nLab hypergeometric construction of KZ solutions



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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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A large class of solutions of the Knizhnik-Zamolodchikov equation, hence a large class of braid group representations, arises as the holomorphic twisted de Rham cohomology (twisted by a “local system”) of configuration spaces of points in the punctured plane (Schechtman & Varchenko 89, 90, 91). This subsumes in particular the construction of conformal blocks (on the punctured Riemann sphere) for 2d CFTs of affine Lie algebra/WZW model-type (Feigin, Schechtman & Varchenko 90, 94, 95).

Since the canonical differential form-representatives in this construction may be regarded as generalized hypergeometric functions, the original authors referred to this approach as “hypergeometric solutions” to the KZ equation (Schechtman & Varchenko 90). In hindsight, it is not the theory of hypergeometric functions that drives the theory, but the peculiarities of (holomorphic) local systems and their twisted de Rham cohomology of configuration spaces of points.


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 May 11, 2022 at 18:38:08. See the history of this page for a list of all contributions to it.