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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Knizhnik-Zamolodchikov equation} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{geometric_quantization}{}\paragraph*{{Geometric quantization}}\label{geometric_quantization} [[!include geometric quantization - contents]] \begin{quote}% under construction \end{quote} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{FromGeometicQuantization}{From geometric quantization of Chern-Simons theory}\dotfill \pageref*{FromGeometicQuantization} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Given a suitable [[Lie algebra]] $\mathfrak{g}$ a \textbf{Knizhnik-Zamolodchikov equation} is the [[equation]] expressing [[flat connection|flatness]] of certain class of [[vector bundles]] [[connection on a bundle|with connection]] on [[Fadell's configuration space]] of $N$ distinct points in $\mathbf{C}^N$. It appeared in the study of [[Wess-Zumino-Novikov-Witten model]] (WZNW model) of [[2d CFT]] in (\hyperlink{KnizhnikZamolodchikov84}{Knizhnik-Zamolodchikov 84}). The Knizhnik-Zamolodchikov equation involves what is called the \textbf{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 \emph{[[Killing form]]}) and $N$ (not necessarily finite-dimensional) [[representations]] $V_1,\ldots, V_n$ of $\mathfrak{g}$. Let $V = V_1\otimes \ldots\otimes V_N$. Consder the [[Fadell's configuration space]] $Conf_N(\mathbf{C}P^1)$ of $N$ distinct points in $\mathbf{C}P^1$ and its subset $Conf_N(\mathbf{C})$. (\ldots{}) \hypertarget{FromGeometicQuantization}{}\subsubsection*{{From geometric quantization of Chern-Simons theory}}\label{FromGeometicQuantization} The existence of the Knizhnik-Zamolodchikov connection can naturally be understood from the [[holographic principle|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 (\hyperlink{Witten89}{Witten 89}). \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[Kohno-Drinfeld theorem]] \item [[Hitchin connection]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The original articles are \begin{itemize}% \item [[Vadim Knizhnik]], [[Alexander Zamolodchikov]], \emph{Current algebra and Wess--Zumino model in two-dimensions}, Nucl. Phys. \textbf{B247}, 83--103 (1984) , \href{http://www.ams.org/mathscinet-getitem?mr=87h:81129}{MR87h:81129} \end{itemize} \begin{itemize}% \item A. Belavin , [[Alexander Polyakov]] , [[Alexander Zamolodchikov]], \emph{Infinite conformal symmetry in two-dimensional quantum field theory} (1984) Nucl. Phys. B 241 (2): 333--80. \item [[Daniel Friedan]], S. Shenker, \emph{The analytic geometry of two-dimensional conformal field theory}, Nuclear Physics B281 (1987) (\href{http://www.physics.rutgers.edu/~friedan/papers/Nucl_Phys_B281_509_1987.pdf}{pdf}) \end{itemize} The interpretation of this structure in terms of a [[flat connection]] on the [[moduli space of conformal structures]] was given in \begin{itemize}% \item [[Graeme Segal]], \emph{Conformal field theory}, Oxford preprint and lecture at the IAMP Congress, Swansea July 1988. \end{itemize} The generalization to higher [[genus]] surfaces is due to \begin{itemize}% \item D. Bernard, \emph{On the Wess-Zumino-Witten models on the torus}, Nucl. Phys. B 303 77-93 (1988) \item D. Bernard, \emph{On the Wess-Zumino-Witten models on Riemann surfaces, Nucl. Phys. B 309 145-174 (1988)} \end{itemize} Finally the interpreation of this connection in terms of the [[geometric quantization]] of [[Chern-Simons theory]] is due to the discussion on p. 20 of \begin{itemize}% \item [[Edward Witten]] \emph{Quantum Field Theory and the Jones Polynomial} Commun. Math. Phys. 121 (3) (1989) 351--399. MR0990772 (\href{http://projecteuclid.org/euclid.cmp/1104178138}{EUCLID}) \end{itemize} A quick review of the Knizhnik-Zamolodchikov equation in the context of an introduction to [[WZW model]] [[CFT]] is in section 5.6 of \begin{itemize}% \item [[Krzysztof GawÄ™dzki]], \emph{Conformal field theory: a case study} (\href{http://arxiv.org/abs/hep-th/9904145}{arXiv:hep-th/9904145}) \end{itemize} 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 \begin{itemize}% \item Shu Oi, Kimio Ueno, \emph{Connection Problem of Knizhnik-Zamolodchikov Equation on Moduli Space $\mathcal{M}_{0,5}$} (\href{http://arxiv.org/abs/1109.0715}{arXiv:1109.0715}) \end{itemize} See also \begin{itemize}% \item wikipedia \href{http://en.wikipedia.org/wiki/Knizhnik–Zamolodchikov_equations}{Knizhnik-Zamolodchikov equations} \item Philippe Di Francesco,Pierre Mathieu,David S\'e{}n\'e{}chal, \emph{Conformal field theory}, Springer 1997 \item [[P. Etingof]], I. Frenkel, \emph{Lectures on representation theory and Knizhnik-Zamolodchikov equations}, book; V. Chari, review in Bull. AMS: \href{http://www.ams.org/journals/bull/2000-37-02/S0273-0979-00-00853-3/S0273-0979-00-00853-3.pdf}{pdf} \item I. B. Frenkel, N. Yu. Reshetikihin, \emph{Quantum affine algebras and holonomic diference equations}, Comm. Math. Phys. \textbf{146} (1992), 1-60, \href{http://www.ams.org/mathscinet-getitem?mr=94c:17024}{MR94c:17024} \item [[Valerio Toledano-Laredo]], \emph{Flat connections and quantum groups}, Acta Appl. Math. 73 (2002), 155-173, \href{http://arxiv.org/abs/math.QA/0205185}{math.QA/0205185} \item [[Toshitake Kohno]], \emph{Conformal field theory and topology}, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs \textbf{210}. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002. x+172 pp. \item [[P. Etingof]], N. Geer, \emph{Monodromy of trigonometric KZ equations}, \href{http://arxiv.org/abs/math.QA/0611003}{math.QA/0611003} \item [[Valerio Toledano-Laredo]], \emph{A Kohno-Drinfeld theorem for quantum Weyl groups}, \href{http://arxiv.org/abs/math.QA/0009181}{math.QA/0009181} \item A. Tsuchiya, Y. Kanie, \emph{Vertex operators in conformal field theory on $\mathbf{P}^1$ and monodromy representations of braid group}, Adv. Stud. Pure Math. \textbf{16}, pp. 297--372 (1988); Erratum in vol. 19, 675--682 \item C. Kassel, \emph{Quantum groups}, Grad. Texts in Math. \textbf{155}, Springer 1995 \item V. Chari, , A. Pressley, \emph{A guide to quantum groups}, Camb. Univ. Press 1994. \item . , . . , \emph{ -- $B_n$}, , , . , 238, , ., 2002, 124--143, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=tm&paperid=349&what=fullt&option_lang=rus}{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 \item V. A. Golubeva, V. P. Leksin, \emph{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, \href{http://www.ams.org/mathscinet-getitem?mr=2317452}{MR2317452} \item V. A. Golubeva, \emph{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 \item V. G. Drinfeld, \emph{Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations}, Problems of modern quantum field theory (Alushta, 1989), 1--13, Res. Rep. Phys., Springer 1989. \item R. Rim\'a{}nyi, V. Tarasov, A. Varchenko, P. Zinn-Justin, \emph{Extended Joseph polynomials, quantized conformal blocks, and a $q$-Selberg type integral}, \href{http://arxiv.org/abs/1110.2187}{arxiv/1110.2187} \item E. Mukhin, V. Tarasov, A. Varchenko, \emph{KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians}, \href{http://arxiv.org/abs/1201.3990}{arxiv/1201.3990} \end{itemize} [[!redirects Knizhnik-Zamolodchikov connection]] [[!redirects KZ equation]] [[!redirects KZ connection]] \end{document}