Given a suitable Lie algebra a Knizhnik-Zamolodchikov equation is the equation expressing flatness of certain class of vector bundles with connection on Fadell's configuration space of distinct points in . It appeared in the study of Wess-Zumino-Novikov-Witten model (WZNW model) of 2d CFT in (Knizhnik-Zamolodchikov 84).
In the standard variant, its basic data involve a given complex simple Lie algebra with a fixed bilinear invariant polynomial (the Killing form) and (not necessarily finite-dimensional) representations of . Let . Consder the Fadell's configuration space of distinct points in and its subset .
The existence of the Knizhnik-Zamolodchikov connection can naturally be understood from the holographic quantization of the WZW model on the Lie group by geometric quantization of -Chern-Simons theory:
as discussed there, for a 2-dimensional manifold , a choice of polarization of the phase space of 3d Chern-Simons theory on is naturally induced by a choice of conformal structure on . Once such a choice is made, the resulting space of quantum states of the Chern-Simons theory over is naturally identified with the space of conformal blocks of the WZW model 2d CFT on the Riemann surface .
But since from the point of view of the 3d Chern-Simons theory the polarization is an arbitrary choice, the space of quantum states should not depend on this choice, up to specified equivalence. Formally this means that as varies (over the moduli space of conformal structures on ) the 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 of this vector bundle.
This flat connection is the Knizhnik-Zamolodchikov connection. This was maybe first realized and explained in (Witten 89).
The original articles are
A. Belavin , Alexander Polyakov , A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory (1984) Nucl. Phys. B 241 (2): 333–80.
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)
A review of the definition of the Knizhnik-Zamolodchikov connection on the moduli space of genus-0 surfaces with marked points is in section 2 of
wikipedia Knizhnik-Zamolodchikov equations
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