nLab
Knizhnik-Zamolodchikov equation

under construction

Idea

General

Given a suitable Lie algebra $\mathfrak{g}$ a Knizhnik-Zamolodchikov equation is the equation expressing flatness of certain class of vector bundles with connection on the configuration space of points of $N$ 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 the Artin’s braid group.

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

(…)

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 $G$ by geometric quantization of $G$-Chern-Simons theory:

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

But since from the point of view of the 3d Chern-Simons theory the polarization $J$ is an arbitrary choice, the space of quantum states $\mathcal{H}_\Sigma^{(J)}$ should not depend on this choice, up to specified equivalence. Formally this means that as $J$ varies (over the moduli space of conformal structures on $\Sigma$) the $\mathcal{H}_{\Sigma}^{(J)}$ 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 $\mathcal{H}_{\Sigma}^{(J)}$ of this vector bundle.

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

Definition

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

1. configuration spaces of points

   For $N_{\mathrm{f}} \in \mathbb{N}$ write

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

   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)$$(z_1, \cdots, z_{N_{\mathrm{f}}}) \;\colon\; \underset{{}^{\{1,\cdots,n\}}}{Conf}(\mathbb{R}^2) \longrightarrow \mathbb{C}^{N_{\mathrm{f}}} \,.$$

2. horizontal chord diagrams

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

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

Definition

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

given in the canonical coordinates (2) by:

(5)$$\omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes t_{i j} \,,$$

where

is the horizontal chord diagram with exactly one chord, which stretches between the $i$th and the $j$th strand.

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

$$\nabla \phi \;\coloneqq\; d \phi + \omega_{KV} \wedge \phi$$

for any smooth function

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

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

The condition of covariant constancy

$$\nabla_{KZ} \phi \;=\; 0$$

is called the Knizhnik-Zamolodchikov equation.

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

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

the corresponding endomorphism-valued Lie algebra weight system

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

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

(6)$$\omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes w_V(t_{i j}) \;\in\; \Omega \big( \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \,, End\big(V_1 \otimes \cdot V_{N_{\mathrm{f}}} \big) \big) \,.$$

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.

Proposition

(Knizhnik-Zamolodchikov connection is flat)

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

$$d \omega_{ZK} + \omega_{ZK} \wedge \omega_{ZK} \;=\; 0 \,.$$

Proposition

(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)

Related entries

• Kohno-Drinfeld theorem

• Hitchin connection

References

The original articles are

• Vadim Knizhnik, Alexander Zamolodchikov, Current algebra and Wess–Zumino model in two-dimensions, Nucl. Phys. B247, 83–103 (1984) doi, MR87h:81129

• A. Belavin , Alexander Polyakov , Alexander Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory (1984) Nucl. Phys. B 241 (2): 333–80.

• Daniel Friedan, S. Shenker, The analytic geometry of two-dimensional conformal field theory, Nuclear Physics B281 (1987) (pdf)

The interpretation of this structure in terms of a flat connection on the moduli space of conformal structures was given in

• Graeme Segal, Conformal field theory, Oxford preprint and lecture at the IAMP Congress, Swansea July 1988.

The generalization to higher genus surfaces is due to

• 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)

Finally the interpretation of this connection in terms of the geometric quantization of Chern-Simons theory is due to the discussion on p. 20 of

• Edward Witten, Quantum Field Theory and the Jones Polynomial Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 (euclid.cmp/1104178138)

A quick review of the Knizhnik-Zamolodchikov equation in the context of an introduction to WZW model CFT is in section 5.6 of

• Krzysztof Gawędzki, Conformal field theory: a case study (arXiv:hep-th/9904145)

A review of the definition of the Knizhnik-Zamolodchikov connection on the moduli space of genus-0 surfaces with $n$ marked points is in section 2 of

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

In relation to hypergeometric functions and quantum groups:

• Alexander Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Adv. Ser. in Math. Phys. 21, World Sci. Publ. 1995. x+371 pp. (doi:10.1142/2467)

• V. Tarasov, Alexander Varchenko, Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), vi+135 pp. (arXiv:q-alg/9703044, numdam:AST_1997__246__R1_0)

See also

• wikipedia Knizhnik-Zamolodchikov equations

• Philippe Di Francesco,Pierre Mathieu,David Sénéchal, Conformal field theory, Springer 1997

• P. Etingof, I. Frenkel, Lectures on representation theory and Knizhnik-Zamolodchikov equations, book; V. Chari, review in Bull. AMS: pdf

• 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. x+172 pp.

• 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 $\mathbf{P}^1$ 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 $B_n$, 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 $B_n$ 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 $q$-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