nLab Knizhnik-Zamolodchikov equation


under construction




Given a suitable Lie algebra 𝔤\mathfrak{g} a Knizhnik-Zamolodchikov equation is the equation expressing flatness of certain class of vector bundles with connection on the configuration space of points of NN distinct points in the plane \mathbb{C}. It appeared in the study of Wess-Zumino-Novikov-Witten model (WZNW model) of 2d CFT in (Knizhnik-Zamolodchikov 84).

The Knizhnik-Zamolodchikov equation involves what is called the Knizhnik-Zamolodchikov connection and it is related to monodromy representations of Artin’s braid group.

In the standard variant, the basic data involves a given complex simple Lie algebra 𝔤\mathfrak{g} with a fixed bilinear invariant polynomial (,)(-,-) (the Killing form) and NN (not necessarily finite-dimensional) representations V 1,,V nV_1,\ldots, V_n of 𝔤\mathfrak{g}. Let V=V 1V NV \,=\, V_1 \otimes \ldots \otimes V_N.


From geometric quantization of Chern-Simons theory

The existence of the Knizhnik-Zamolodchikov connection can naturally be understood from the holographic quantization of the WZW model on the Lie group GG by geometric quantization of GG-Chern-Simons theory:

As discussed there, for a 2-dimensional manifold Σ\Sigma, a choice of polarization of the phase space of 3d Chern-Simons theory on Σ\Sigma is naturally induced by a choice JJ of conformal structure on Σ\Sigma. Once such a choice is made, the resulting space of quantum states Σ (J)\mathcal{H}_\Sigma^{(J)} of the Chern-Simons theory over Σ\Sigma is naturally identified with the space of conformal blocks of the WZW model 2d CFT on the Riemann surface (Σ,J)(\Sigma, J).

But since, from the point of view of the 3d Chern-Simons theory, the polarization JJ is an arbitrary choice, the space of quantum states Σ (J)\mathcal{H}_\Sigma^{(J)} should not depend on this choice, up to specified equivalence. Formally this means that as JJ varies (over the moduli space of conformal structures on Σ\Sigma) the Σ (J)\mathcal{H}_{\Sigma}^{(J)} should form a vector bundle on this moduli space of conformal structures which is equipped with a flat connection whose parallel transport hence provides equivalences between between the fibers Σ (J)\mathcal{H}_{\Sigma}^{(J)} of this vector bundle.

This flat connection is the Knizhnik-Zamolodchikov connection. This was maybe first realized and explained in (Witten 89).


For the Definition of the Knizhnik-Zamolodchikov connection we need the following notation:

  1. configuration spaces of points

    For N fN_{\mathrm{f}} \in \mathbb{N} write

    (1)Conf {1,,N f}( 2)( 2) n\FatDiagonal \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{R}^2) \;\coloneqq\; (\mathbb{R}^2)^n \backslash FatDiagonal

    for the ordered configuration space of n points in the plane, regarded as a smooth manifold.

    Identifying the plane with the complex plane \mathbb{C}, we have canonical holomorphic coordinate functions

    (2)(z 1,,z N f):Conf {1,,n}( 2) N f. (z_1, \cdots, z_{N_{\mathrm{f}}}) \;\colon\; \underset{{}^{\{1,\cdots,n\}}}{Conf}(\mathbb{R}^2) \longrightarrow \mathbb{C}^{N_{\mathrm{f}}} \,.
  2. horizontal chord diagrams

    (3)𝒜 N f pbSpan(𝒟 N f pb)/4T \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} \;\coloneqq\; Span\big(\mathcal{D}^{{}^{pb}}_{N_{\mathrm{f}}}\big)/4T

    for the quotient vector space of the linear span of horizontal chord diagrams on nn strands by the 4T relations (infinitesimal braid relations), regarded as an associative algebra under concatenation of strands (here).


(Knizhnik-Zamolodchikov form)

The universal Knizhnik-Zamolodchikov form is the horizontal chord diagram-algebra valued differential form (3) on the configuration space of points (1)

(4)ω KZΩ(Conf {1,,N f}(),𝒜 N f pb) \omega_{KZ} \;\in\; \Omega \big( \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \,, \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} \big)

given in the canonical coordinates (2) by:

(5)ω KZi<j{1,,n}d dRlog(z iz j)t ij, \omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes t_{i j} \,,


is the horizontal chord diagram with exactly one chord, which stretches between the iith and the jjth strand.

Regarded as a connection form for a connection on a vector bundle, this defines the universal Knizhnik-Zamolodchikov connection KZ\nabla_{KZ}, with covariant derivative

ϕdϕ+ω KVϕ \nabla \phi \;\coloneqq\; d \phi + \omega_{KV} \wedge \phi

for any smooth function

ϕ:Conf {1,,N f}()𝒜 N f pbMod \phi \;\colon\; \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \longrightarrow \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} Mod

with values in modules over the algebra of horizontal chord diagrams modulo 4T relations.

The condition of covariant constancy

KZϕ=0 \nabla_{KZ} \phi \;=\; 0

is called the Knizhnik-Zamolodchikov equation.

Finally, given a metric Lie algebra 𝔤\mathfrak{g} and a tuple of Lie algebra representations

(V 1,,V N f)(𝔤Rep /) N f, ( V_1, \cdots, V_{N_{\mathrm{f}}} ) \;\in\; (\mathfrak{g} Rep_{/\sim})^{N_{\mathrm{f}}} \,,

the corresponding endomorphism-valued Lie algebra weight system

w V:𝒜 N f pfEnd 𝔤(V 1V N f) w_{V} \;\colon\; \mathcal{A}^{{}^{pf}}_{N_{\mathrm{f}}} \longrightarrow End_{\mathfrak{g}}\big( V_1 \otimes \cdots V_{N_{\mathrm{f}}} \big)

turns the universal Knizhnik-Zamolodchikov form (4) into a endomorphism ring-valued differential form

(6)ω KZi<j{1,,n}d dRlog(z iz j)w V(t ij)Ω(Conf {1,,N f}(),End(V 1V N f)). \omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes w_V(t_{i j}) \;\in\; \Omega \big( \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \,, End\big(V_1 \otimes \cdot V_{N_{\mathrm{f}}} \big) \big) \,.

The universal formulation (4) is highlighted for instance in Bat-Natan 95, Section 4.2, Lescop 00, p. 7. Most authors state the version after evaluation in a Lie algebra weight system, e.g. Kohno 14, Section 5.


(Knizhnik-Zamolodchikov connection is flat)

The Knizhnik-Zamolodchikov connection ω ZK\omega_{ZK} (Def. ) is flat:

dω ZK+ω ZKω ZK=0. d \omega_{ZK} + \omega_{ZK} \wedge \omega_{ZK} \;=\; 0 \,.

(Kontsevich integral for braids)

The Dyson formula for the holonomy of the Knizhnik-Zamolodchikov connection (Def. ) is called the Kontsevich integral on braids.

(e.g. Lescop 00, side-remark 1.14)



The original articles:

Textbook account, specifically for the 𝔰𝔲 ( 2 ) \mathfrak{su}(2) / 𝔰𝔩 ( 2 , ) \mathfrak{sl}(2,\mathbb{Z}) -WZW model:

Relation to braid representations:

and understood in terms of anyon statistics:

The generalization to higher genus surfaces:

  • D. Bernard, On the Wess-Zumino-Witten models on the torus, Nucl. Phys. B 303 77-93 (1988)

  • D. Bernard, On the Wess-Zumino-Witten models on Riemann surfaces, Nucl. Phys. B 309 145-174 (1988)

The interpretation of this connection in terms of the geometric quantization of Chern-Simons theory/Wess-Zumino-Witten model:


A quick review of the Knizhnik-Zamolodchikov equation in the context of an introduction to WZW model CFT is in

A review of the definition of the Knizhnik-Zamolodchikov connection on the moduli space of genus=0 surfaces with nn marked points:

  • Shu Oi, Kimio Ueno, section 2 of: Connection Problem of Knizhnik-Zamolodchikov Equation on Moduli Space 0,5\mathcal{M}_{0,5} (arXiv:1109.0715)

In relation to hypergeometric functions and quantum groups (for more see the references below):

On KZ-equations controlling codimension=2=2 defects in D=4 super Yang-Mills theory:

See also

  • Wikipedia, Knizhnik-Zamolodchikov equations

  • I. B. Frenkel, N. Yu. Reshetikihin, Quantum affine algebras and holonomic diference equations, Comm. Math. Phys. 146 (1992), 1-60, MR94c:17024

  • Valerio Toledano-Laredo, Flat connections and quantum groups, Acta Appl. Math. 73 (2002), 155-173, math.QA/0205185

  • Toshitake Kohno, Conformal field theory and topology, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs 210. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002 [[AMS:mmono-210]]

  • P. Etingof, N. Geer, Monodromy of trigonometric KZ equations, math.QA/0611003

  • Valerio Toledano-Laredo, A Kohno-Drinfeld theorem for quantum Weyl groups, math.QA/0009181

  • A. Tsuchiya, Y. Kanie, Vertex operators in conformal field theory on P 1\mathbf{P}^1 and monodromy representations of braid group, Adv. Stud. Pure Math. 16, pp. 297–372 (1988); Erratum in vol. 19, 675–682

  • C. Kassel, Quantum groups, Grad. Texts in Math. 155, Springer 1995

  • V. Chari, , A. Pressley, A guide to quantum groups, Camb. Univ. Press 1994В.

  • А. Голубева, В. П. Лексин, Алгебраическая характеризация монодромии обобщенных уравнений Книжника–Замолодчикова типа B nB_n, Монодромия в задачах алгебраической геометрии и дифференциальных уравнений, Сборник статей, Тр. МИАН, 238, Наука, М., 2002, 124–143, pdf; V. A. Golubeva, V. P. Leksin, “Algebraic Characterization of the Monodromy of Generalized Knizhnik–Zamolodchikov Equations of Bn Type”, Proc. Steklov Inst. Math., 238 (2002), 115–133

  • V. A. Golubeva, V. P. Leksin, Rigidity theorems for multiparametric deformations of algebraic structures, associated with the Knizhnik-Zamolodchikov equations, Journal of Dynamical and Control Systems, 13:2 (2007), 161–171, MR2317452

  • V. A. Golubeva, Integrability conditions for two–parameter Knizhnik–Zamolodchikov equations of type B nB_n in the tensor and spinor cases, Doklady Mathematics, 79:2 (2009), 147–149

  • V. G. Drinfelʹd, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, Problems of modern quantum field theory (Alushta, 1989), 1–13, Res. Rep. Phys., Springer 1989.

  • R. Rimányi, V. Tarasov, A. Varchenko, P. Zinn-Justin, Extended Joseph polynomials, quantized conformal blocks, and a qq-Selberg type integral, arxiv/1110.2187

  • E. Mukhin, V. Tarasov, A. Varchenko, KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians, arxiv/1201.3990

On the flat vector bundles underlying the KZ-equation:

Braid representations via twisted cohomology of configuration spaces

The “hypergeometric integral” construction of conformal blocks for affine Lie algebra/WZW model-2d CFTs and of more general solutions to the Knizhnik-Zamolodchikov equation, via twisted de Rham cohomology of configuration spaces of points, originates with:

following precursor observations due to:

The proof that for rational levels this construction indeed yields conformal blocks is due to:


See also:

This “hypergeometric” construction uses results on the twisted de Rham cohomology of configuration spaces of points due to:


reviewed in:

  • Yukihito Kawahara, The twisted de Rham cohomology for basic constructions of hyperplane arrangements and its applications, Hokkaido Math. J. 34 2 (2005) 489-505 [[doi:10.14492/hokmj/1285766233]]

Discussion for the special case of level=0=0 (cf. at logarithmic CFT – Examples):

Interpretation of the hypergeometric construction as happening in twisted equivariant differential K-theory, showing that the K-theory classification of D-brane charge and the K-theory classification of topological phases of matter both reflect braid group representations as expected for defect branes and for anyons/topological order, respectively:

Last revised on March 1, 2024 at 10:13:27. See the history of this page for a list of all contributions to it.