View source for Knizhnik–Zamolodchikov equations

{{Short description|Partial differential equations of correlation functions}}
In [[mathematical physics]] the '''Knizhnik–Zamolodchikov equations''', or '''KZ equations''', are linear differential equations satisfied by the [[Correlation function (quantum field theory)|correlation functions]] (on the Riemann sphere) of [[two-dimensional conformal field theory|two-dimensional conformal field theories]] associated with an [[affine Lie algebra]] at a fixed level. They form a system of complex [[partial differential equation]]s with [[regular singular point]]s satisfied by the ''N''-point functions of [[Wess–Zumino–Witten_model#Fields_and_correlation_functions|affine primary field]]s and can be derived using either the formalism of [[Lie algebra]]s or that of [[vertex algebra]]s. 

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 [[Carl Friedrich Gauss|Gauss]] for the [[connection coefficients]] of the [[hypergeometric differential equation]].

==Definition==
Let <math>\hat{\mathfrak{g}}_k</math> denote the affine Lie algebra with level {{mvar|k}} and dual [[Coxeter number]] {{mvar|h}}. Let {{mvar|v}} be a vector from a zero mode representation of <math>\hat{\mathfrak{g}}_k</math> and <math>\Phi(v,z)</math> the primary field associated with it. Let <math>t^a</math> be a basis of the underlying [[Lie algebra]] <math>\mathfrak{g}</math>, <math>t^a_i</math> their representation on the primary field <math>\Phi(v_i,z)</math> and {{mvar|η}} the [[Killing form]]. Then for <math>i,j=1,2,\ldots,N</math> the '''Knizhnik–Zamolodchikov equations''' read

: <math>\left( (k+h)\partial_{z_i} + \sum_{j \neq i} \frac{\sum_{a,b} \eta_{ab} t^a_i \otimes t^b_j}{z_i-z_j} \right) \left \langle \Phi(v_N,z_N)\dots\Phi(v_1,z_1) \right \rangle = 0. </math>

==Informal derivation==
The Knizhnik–Zamolodchikov equations result from 
the [[Wess%E2%80%93Zumino%E2%80%93Witten_model#Sugawara_construction|Sugawara construction]] of the Virasoro algebra from the affine Lie algebra. More specifically, they result from applying the identity 

: <math> L_{-1} = \frac{1}{2(k+h)} \sum_{k \in \mathbf{Z}} \sum_{a, b}  \eta_{ab}J^a_{-k}J^b_{k-1} </math>

to the affine primary field <math>\Phi(v_i,z_i)</math> in a correlation function of affine primary fields. In this context, only the terms <math>k=0,1</math> are non-vanishing. The action of <math> J^a_{-1}</math> can then be rewritten using global [[Ward identities]], 

:<math> 
\left( \left(J^a_{-1}\right)_i 
+ \sum_{j\neq i} \frac{t^a_j}{z_i-z_j}
\right) 
\left\langle \Phi(v_N,z_N)\dots \Phi(v_1,z_1)\right\rangle = 0, 
</math>

and <math>L_{-1}</math> can be identified with the infinitesimal translation operator <math>\frac{\partial}{\partial z}</math>.

==Mathematical formulation==
{{main article|Vertex algebra}}

Since the treatment in {{harvtxt|Tsuchiya|Kanie|1988}}, the Knizhnik–Zamolodchikov equation has been formulated mathematically in the language of [[vertex algebra]]s due to {{harvtxt|Borcherds|1986}} and {{harvtxt|Frenkel|Lepowsky|Meurman|1988}}. This approach was popularized amongst theoretical physicists by {{harvtxt|Goddard|1989}} and amongst mathematicians by {{harvtxt|Kac|1997}}.

The vacuum representation ''H''<sub>0</sub> of an [[affine Kac–Moody algebra]] at a fixed level can be encoded in a [[vertex algebra]].
The derivation {{mvar|d}} acts as the energy operator ''L''<sub>0</sub> on ''H''<sub>0</sub>, which can be written as a direct sum of the non-negative integer eigenspaces of ''L''<sub>0</sub>, the zero energy space being generated by the vacuum vector  Ω. The eigenvalue of an eigenvector of ''L''<sub>0</sub> 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

:<math>V(a,0)\Omega = a.</math>

The vertex operators of energy 1 correspond to the generators of the affine algebra

:<math> X(z)=\sum X(n) z^{-n-1}</math>

where ''X'' ranges over the elements of the underlying finite-dimensional simple complex Lie algebra <math>\mathfrak{g}</math>.

There is an energy 2 eigenvector {{math|''L''<sub>−2</sub>Ω}} which give the generators ''L''<sub>''n''</sub> of the [[Virasoro algebra]] associated to the Kac–Moody algebra by the ''Segal–Sugawara construction''

:<math> T(z) = \sum L_n z^{-n-2}.</math>

If ''a'' has energy {{mvar|α}}, then the corresponding vertex operator has the form

:<math> V(a,z) = \sum V(a,n)z^{-n-\alpha}.</math>

The vertex operators satisfy

:<math>\begin{align}
\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{align}</math>

as well as the locality and associativity relations

:<math>V(a,z)V(b,w) = V(b,w) V(a,z) = V(V(a,z-w)b,w).</math>

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 {{math|''z''<sup>±1</sup>, ''w''<sup>±1</sup>}} and {{math|(''z'' − ''w'')<sup>−1</sup>}} 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<sub>i</sub>'' at the same level becomes a module for the vertex algebra, in the sense that for each ''a'' there is a vertex operator {{math|''V<sub>i</sub>''(''a'', ''z'')}} on ''H<sub>i</sub>'' such that

:<math>V_i(a,z)V_i(b,w) = V_i(b,w) V_i(a,z)=V_i(V(a,z-w)b,w).</math>

The most general vertex operators at a given level are [[intertwining operator]]s {{math|Φ(''v'', ''z'')}} between representations ''H<sub>i</sub>'' and ''H''<sub>''j''</sub> where ''v'' lies in ''H''<sub>''k''</sub>. These operators can also be written as

:<math> \Phi(v,z)=\sum \Phi(v,n) z^{-n-\delta}</math>

but δ can now be [[rational number]]s. Again these intertwining operators are characterized by properties

:<math> V_j(a,z) \Phi(v,w)= \Phi(v,w) V_i(a,w) = \Phi \left (V_k(a,z-w)v,w \right)</math>

and relations with ''L''<sub>0</sub> and ''L''<sub>−1</sub> similar to those above.

When ''v'' is in the lowest energy subspace for ''L''<sub>0</sub> on ''H<sub>k</sub>'', an irreducible representation of <math>\mathfrak{g}</math>, the operator {{math|Φ(''v'', ''w'')}} is called a [[primary field]] of charge ''k''.

Given a chain of ''n'' primary fields starting and ending at ''H''<sub>0</sub>, their correlation or ''n''-point function is defined by

:<math> \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 ).</math>

In the physics literature the ''v''<sub>''i''</sub> are often suppressed and the primary field written Φ<sub>''i''</sub>(''z<sub>i</sub>''), with the understanding that it is labelled by the corresponding irreducible representation of <math>\mathfrak{g}</math>.

===Vertex algebra derivation===
If (''X''<sub>''s''</sub>) is an orthonormal basis of <math>\mathfrak{g}</math> for the Killing form, the Knizhnik–Zamolodchikov equations may be deduced by integrating the correlation function

:<math>\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}</math>

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''<sub>''j''</sub>'s:

:<math>{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) \cdots \Phi(v_n,z_n) \right \rangle (z-z_j)^{-1}.</math>

Integrating both sides in the ''z'' variable about a small circle centred on ''z<sub>i</sub>'' yields the ''i''<sup>th</sup> Knizhnik–Zamolodchikov equation.

===Lie algebra derivation===
It is also possible to deduce the Knizhnik–Zamodchikov equations without explicit use of vertex algebras. The term{{math|Φ(''v<sub>i</sub>'', ''z<sub>i</sub>'')}} may be replaced in the correlation function by its commutator with ''L<sub>r</sub>'' where ''r'' = 0, ±1. The result can be expressed in terms of the derivative with respect to ''z<sub>i</sub>''. On the other hand, ''L<sub>r</sub>'' is also given by the Segal–Sugawara formula:

:<math>\begin{align}
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\ge 0} X_s(-m\pm 1)X_s(m)
\end{align}</math>

After substituting these formulas for ''L<sub>r</sub>'', the resulting expressions can be simplified using the commutator formulas

: <math> [X(m),\Phi(a,n)]= \Phi(Xa,m+n).</math>

===Original derivation===
The original proof of {{harvtxt|Knizhnik|Zamolodchikov|1984}}, reproduced in {{harvtxt|Tsuchiya|Kanie|1988}}, uses a combination of both of the above methods. First note that for ''X'' in <math>\mathfrak{g}</math>

:<math> \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}.</math>

Hence

:<math> \sum_s \langle X_s(z)\Phi(z_1,v_1) \cdots \Phi(X_sv_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}.</math>

On the other hand,

:<math>\sum_s X_s(z)\Phi \left (X_sv_i,z_i \right ) = (z-z_i)^{-1} \Phi \left (\sum_s X_s^2v_i,z_i \right ) + (k+g){\partial\over \partial z_i} \Phi(v_i,z_i) +O(z-z_i)</math>

so that

:<math>(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_sv_i,z_i \right ) -(z-z_i)^{-1}\Phi \left (\sum_s X_s^2 v_i,z_i \right )\right].</math>

The result follows by using this limit in the previous equality.

== Monodromy representation of KZ equation ==
In [[two-dimensional conformal field theory|conformal field theory]] along the [[#definition|above definition]] the ''n''-point correlation function of the primary field satisfies KZ equation. In particular, for <math>\mathfrak{sl}_2</math> and non negative integers ''k'' there are <math>k + 1</math> primary fields <math>\Phi_j(z_j)</math> 's corresponding to [[spin representation|spin <sub>j</sub> representation]] (<math>j = 0, 1/2, 1, 3/2,\ldots , k/2</math>). The correlation function <math>\Psi(z_1,\dots,z_n)</math> of the primary fields <math>\Phi_j(z_j)</math> 's for the representation <math>(\rho,V_i)</math> takes values in the tensor product <math>V_1\otimes\cdots\otimes V_n</math> and its KZ equation is
:<math> (k+2)\frac{\partial}{\partial z_i}\Psi=\sum_{i,j\ne i}\frac{\Omega_{ij}}{z_i-z_j}\Psi</math>,
where <math>\Omega_{ij}=\sum_a\rho_i(J^a)\otimes\rho_j(J_a)</math> as the above [[#informal derivation|informal derivation]].

This ''n''-point correlation function can be analytically continued as multi-valued holomorphic function to the domain <math>X_n \subset \Complex^n</math> with <math>z_i\ne z_j</math> for <math>i\ne j</math>. Due to this analytic continuation, the [[holonomy]] of the KZ equation can be described by the [[braid group]] <math>B_n</math> introduced by [[Emil Artin]].<ref>{{harvnb|Kohno|2002}}</ref> In general, A complex semi-simple Lie algebra <math>\mathfrak{g}</math> and its representations <math>(\rho,V_i)</math> give the [[linear representation]] of braid group
:<math>\theta \colon B_n \rightarrow V_1\otimes\cdots\otimes V_n</math>
as the holonomy of KZ equation. Oppositely, a KZ equation gives the linear representation of braid groups as its holonomy.

The action on <math>V_1\otimes\dots\otimes V_n</math> by the analytic continuation of KZ equation is called '''monodromy representation of KZ equation'''. In particular, if all <math>V_i</math> 's have spin <sub>1/2</sub> representation then the linear representation obtained from KZ equation agrees with the representation constructed from [[operator algebra|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 <math>\mathfrak{g}=\mathfrak{sl}(2)</math>, the KZ equations are mapped to [[Two-dimensional_conformal_field_theory#BPZ_equations|BPZ equations]] by [[Evgeny Sklyanin|Sklyanin]]'s separation of variables for the <math>\mathfrak{sl}(2)</math> [[Gaudin model]].<ref name="Ribault 2014 h431">{{cite arXiv | last=Ribault | first=Sylvain | title=Conformal field theory on the plane | date=2014-06-17 | class=hep-th | eprint=1406.4290 }}</ref>

==Applications==
*[[Representation theory]] of [[affine Lie algebra]] and [[quantum groups]]
*[[Braid groups]]
*[[Topology]] of [[Arrangement of hyperplanes|hyperplane complements]]
*[[Knot theory]] and [[3-fold]]s

==See also==
*[[Quantum KZ equations]]

==References==
<references/>
{{refbegin}}
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* {{citation |first1=V.G. |last1=Knizhnik |author-link=Vadim Knizhnik |first2=A.B. |last2=Zamolodchikov |title=Current Algebra and Wess–Zumino Model in Two-Dimensions |year=1984 |journal=Nucl. Phys. B |volume=247 |issue=1 |pages=83–103 |doi=10.1016/0550-3213(84)90374-2 |bibcode=1984NuPhB.247...83K}}
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*{{citation |first=Peter |last=Goddard |author-link=Peter Goddard (physicist) |chapter=Meromorphic conformal field theory |editor1-first=Victor G. |editor1-last=Kac |title=Infinite Dimensional Lie Algebras And Groups |volume=7 |year=1989 |publisher=World Scientific |pages=556–587 |chapter-url=https://books.google.com/books?id=uo0GCwAAQBAJ&pg=PA557 |isbn=978-981-4663-17-5 |series=Advanced Series In Mathematical Physics}}
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* {{citation |title=Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations |first1=Pavel I. |last1=Etingof |authorlink1=Pavel Etingof |first2=Igor |last2=Frenkel |author-link2=Igor Frenkel |first3=Alexander A. |last3=Kirillov |publisher=American Mathematical Society |year=1998 |series=Mathematical Surveys and Monographs |volume=58 |isbn=0821804960}}
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{{refend}}

{{DEFAULTSORT:Knizhnik-Zamolodchikov equations}}
[[Category:Lie algebras]]
[[Category:Conformal field theory]]