Knizhnik–Zamolodchikov equations


In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions 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 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 Wess–Zumino–Witten model using the classical formulas of Gauss for the connection coefficients of the hypergeometric differential equation.

Definition

Let denote the affine Lie algebra with level and dual Coxeter number. Let be a vector from a zero mode representation of and the primary field associated with it. Let be a basis of the underlying Lie algebra, their representation on the primary field and the Killing form. Then for the Knizhnik–Zamolodchikov equations read

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
to the affine primary field in a correlation function of affine primary fields. In this context, only the terms are non-vanishing. The action of can then be rewritten using global Ward identities,
and can be identified with the infinitesimal translation operator.

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 H0 of an affine Kac–Moody algebra at a fixed level can be encoded in a vertex algebra.
The derivation 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. The eigenvalue of an eigenvector of L0 is called its energy. For every state a in L there is a vertex operator V 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 which give the generators Ln of the Virasoro algebra associated to the Kac–Moody algebra by the Segal–Sugawara construction
If a has energy, 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 and 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 on Hi such that
The most general vertex operators at a given level are intertwining operators 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 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, with the understanding that it is labelled by the corresponding irreducible representation of.

Vertex algebra derivation

If is an orthonormal basis 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.

Lie algebra derivation

It is also possible to deduce the Knizhnik–Zamodchikov equations without explicit use of vertex algebras. The term 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

Original derivation

The original proof of, reproduced in, uses a combination of both of the above methods. First note that for X in
Hence
On the other hand,
so that
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 and non negative integers k there are k + 1 primary fields 's corresponding to spin j representation. The correlation function of the primary fields 's for the representation takes values in the tensor product and its KZ equation is
where as the above [|informal derivation].
This n-point correlation function can be analytically continued as multi-valued holomorphic function to the domain with . Due to this analytic continuation the holonomy of KZ equation can be described by the braid group introduced by Emil Artin. In general, A complex semi-simple Lie algebra and its representations 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.
The action on by the analytic continuation of KZ equation is called monodromy representation of KZ equation. In particular, if all '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.

Applications