In mathematical physics the Knizhnik–Zamolodchikov equations or KZ 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 Npoint 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 fourpoint functions, for which the equations reduce to a single matrixvalued 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 $\backslash hat\{\backslash mathfrak\{g\}\}\_k$ denote the affine Lie algebra with level $k$ and dual Coxeter number $h$. Let $v$ be a vector from a zero mode representation of $\backslash hat\{\backslash mathfrak\{g\}\}\_k$ and $\backslash Phi(v,z)$ the primary field associated with it. Let $t^a$ be a basis of the underlying Lie algebra $\backslash mathfrak\{g\}$, $t^a\_i$ their representation on the primary field $\backslash Phi(v\_i,z)$ and $\backslash eta$ the Killing form. Then for $i,j=1,2,\backslash ldots,N$ the Knizhnik–Zamolodchikov equations read
 $\backslash left(\; (k+h)\backslash partial\_\{z\_i\}\; +\; \backslash sum\_\{j\; \backslash neq\; i\}\; \backslash frac\{\backslash sum\_\{a,b\}\; \backslash eta\_\{ab\}\; t^a\_i\; \backslash otimes\; t^b\_j\}\{z\_iz\_j\}\; \backslash right)\; \backslash langle\; \backslash Phi(v\_N,z\_N)\backslash dots\backslash Phi(v\_1,z\_1)\; \backslash rangle\; =\; 0.$
Informal derivation
The Knizhnik–Zamolodchikov equations result from the existence of null vectors in the $\backslash hat\{\backslash mathfrak\{g\}\}\_k$ 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 $\backslash hat\{\backslash mathfrak\{g\}\}\_k$ module are of the form
 $\backslash left(\; L\_\{1\}\; \; \backslash frac\{1\}\{2(k+h)\}\; \backslash sum\_\{k\; \backslash in\; \backslash mathbf\{Z\}\}\; \backslash sum\_\{a,b\}\; \backslash eta\_\{ab\}\; J^a\_\{k\}J^b\_\{k1\}\; \backslash right)v\; =\; 0,$
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 operatorstate correspondence then leads directly to the Knizhnik–Zamolodchikov equations as given above.
Mathematical formulation
Main article:
Vertex algebra
Since the treatment in Tsuchiya & Kanie (1988), the Knizhnik–Zamolodchikov equation has been formulated mathematically in the language of vertex algebras due to Borcherds (1986) and Frenkel, Lepowsky & Meurman (1988). This approach was popularized amongst theoretical physicists by Goddard (1988) and amongst mathematicians by Kac (1996).
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 nonnegative 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
 $V(a,0)\backslash Omega\; =\; a.\backslash ,$
The vertex operators of energy 1 correspond to the generators of the affine algebra
 $X(z)=\backslash sum\; X(n)\; z^\{n1\}$
where X ranges over the elements of the underlying finitedimensional simple complex Lie algebra $\backslash mathfrak\{g\}$.
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
 $T(z)\; =\; \backslash sum\; L\_n\; z^\{n2\}.$
If a has energy α, then the corresponding vertex operator has the form
 $V(a,z)\; =\; \backslash sum\; V(a,n)z^\{n\backslash alpha\}.$
The vertex operators satisfy
 $\{d\backslash over\; dz\}\; V(a,z)\; =\; [L\_\{1\},V(a,z)]=\; V(L\_\{1\}a,z),\backslash ,\backslash ,\; [L\_0,V(a,z)]=(z^\{1\}\; \{d\backslash over\; dz\}\; +\; \backslash alpha)V(a,z)$
as well as the locality and associativity relations
 $V(a,z)V(b,w)\; =\; V(b,w)\; V(a,z)\; =\; V(V(a,zw)b,w).\backslash ,$
These last two relations are understood as analytic continuations: the inner products with finite energy vectors of the three expressions define the same polynomials 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
 $V\_i(a,z)V\_i(b,w)\; =\; V\_i(b,w)\; V\_i(a,z)=V\_i(V(a,zw)b,w).\backslash ,$
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
 $\backslash Phi(v,z)=\backslash sum\; \backslash Phi(v,n)\; z^\{n\backslash delta\}\backslash ,$
but δ can now be rational numbers. Again these intertwining operators are characterized by properties
 $V\_j(a,z)\; \backslash Phi(v,w)=\; \backslash Phi(v,w)\; V\_i(a,w)\; =\; \backslash Phi(V\_k(a,zw)v,w)\backslash ,$
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
$\backslash mathfrak\{g\}$, 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 npoint function is defined by
 $\backslash langle\; \backslash Phi(v\_1,z\_1)\; \backslash Phi(v\_2,z\_2)\; \backslash dots\; \backslash Phi(v\_n,z\_n)\backslash rangle\; =\; (\backslash Phi(v\_1,z\_1)\; \backslash Phi(v\_2,z\_2)\; \backslash dots\; \backslash Phi(v\_n,z\_n)\backslash Omega,\backslash Omega).$
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 $\backslash mathfrak\{g\}$.
Vertex algebra derivation
If (X_{s}) is an orthonormal basis of $\backslash mathfrak\{g\}$ for the Killing form, the Knizhnik–Zamolodchikov equations may be deduced by integrating the correlation function
 $\backslash sum\_s\; \backslash langle\; X\_s(w)X\_s(z)\backslash Phi(v\_1,z\_1)\; \backslash cdots\; \backslash Phi(v\_n,z\_n)\; \backslash rangle\; (wz)^\{1\}$
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:
 $\{1\backslash over\; 2\}(k+h)\; \backslash langle\; T(z)\backslash Phi(v\_1,z\_1)\backslash cdots\; \backslash Phi(v\_n,z\_n)\; \backslash rangle\; =\; \; \backslash sum\_\{j,s\}\; \backslash langle\; X\_s(z)\backslash Phi(v\_1,z\_1)\; \backslash cdots\; \backslash Phi(X\_s\; v\_j,z\_j)\; \backslash Phi(X\_n,z\_n)\backslash rangle\; (zz\_j)^\{1\}.$
Integrating both sides in the z variable about a small circle centred on z_{i} yields the i^{th} Knizhnik–Zamolodchikov equation.
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:
 $L\_0\; =\; (k+h)^\{1\}\backslash sum\_s\backslash left[\; \{1\backslash over\; 2\}X\_s(0)^2\; +\; \backslash sum\_\{m>0\}\; X\_s(m)X\_s(m)\backslash right],\; \backslash ,\backslash ,\backslash ,\; L\_\{\backslash pm\; 1\; \}\; =(k+h)^\{1\}\; \backslash sum\_s\backslash sum\_\{\; m\backslash ge\; 0\}\; X\_s(m\backslash pm\; 1)X\_s(m).$
After substituting these formulas for L_{r}, the resulting expressions can be simplified using the commutator formulas
 $[X(m),\backslash Phi(a,n)]=\; \backslash Phi(Xa,m+n).\backslash ,$
Original derivation
The original proof of Knizhnik & Zamolodchikov (1984), reproduced in Tsuchiya & Kanie (1988), uses a combination of both of the above methods. First note that for X in $\backslash mathfrak\{g\}$
 $\backslash langle\; X(z)\backslash Phi(v\_1,z\_1)\; \backslash cdots\; \backslash Phi(v\_n,z\_n)\; \backslash rangle\; =\; \backslash sum\_j\; \backslash langle\; \backslash Phi(v\_1,z\_1)\; \backslash cdots\; \backslash Phi(Xv\_j,z\_j)\; \backslash cdots\; \backslash Phi(v\_n,z\_n)\; \backslash rangle\; (zz\_j)^\{1\}.$
Hence
 $\backslash sum\_s\; \backslash langle\; X\_s(z)\backslash Phi(z\_1,v\_1)\; \backslash cdots\; \backslash Phi(X\_sv\_i,z\_i)\; \backslash cdots\; \backslash Phi(v\_n,z\_n)\backslash rangle$
= \sum_j\sum_s \langle\cdots \Phi(X_s v_j, z_j) \cdots \Phi(X_s v_i,z_i) \cdots\rangle (zz_j)^{1}.
On the other hand
 $\backslash sum\_s\; X\_s(z)\backslash Phi(X\_sv\_i,z\_i)\; =\; (zz\_i)^\{1\}\backslash Phi(\backslash sum\_s\; X\_s^2v\_i,z\_i)\; +\; (k+g)\{\backslash partial\backslash over\; \backslash partial\; z\_i\}\; \backslash Phi(v\_i,z\_i)\; +O(zz\_i)$
so that
 $(k+g)\{\backslash partial\backslash over\; \backslash partial\; z\_i\}\; \backslash Phi(v\_i,z\_i)\; =\; \backslash lim\_\{z\backslash rightarrow\; z\_i\}\; \backslash left[\backslash sum\_s\; X\_s(z)\backslash Phi(X\_sv\_i,z\_i)\; (zz\_i)^\{1\}\backslash Phi(\backslash sum\_s\; X\_s^2\; v\_i,z\_i)\backslash right].$
The result follows by using this limit in the previous equality.
Applications
See also
References


 (Erratum in volume 19, pp. 675–682.)






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