nLab hypergeometric KZ-solutions -- references

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$]$

  (for $\mathfrak{sl}(2,\mathbb{C})$)

• 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:

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 $[$doi:10.1007/BF00626525$]$

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 $[$doi:10.1007/BF02101739$]$

  (for $\mathfrak{sl}(2, \mathbb{C})$)

• Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 [euclid:cmp/1104272957]

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)

See also:

• Alexander Varchenko, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 1 (1995) 99-137 $[$arXiv:hep-th/9403102, doi:10.1007/BF02103772$]$

• Edward Frenkel, David Ben-Zvi, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, AMS 2004 $[$ISBN:978-1-4704-1315-6, web$]$

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

• Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56 (1980) 167–189 $[$doi:10.1007/BF01392549$]$

• Kazuhiko Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 2 (1987) 191-208 $[$doi:10.2969/jmsj/03920191$]$

• Hélène Esnault, Vadim Schechtman, Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992) 557-561 $[$pdf$]$

• Vadim Schechtman, H. Terao, Alexander Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, Journal of Pure and Applied Algebra 100 1–3 (1995) 93-102 $[$arXiv:hep-th/9411083, doi:10.1016/0022-4049(95)00014-N$]$

also:

• Peter Orlik, Hypergeometric integrals and arrangements, Journal of Computational and Applied Mathematics 105 (1999) 417–424 $[$doi:10.1016/S0377-0427(99)00036-9, pdf$]$

• Daniel C. Cohen, Peter Orlik, Arrangements and local systems, Math. Res. Lett. 7 (2000) 299-316 $[$arXiv:math/9907117, doi:10.4310/MRL.2000.v7.n3.a5$]$

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:

• Hisham Sati, Urs Schreiber, Anyonic defect branes in TED-K-theory $[$arXiv:2203.11838$]$