# nLab hypergeometric construction of KZ solutions

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

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:

• Vadim Schechtman, Alexander Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$cds:1044951$]$

• Etsuro Date, Michio Jimbo, Atsushi Matsuo, Tetsuji Miwa, Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation, International Journal of Modern Physics B 04 05 (1990) 1049-1057 $[$doi:10.1142/S0217979290000528$]$

• Atsushi Matsuo, An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation, Communications in Mathematical Physics 134 (1990) 65–77 $[$doi:10.1007/BF02102089$]$

• Vadim Schechtman, Alexander Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 $[$doi:10.1007/BF00626523$]$

• Vadim Schechtman, Alexander Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 $[$dml:143938, pdf$]$

following precursor observations due to:

• Vladimir S. Dotsenko, Vladimir A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B 240 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$

• Philippe Christe, Rainald Flume, The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term, Nuclear Physics B

282 (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$

The proof that for rational levels this construction indeed yields conformal blocks is due to:

Review:

• Alexander Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (doi:10.1142/2467)

• Ivan Cherednik, Section 8.2 of: Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 $[$doi:10.2969/msjmemoirs/00101C010$]$

• Pavel Etingof, Igor Frenkel, Alexander Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) $[$ISBN:978-1-4704-1285-2, review pdf$]$

• Toshitake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 $[$doi:10.5427/jsing.2012.5g, pdf$]$

• Toshitake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39 (2014) 575–598 $[$doi:10.1007%2Fs40306-014-0088-6, pdf$]$

• Toshitake Kohno, Introduction to representation theory of braid groups, Peking 2018 $[$pdf, pdf$]$

(motivation from braid representations)

This “hypergeometric” construction uses results on the twisted de Rham cohomology of configuration spaces of points due to:

also:

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$=0$ (cf. at logarithmic CFT – Examples):

• Fedor A. Smirnov, Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations, $[$arXiv:hep-th/9210051$]$

• Fedor A. Smirnov, Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration, Communications in Mathematical Physics 155 (1993) 459–487 $[$doi:10.1007/BF02096723, arXiv:hep-th/9210052$]$

• S. Pakuliak, A. Perelomov, Relation Between Hyperelliptic Integrals, Mod. Phys. Lett. 9 19 (1994) 1791-1798 $[$doi:10.1142/S0217732394001647$]$

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: