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| In [[mathematical physics]] the '''Knizhnik–Zamolodchikov equations''' or '''KZ equations''' are a set of additional constraints satisfied by the [[Correlation function (quantum field theory)|correlation functions]] of the [[conformal field theory]] associated with an [[affine Lie algebra]] at a fixed level. They form a system of [[Complex differential equation|complex]] [[partial differential equation]]s with [[regular singular point]]s satisfied by the ''N''-point functions of [[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]] deduced the theory for [[SU(2)]] using the classical formulas of [[Carl Friedrich Gauss|Gauss]] for the connection coefficients of the [[hypergeometric differential equation]].
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| ==Definition==
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| Let <math>\hat{\mathfrak{g}}_k</math> denote the affine Lie algebra with level <math>k</math> and dual [[Coxeter number]] <math>h</math>. Let <math>v</math> 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 <math>\eta</math> the [[Killing form]]. Then for <math>i,j=1,2,\ldots,N</math> the '''Knizhnik–Zamolodchikov equations''' read
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| : <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) \langle \Phi(v_N,z_N)\dots\Phi(v_1,z_1) \rangle = 0. </math>
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| ==Informal derivation==
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| The Knizhnik–Zamolodchikov equations result from the existence of null vectors in the <math>\hat{\mathfrak{g}}_k</math> 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. | |
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| The null vectors of a <math>\hat{\mathfrak{g}}_k</math> module are of the form
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| : <math> \left( L_{-1} - \frac{1}{2(k+h)} \sum_{k \in \mathbf{Z}} \sum_{a,b} \eta_{ab} J^a_{-k}J^b_{k-1} \right)v = 0,</math>
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| where <math>v</math> is a highest weight vector and <math>J^a_k</math> the [[Noether's theorem|conserved current]] associated with the affine generator <math>t^a</math>. Since <math>v</math> is of highest weight, the action of most <math>J^a_k</math> on it vanish and only <math>J^a_{-1}J^b_{0}</math> remain. The operator-state correspondence then leads directly to the Knizhnik–Zamolodchikov equations as given above.
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| ==Mathematical formulation==
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| {{main|Vertex algebra}}
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| 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|1988}} and amongst mathematicians by {{harvtxt|Kac|1996}}.
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| The vacuum representation ''H''<sub>0</sub> of an [[affine Kac–Moody algebra]] at a fixed level can be encoded in a [[vertex algebra]].
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| The derivation ''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
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| :<math>V(a,0)\Omega = a.\,</math>
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| The vertex operators of energy 1 correspond to the generators of the affine algebra
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| :<math> X(z)=\sum X(n) z^{-n-1}</math>
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| where ''X'' ranges over the elements of the underlying finite-dimensional simple complex Lie algebra <math>\mathfrak{g}</math>.
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| There is an energy 2 eigenvector ''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''
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| :<math> T(z) = \sum L_n z^{-n-2}.</math>
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| If ''a'' has energy α, then the corresponding vertex operator has the form
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| :<math> V(a,z) = \sum V(a,n)z^{-n-\alpha}.</math>
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| The vertex operators satisfy
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| :<math>{d\over dz} V(a,z) = [L_{-1},V(a,z)]= V(L_{-1}a,z),\,\, [L_0,V(a,z)]=(z^{-1} {d\over dz} + \alpha)V(a,z)</math>
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| as well as the locality and associativity relations
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| :<math>V(a,z)V(b,w) = V(b,w) V(a,z) = V(V(a,z-w)b,w).\,</math>
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| 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''<sup>±1</sup>, ''w''<sup>±1</sup> and (''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.
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| 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 ''V''<sub>''i''</sub>(''a'',''z'') on ''H''<sub>''i''</sub> such that
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| :<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>
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| The most general vertex operators at a given level are [[intertwining operator]]s 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
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| :<math> \Phi(v,z)=\sum \Phi(v,n) z^{-n-\delta}\,</math>
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| but δ can now be [[rational number]]s. Again these intertwining operators are characterized by properties
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| :<math> V_j(a,z) \Phi(v,w)= \Phi(v,w) V_i(a,w) = \Phi(V_k(a,z-w)v,w)\,</math>
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| and relations with ''L''<sub>0</sub> and ''L''<sub>–1</sub> similar to those above. | |
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| When ''v'' is in the lowest energy subspace for ''L''<sub>0</sub> on ''H''<sub>''k''</sub>, an irreducible representation of
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| <math>\mathfrak{g}</math>, the operator Φ(''v'',''w'') is called a [[primary field]] of charge ''k''.
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| Given a chain of ''n'' primary fields starting and ending at ''H''<sub>0</sub>, their correlation or ''n''-point function is defined by
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| :<math> \langle \Phi(v_1,z_1) \Phi(v_2,z_2) \dots \Phi(v_n,z_n)\rangle = (\Phi(v_1,z_1) \Phi(v_2,z_2) \dots \Phi(v_n,z_n)\Omega,\Omega).</math>
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| 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>.
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| ===Vertex algebra derivation===
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| 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
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| :<math>\sum_s \langle X_s(w)X_s(z)\Phi(v_1,z_1) \cdots \Phi(v_n,z_n) \rangle (w-z)^{-1}</math> | |
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| 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:
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| :<math>{1\over 2}(k+h) \langle T(z)\Phi(v_1,z_1)\cdots \Phi(v_n,z_n) \rangle = - \sum_{j,s} \langle X_s(z)\Phi(v_1,z_1) \cdots \Phi(X_s v_j,z_j) \Phi(X_n,z_n)\rangle (z-z_j)^{-1}.</math>
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| 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.
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| ===Lie algebra derivation===
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| It is also possible to deduce the Knizhnik–Zamodchikov equations without explicit use of vertex algebras. The term 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 or ±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:
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| :<math>L_0 = (k+h)^{-1}\sum_s\left[ {1\over 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).</math>
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| After substituting these formulas for ''L''<sub>''r''</sub>, the resulting expressions can be simplified using the commutator formulas
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| : <math> [X(m),\Phi(a,n)]= \Phi(Xa,m+n).\,</math>
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| ===Original derivation===
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| 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>
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| :<math> \langle X(z)\Phi(v_1,z_1) \cdots \Phi(v_n,z_n) \rangle = \sum_j \langle \Phi(v_1,z_1) \cdots \Phi(Xv_j,z_j) \cdots \Phi(v_n,z_n) \rangle (z-z_j)^{-1}.</math>
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| Hence
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| :<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
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| = \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> | |
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| On the other hand
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| :<math>\sum_s X_s(z)\Phi(X_sv_i,z_i) = (z-z_i)^{-1}\Phi(\sum_s X_s^2v_i,z_i) + (k+g){\partial\over \partial z_i} \Phi(v_i,z_i) +O(z-z_i)</math>
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| so that
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| :<math>(k+g){\partial\over \partial z_i} \Phi(v_i,z_i) = \lim_{z\rightarrow z_i} \left[\sum_s X_s(z)\Phi(X_sv_i,z_i) -(z-z_i)^{-1}\Phi(\sum_s X_s^2 v_i,z_i)\right].</math> | |
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| The result follows by using this limit in the previous equality.
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| ==Applications==
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| *[[Representation theory]] of [[Affine Lie algebra]] and [[quantum groups]]
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| *[[Braid groups]]
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| *[[Topology]] of [[Arrangement of hyperplanes|hyperplane complements]]
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| *[[Knot theory]] and [[3-fold]]s
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| ==See also==
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| *[[Conformal field theory]]
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| *[[Correlation function (quantum field theory)|Correlation functions]]
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| *[[Quantum KZ equations]]
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| ==References==
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| <references/>
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| *{{cite journal|last=Baik|first=Jinho|coauthors=Deift, Percy, and Johansson, Kurt|title=On the distribution of the length of the longest increasing subsequence of random permutations|journal=J. Amer. Math. Soc.|year=1999|month=June|volume=12|issue=4|pages=1119–1178|url=http://www.ams.org/journals/bull/2000-37-02/S0273-0979-00-00853-3/S0273-0979-00-00853-3.pdf|accessdate=5 December 2012}}
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| * {{citation|first=V.G.|last= Knizhnik|authorlink=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|pages=83–103|doi=10.1016/0550-3213(84)90374-2}}
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| * {{citation|title=Vertex operators in conformal field theory on P(1) and monodromy representations of braid group|first=A. |last=Tsuchiya| first2=Y.|last2= Kanie|year= 1988|series=Adv. Stud. Pure Math.|volume=16|pages=297–372}} (Erratum in volume 19, pp. 675–682.)
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| * {{citation|first=Richard|last= Borcherds|authorlink=Richard Borcherds|title=Vertex algebras, Kac–Moody algebras, and the Monster|journal=Proc. Natl. Acad. Sci. USA.|volume=83| year=1986|pages=3068–3071|doi=10.1073/pnas.83.10.3068|pmid=16593694|pmc=323452}}
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| * {{citation|first1=Igor|last1= Frenkel|authorlink1=Igor Frenkel|first2=James|last2= Lepowsky|authorlink2=James Lepowsky|first3= Arne|last3= Meurman|title=Vertex operator algebras and the Monster|series=Pure and Applied Mathematics|volume= 134|publisher= Academic Press|year= 1988|isbn= 0-12-267065-5}}
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| *{{citation|first=Peter|last=Goddard|authorlink=Peter Goddard (physicist)|title=Meromorphic conformal field theory|series=Adv. Series in Mathematical Physics|volume=7|year=1989|publisher=World Scientific|pages=556–587|url=http://ccdb4fs.kek.jp/cgi-bin/img_index?198903335}}
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| * {{citation|first=Victor|last= Kac|authorlink=Victor Kac|title= Vertex algebras for beginners|series=University Lecture Series|volume= 10|publisher=American Mathematical Society|year= 1998|id= ISBN 0-8218-0643-2}}
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| *{{citation|title=Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations|first=Pavel I.|last= Etingof|first2= Igor|last2= Frenkel|authorlink2=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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| *{{citation|first= Edward|last= Frenkel|first2= David|last2= Ben-Zvi|title=Vertex algebras and Algebraic Curves|series= Mathematical Surveys and Monographs|volume= 88|publisher=American Mathematical Society|year= 2001|isbn= 0-8218-2894-0}}
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| {{DEFAULTSORT:Knizhnik-Zamolodchikov equations}}
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| [[Category:Lie algebras]]
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| [[Category:Conformal field theory]]
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