Knizhnik–Zamolodchikov equations

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 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

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 ${\mathfrak {g}}$ , $t_{i}^{a}$ their representation on the primary field $\Phi (v_{i},z)$ and $η$ 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_{i}^{a}\otimes t_{j}^{b}}{z_{i}-z_{j}}}\right)\left\langle \Phi (v_{N},z_{N})\dots \Phi (v_{1},z_{1})\right\rangle =0.

Informal derivation

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_{-k}^{a}J_{k-1}^{b}\right)v=0,

where $v$ is a highest weight vector and $J_{k}^{a}$ the conserved current associated with the affine generator $t^{a}$ . Since $v$ is of highest weight, the action of most $J_{k}^{a}$ on it vanish and only $J_{-1}^{a}J_{0}^{b}$ remain. The operator-state correspondence then leads directly to the Knizhnik–Zamolodchikov equations as given above.

Mathematical formulation

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 H₀ 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 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(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

{\begin{aligned}{\frac {d}{dz}}V(a,z)&=\left[L_{-1},V(a,z)\right]=V\left(L_{-1}a,z\right)\\\left[L_{0},V(a,z)\right]&=\left(z^{-1}{\frac {d}{dz}}+\alpha \right)V(a,z)\end{aligned}}

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 polynomials in $z \pm1, w \pm1$ 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 \left(V_{k}(a,z-w)v,w\right)

and relations with L₀ and L₋₁ similar to those above.

When v is in the lowest energy subspace for L₀ 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 H₀, their correlation or n-point function is defined by

\left\langle \Phi (v_{1},z_{1})\Phi (v_{2},z_{2})\cdots \Phi (v_{n},z_{n})\right\rangle =\left(\Phi \left(v_{1},z_{1}\right)\Phi \left(v_{2},z_{2}\right)\cdots \Phi \left(v_{n},z_{n}\right)\Omega ,\Omega \right).

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 ${\mathfrak {g}}$ .

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

\sum _{s}\left\langle X_{s}(w)X_{s}(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\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:

{1 \over 2}(k+h)\left\langle T(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle =-\sum _{j,s}\left\langle X_{s}(z)\Phi (v_{1},z_{1})\cdots \Phi (X_{s}v_{j},z_{j})\Phi (X_{n},z_{n})\right\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.

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, ±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:

{\begin{aligned}L_{0}&=(k+h)^{-1}\sum _{s}\left[{\frac {1}{2}}X_{s}(0)^{2}+\sum _{m>0}X_{s}(-m)X_{s}(m)\right]\\L_{\pm 1}&=(k+h)^{-1}\sum _{s}\sum _{m\geq 0}X_{s}(-m\pm 1)X_{s}(m)\end{aligned}}

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).

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 ${\mathfrak {g}}$

\left\langle X(z)\Phi (v_{1},z_{1})\cdots \Phi (v_{n},z_{n})\right\rangle =\sum _{j}\left\langle \Phi (v_{1},z_{1})\cdots \Phi (Xv_{j},z_{j})\cdots \Phi (v_{n},z_{n})\right\rangle (z-z_{j})^{-1}.

Hence

\sum _{s}\langle X_{s}(z)\Phi (z_{1},v_{1})\cdots \Phi (X_{s}v_{i},z_{i})\cdots \Phi (v_{n},z_{n})\rangle =\sum _{j}\sum _{s}\langle \cdots \Phi (X_{s}v_{j},z_{j})\cdots \Phi (X_{s}v_{i},z_{i})\cdots \rangle (z-z_{j})^{-1}.

On the other hand

\sum _{s}X_{s}(z)\Phi \left(X_{s}v_{i},z_{i}\right)=(z-z_{i})^{-1}\Phi \left(\sum _{s}X_{s}^{2}v_{i},z_{i}\right)+(k+g){\partial  \over \partial z_{i}}\Phi (v_{i},z_{i})+O(z-z_{i})

so that

(k+g){\frac {\partial }{\partial z_{i}}}\Phi (v_{i},z_{i})=\lim _{z\to z_{i}}\left[\sum _{s}X_{s}(z)\Phi \left(X_{s}v_{i},z_{i}\right)-(z-z_{i})^{-1}\Phi \left(\sum _{s}X_{s}^{2}v_{i},z_{i}\right)\right].

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 ${\mathfrak {sl}}_{2}$ and non negative integers k there are k + 1 primary fields $\Phi _{j}(z_{j})$ 's corresponding to spin _j representation ( j = 0, 1/2, 1, 3/2,.., k/2). The correlation function $\Psi (z_{1},\dots ,z_{n})$ of the primary fields $\Phi _{j}(z_{j})$ 's for the representation $(\rho ,V_{i})$ takes values in the tensor product $V_{1}\otimes \cdots \otimes V_{n}$ and its KZ equation is

(k+2){\frac {\partial }{\partial z_{i}}}\Psi =\sum _{i,j\neq i}{\frac {\Omega _{ij}}{z_{i}-z_{j}}}\Psi \ ,

where $\Omega _{ij}=\sum _{a}\rho _{i}(J^{a})\otimes \rho (J_{a})$ as the above informal derivation.

This n-point correlation function can be analytically continued as multi-valued holomorphic function to the domain $X_{n}\subset \mathbb {C} ^{n}$ with $z_{i}\neq z_{j},i\neq j$ . Due to this analytic continuation the holonomy of KZ equation can be described by the braid group $B_{n}$ introduced by Emil Artin.Kohno (2002) In general, A complex semi-simple Lie algebra ${\mathfrak {g}}$ and its representations $(\rho ,V_{i})$ give the linear representation of braid group

\theta :B_{n}\rightarrow V_{1}\otimes \cdots \otimes V_{n}

as the holonomy of KZ equation. Oppositely, a KZ equation gives the linear representation of braid groups as its holonomy.

The action on $V_{1}\otimes \dots \otimes V_{n}$ by the analytic continuation of KZ equation is called monodromy representation of KZ equation. In particular, if all $V_{i}$ '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

Representation theory of affine Lie algebra and quantum groups
Braid groups
Topology of hyperplane complements
Knot theory and 3-folds

References

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