d acts as the energy operator L_{0} on H_{0}, which can be written as a direct sum of the non-negative integer eigenspaces of L_{0}, the zero energy space being generated by the vacuum vector Ω. The eigenvalue of an eigenvector of L_{0} is called its energy. For every state a in L there is a vertex operator V(a,z) which creates a from the vacuum vector Ω, in the sense that

- $V(a,0)Omega\; =\; a.,$

The vertex operators of energy 1 correspond to the generators of the affine algebra

- $X(z)=sum\; X(n)\; z^\{-n-1\}$

where X ranges over the elements of the underlying finite-dimensional simple complex Lie algebra $mathfrak\{g\}$.

There is an energy 2 eigenvector L_{−2}Ω which give the generators L_{n} of the Virasoro algebra associated to the Kac-Moody algebra by the Segal-Sugawara construction

- $T(z)\; =\; sum\; L\_n\; z^\{-n-2\}.$

If a has energy α, then the corresponding vertex operator has the form

- $V(a,z)\; sum\; V(a,n)z^\{-n-alpha\}.$

The vertex operators satisfy

- $\{dover\; dz\}\; V(a,z)\; =\; [L\_\{-1\},V(a,z)]=\; V(L\_\{-1\}a,z),,,\; [L\_0,V(a,z)]=(z^\{-1\}\; \{dover\; dz\}\; +\; alpha)V(a,z)$

as well as the locality and associativity relations

- $V(a,z)V(b,w)\; =\; V(b,w)\; V(a,z)\; =\; V(V(a,z-w)b,w).,$

These last two relations are understood as analytic continuations: the inner products with finite energy vectors of the three expressions define the same poynomials in z^{±1}, w^{±1} and (z – w)^{–1} in the domains |z| < |w|, |z| > |w| and |z – w| < |w|. All the structural relations of the Kac-Moody and Virasoro algebra can be recovered from these relations, including the Segal-Sugawara construction.

Every other integral representation H_{i} at the same level becomes a module for the vertex algebra, in the sense that for each a there is a vertex operator V_{i}(a,z) on H_{i} such that

- $V\_i(a,z)V\_i(b,w)\; =\; V\_i(b,w)\; V\_i(a,z)=V\_i(V(a,z-w)b,w).,$

The most general vertex operators at a given level are intertwining operators Φ(v,z) between representations H_{i} and H_{j} where v lies in H_{k}. These operators can also be written as

- $Phi(v,z)=sum\; Phi(v,n)\; z^\{-n-delta\},$

but δ can now be rational numbers. Again these intertwining operators are characterized by properties

- $V\_j(a,z)\; Phi(v,w)=\; Phi(v,w)\; V\_i(a,w)\; =\; Phi(V\_k(a,z-w)v,w),$

and relations with L_{0} and L_{–1} similar to those above.

When v is in the lowest energy subspace for L_{0} on H_{k}, an irreducible representation of
$mathfrak\{g\}$, the operator Φ(v,w) is called a primary field of charge k.

Given a chain of n primary fields starting and ending at H_{0}, their correlation or n-point function is defined by

- $langle\; Phi(v\_1,z\_1)\; Phi(v\_2,z\_2)\; dots\; Phi(v\_n,z\_n)rangle\; =\; (Phi(v\_1,z\_1)\; Phi(v\_2,z\_2)\; dots\; Phi(v\_n,z\_n)Omega,Omega).$

In the physics literature the v_{i} are often suppressed and the primary field written Φ_{i}(z_{i}), with the understanding that it is labelled by the corresponding irreducible representation of $mathfrak\{g\}$.

- $sum\_s\; langle\; X\_s(w)X\_s(z)Phi(v\_1,z\_1)\; cdots\; Phi(v\_n,z\_n)\; rangle\; (w-z)^\{-1\}$

first in the w variable around a small circle centred at z; by Cauchy's theorem the result can be expressed as sum of integrals around n small circles centred at the z_{j}'s:

- $\{1over\; 2\}(k+h)\; langle\; T(z)Phi(v\_1,z\_1)cdots\; Phi(v\_n,z\_n)\; rangle\; =\; -\; sum\_\{j,s\}\; langle\; X\_s(z)Phi(v\_1,z\_1)\; cdots\; Phi(X\_s\; v\_j,z\_j)\; Phi(X\_n,z\_n)rangle\; (z-z\_j)^\{-1\}.$

Integrating both sides in the z variable about a small circle centred on z_{i} yields the i^{th} Knizhnik-Zamolodchikov equation.

- $L\_0\; =\; (k+h)^\{-1\}sum\_sleft[\{1over\; 2\}X\_s(0)^2\; +\; sum\_\{m>0\}\; X\_s(-m)X\_s(m)right],\; ,,,\; L\_\{pm\; 1\; \}\; =(k+h)^\{-1\}\; sum\_ssum\_\{\; mge\; 0\}\; X\_s(-mpm\; 1)X\_s(m).$

After substituting these formulas for L_{r}, the resulting expressions can be simplified using the commutator formulas

- $[X(m),Phi(a,n)]=\; Phi(Xa,m+n).,$

- $langle\; X(z)Phi(v\_1,z\_1)\; cdots\; Phi(v\_n,z\_n)\; rangle\; =\; sum\_j\; langle\; Phi(v\_1,z\_1)\; cdots\; Phi(Xv\_j,z\_j)\; cdots\; Phi(v\_n,z\_n)\; rangle\; (z-z\_j)^\{-1\}.$

Hence

- $sum\_s\; langle\; X\_s(z)Phi(z\_1,v\_1)\; cdots\; Phi(X\_sv\_i,z\_i)\; cdots\; Phi(v\_n,z\_n)rangle$

On the other hand

- $sum\_s\; X\_s(z)Phi(X\_sv\_i,z\_i)\; =\; (z-z\_i)^\{-1\}Phi(sum\_s\; X\_s^2v\_i,z\_i)\; +\; (k+g)\{partialover\; partial\; z\_i\}\; Phi(v\_i,z\_i)\; +O(z-z\_i)$

so that

- $(k+g)\{partialover\; partial\; z\_i\}\; Phi(v\_i,z\_i)\; =\; lim\_\{zrightarrow\; z\_i\}\; left[sum\_s\; X\_s(z)Phi(X\_sv\_i,z\_i)\; -(z-z\_i)^\{-1\}Phi(sum\_s\; X\_s^2\; v\_i,z\_i)right].$

The result follows by using this limit in the previous equality.

- (Erratum in volume 19, pp. 675-682.)

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