Knizhnik–Zamolodchikov equations

In mathematical physics the Knizhnik–Zamolodchikov equations or KZ equations are a 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 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. In mathematical physics the Knizhnik–Zamolodchikov equations or KZ equations are a 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 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. Let g ^ k {displaystyle {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 g ^ k {displaystyle {hat {mathfrak {g}}}_{k}} and Φ ( v , z ) {displaystyle Phi (v,z)} the primary field associated with it. Let t a {displaystyle t^{a}} be a basis of the underlying Lie algebra g {displaystyle {mathfrak {g}}} , t i a {displaystyle t_{i}^{a}} their representation on the primary field Φ ( v i , z ) {displaystyle Phi (v_{i},z)} and η the Killing form. Then for i , j = 1 , 2 , … , N {displaystyle i,j=1,2,ldots ,N} the Knizhnik–Zamolodchikov equations read The Knizhnik–Zamolodchikov equations result from the existence of null vectors in the g ^ k {displaystyle {hat {mathfrak {g}}}_{k}} module. This is quite similar to the case in minimal models, where the existence of null vectors result in BPZ differential equations for correlation functions. However, in CFTs based on the Virasoro algebra, generic correlation functions and conformal blocks do not involve null vectors and do not obey nontrivial differential equations. The null vectors of a g ^ k {displaystyle {hat {mathfrak {g}}}_{k}} module are of the form where v is a highest weight vector and J k a {displaystyle J_{k}^{a}} the conserved current associated with the affine generator t a {displaystyle t^{a}} . Since v is of highest weight, the action of most J k a {displaystyle J_{k}^{a}} on it vanish and only J − 1 a J 0 b {displaystyle J_{-1}^{a}J_{0}^{b}} remain. The operator-state correspondence then leads directly to the Knizhnik–Zamolodchikov equations as given above. 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