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.
denote the affine Lie algebra with level k and dual Coxeter number h. Let v be a vector from a zero mode representation of
More specifically, they result from applying the identity to the affine primary field
in a correlation function of affine primary fields.
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 (1989) and amongst mathematicians by Kac (1997).
The derivation d acts as the energy operator L0 on H0, which can be written as a direct sum of the non-negative integer eigenspaces of L0, the zero energy space being generated by the vacuum vector Ω.
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 The vertex operators of energy 1 correspond to the generators of the affine algebra where X ranges over the elements of the underlying finite-dimensional simple complex Lie algebra
There is an energy 2 eigenvector L−2Ω which give the generators Ln of the Virasoro algebra associated to the Kac–Moody algebra by the Segal–Sugawara construction If a has energy α, then the corresponding vertex operator has the form The vertex operators satisfy as well as the locality and associativity relations 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|.
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 The most general vertex operators at a given level are intertwining operators Φ(v, z) between representations Hi and Hj where v lies in Hk.
These operators can also be written as but δ can now be rational numbers.
Again these intertwining operators are characterized by properties and relations with L0 and L−1 similar to those above.
When v is in the lowest energy subspace for L0 on Hk, an irreducible representation of
, the operator Φ(v, w) is called a primary field of charge k. Given a chain of n primary fields starting and ending at H0, their correlation or n-point function is defined by In the physics literature the vi are often suppressed and the primary field written Φi(zi), with the understanding that it is labelled by the corresponding irreducible representation of
for the Killing form, the Knizhnik–Zamolodchikov equations may be deduced by integrating the correlation function 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 zj's: Integrating both sides in the z variable about a small circle centred on zi yields the ith Knizhnik–Zamolodchikov equation.
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: After substituting these formulas for Lr, the resulting expressions can be simplified using the commutator formulas The original proof of Knizhnik & Zamolodchikov (1984), reproduced in Tsuchiya & Kanie (1988), uses a combination of both of the above methods.
Hence On the other hand, so that The result follows by using this limit in the previous equality.
In conformal field theory along the above definition the n-point correlation function of the primary field satisfies KZ equation.
Due to this analytic continuation, the holonomy of the KZ equation can be described by the braid group
[1] In general, A complex semi-simple Lie algebra
give the linear representation of braid group as the holonomy of KZ equation.
Oppositely, a KZ equation gives the linear representation of braid groups as its holonomy.
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.