nLab
Knizhnik-Zamolodchikov equation

UNDER CONSTRUCTION

Idea

Given a Lie algebra $𝔤$ a Knizhnik-Zamolodchikov equation is the equation expressing flatness of certain class of vector bundles on Fadell's configuration space of $N$ distinct points in $C^N$. It appeared in the study of Wess-Zumino-Novikov-Witten (WZNW) model of 2d CFT in

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

It involves so called 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 $𝔤$ with a fixed invariant bilinear form $(,)$ (cf. Killing form) and $N$ (not necessarily finite-dimensional) representations $V_1,\dots ,V_n$ of $𝔤$. Let $V=V_1\otimes \dots \otimes V_N$. Consder the Fadell's configuration space ${\mathrm{Conf}}_N(C{P}^1)$ of $N$ distinct points in $C{P}^1$ and its subset ${\mathrm{Conf}}_N(C)$.

Literature

Related entries include Kohno-Drinfeld theorem.

• wikipedia Knizhnik-Zamolodhcikov 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 $P^1$ 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