Partial differential equations of correlation functions
In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories 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 affine 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 derived the equations for the SU(2) Wess–Zumino–Witten model using the classical formulas of Gauss for the connection coefficients of the hypergeometric differential equation.
Definition.
Let formula_1 denote the affine Lie algebra with level k and dual Coxeter number h. Let v be a vector from a zero mode representation of formula_1 and formula_3 the primary field associated with it. Let formula_4 be a basis of the underlying Lie algebra formula_5, formula_6 their representation on the primary field formula_7 and η the Killing form. Then for formula_8 the Knizhnik–Zamolodchikov equations read
 formula_9
Informal derivation.
The Knizhnik–Zamolodchikov equations result from 
the Sugawara construction of the Virasoro algebra from the affine Lie algebra. More specifically, they result from applying the identity 
 formula_10
to the affine primary field formula_11 in a correlation function of affine primary fields. In this context, only the terms formula_12 are non-vanishing. The action of formula_13 can then be rewritten using global Ward identities, 
formula_14
and formula_15 can be identified with the infinitesimal translation operator formula_16.
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 .
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
formula_17
The vertex operators of energy 1 correspond to the generators of the affine algebra
formula_18
where "X" ranges over the elements of the underlying finite-dimensional simple complex Lie algebra formula_5.
There is an energy 2 eigenvector "L"−2Ω which give the generators "L""n" of the Virasoro algebra associated to the Kac–Moody algebra by the "Segal–Sugawara construction"
formula_20
If "a" has energy α, then the corresponding vertex operator has the form
formula_21
The vertex operators satisfy
formula_22
as well as the locality and associativity relations
formula_23
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 "Hi" at the same level becomes a module for the vertex algebra, in the sense that for each "a" there is a vertex operator "Vi"("a", "z") on "Hi" such that
formula_24
The most general vertex operators at a given level are intertwining operators Φ("v", "z") between representations "Hi" and "H""j" where "v" lies in "H""k". These operators can also be written as
formula_25
but δ can now be rational numbers. Again these intertwining operators are characterized by properties
formula_26
and relations with "L"0 and "L"−1 similar to those above.
When "v" is in the lowest energy subspace for "L"0 on "Hk", an irreducible representation of formula_5, 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
formula_28
In the physics literature the "v""i" are often suppressed and the primary field written Φ"i"("zi"), with the understanding that it is labelled by the corresponding irreducible representation of formula_5.
Vertex algebra derivation.
If ("X""s") is an orthonormal basis of formula_5 for the Killing form, the Knizhnik–Zamolodchikov equations may be deduced by integrating the correlation function
formula_31
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:
formula_32
Integrating both sides in the "z" variable about a small circle centred on "zi" 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 termΦ("vi", "zi") may be replaced in the correlation function by its commutator with "Lr" where "r" = 0, ±1. The result can be expressed in terms of the derivative with respect to "zi". On the other hand, "Lr" is also given by the Segal–Sugawara formula:
formula_33
After substituting these formulas for "Lr", the resulting expressions can be simplified using the commutator formulas
 formula_34
Original derivation.
The original proof of , reproduced in , uses a combination of both of the above methods. First note that for "X" in formula_5
formula_36
Hence
formula_37
On the other hand,
formula_38
so that
formula_39
The result follows by using this limit in the previous equality.
Monodromy representation of KZ equation.
In conformal field theory along the above definition the "n"-point correlation function of the primary field satisfies KZ equation. In particular, for formula_40 and non negative integers "k" there are formula_41 primary fields formula_42 's corresponding to spin j representation (formula_43). The correlation function formula_44 of the primary fields formula_42 's for the representation formula_46 takes values in the tensor product formula_47 and its KZ equation is
formula_48,
where formula_49 as the above informal derivation.
This "n"-point correlation function can be analytically continued as multi-valued holomorphic function to the domain formula_50 with formula_51 for formula_52. Due to this analytic continuation, the holonomy of the KZ equation can be described by the braid group formula_53 introduced by Emil Artin. In general, A complex semi-simple Lie algebra formula_5 and its representations formula_46 give the linear representation of braid group
formula_56
as the holonomy of KZ equation. Oppositely, a KZ equation gives the linear representation of braid groups as its holonomy.
The action on formula_57 by the analytic continuation of KZ equation is called monodromy representation of KZ equation. In particular, if all formula_58 's have spin 1/2 representation then the linear representation obtained from KZ equation agrees with the representation constructed from operator algebra theory by Vaughan Jones. It is known that the monodromy representation of KZ equation with a general semi-simple Lie algebra agrees with the linear representation of braid group given by R-matrix of the corresponding quantum group.
KZ-BPZ relation.
In the case when the underlying Lie algebra is formula_59, the KZ equations are mapped to BPZ equations by Sklyanin's separation of variables for the formula_60 Gaudin model.
References.
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