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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexa bare list of references, to be `!include`d into the References-subsections of relevant entries <a href="https://ncatlab.org/nlab/revision/hypergeometric+KZ-solutions+--+references/1">v1</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references">current</a>

   a bare list of references, to be !included into the References-subsections of relevant entries

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItextook the liberty of adding this pointer, at the end: *** Interpretation of the [[hypergeometric construction of KZ solutions|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 [[anyons]]/[[topological order]], respectively: * [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Anyonic defect branes in TED-K-theory]]* $[$[arXiv:2203.11838](https://arxiv.org/abs/2203.11838)$]$ *** <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+KZ-solutions+--+references/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+KZ-solutions+--+references/3">v3</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references">current</a>

   took the liberty of adding this pointer, at the end:

   ---

   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 anyons/topological order, respectively:

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

   ---

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 9th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded previously missing pointer to this original reference: * 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](https://doi.org/10.1142/S0217979290000528)$]$ <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+KZ-solutions+--+references/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+KZ-solutions+--+references/5">v5</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references">current</a>

   added previously missing pointer to this original reference:

   • 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$]$

   diff, v5, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 12th 2022
   • (edited Jun 12th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the following pointers: *** Discussion for the special case of [[level (Chern-Simons theory)|level]]$=0$ (cf. at *[logarithmic CFT -- Examples](https://ncatlab.org/nlab/show/logarithmic%20CFT#Examples)*): * Fedor A. Smirnov, *Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations*, $[$[arXiv:hep-th/9210051](https://arxiv.org/abs/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](https://doi.org/10.1007/BF02096723)$]$ * S. Pakuliak, A. Perelomov, *Relation Between Hyperelliptic Integrals*, Mod. Phys. Lett. **9** 19 (1994) 1791-1798 $[$[doi:10.1142/S0217732394001647](https://doi.org/10.1142/S0217732394001647)$]$ <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+KZ-solutions+--+references/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+KZ-solutions+--+references/8">v8</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references">current</a>

   I have added the following pointers:

   ---

   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$]$

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

   diff, v8, current

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJun 13th 2022
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn my memory, Varchenko's book is not only a review, it has quite a few of new results and formulas (but I have to check when I will have time, now in the middle of a heavy exam grading season). Main emphasis in the book is on viewing hypergeometric pairing as relating quantum group (at root of unity) representations and affine Lie algebra representations. Regarding that there are some constructions (from the early days) of quantum groups from monodromies this is quite in accord with other intepretations,

   In my memory, Varchenko’s book is not only a review, it has quite a few of new results and formulas (but I have to check when I will have time, now in the middle of a heavy exam grading season). Main emphasis in the book is on viewing hypergeometric pairing as relating quantum group (at root of unity) representations and affine Lie algebra representations. Regarding that there are some constructions (from the early days) of quantum groups from monodromies this is quite in accord with other intepretations,

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 13th 2022
   • (edited Jun 13th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, maybe. Though there is certainly a lot of review in the book, even an attempt to be expository. To some extent the book seems to be meant as filling in proofs that were left open in previous articles. Concretely, I was told that the book (i.e. [Varchenko 1995](https://ncatlab.org/nlab/show/Knizhnik-Zamolodchikov+equation#Varchenko95)) contains the proof of [Feigin, Schechtman & Varchenko's 1994](https://ncatlab.org/nlab/show/Knizhnik-Zamolodchikov+equation#FeiginSchechtmanVarchenko94) Remark 3.4.3 (the one on [p. 10](https://link.springer.com/content/pdf/10.1007/BF02101739.pdf#page=10)). But I haven't found that proof yet. I have asked about it on MathOverflow ([here](https://mathoverflow.net/q/416486/381)), and others seem not to have found it either. But if anyone knows where the proof of that [Remark 3.4.3](https://link.springer.com/content/pdf/10.1007/BF02101739.pdf#page=10) is spelled out "in print", I'd be most grateful.

   Yes, maybe. Though there is certainly a lot of review in the book, even an attempt to be expository.

   To some extent the book seems to be meant as filling in proofs that were left open in previous articles.

   Concretely, I was told that the book (i.e. Varchenko 1995) contains the proof of Feigin, Schechtman & Varchenko’s 1994 Remark 3.4.3 (the one on p. 10).

   But I haven’t found that proof yet. I have asked about it on MathOverflow (here), and others seem not to have found it either.

   But if anyone knows where the proof of that Remark 3.4.3 is spelled out “in print”, I’d be most grateful.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeJun 13th 2022
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI could ask somebody about it, I know somebody who traced their work when it was fresh.

   I could ask somebody about it, I know somebody who traced their work when it was fresh.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 13th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI'd be interested in hearing what they may have to say!

   I’d be interested in hearing what they may have to say!

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 19th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Alexander Varchenko]], *Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base*, Comm. Math. Phys. **171** 1 (1995) 99-137 $[$[arXiv:hep-th/9403102](https://arxiv.org/abs/hep-th/9403102), [doi:10.1007/BF02103772](https://doi.org/10.1007/BF02103772)$]$ <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+KZ-solutions+--+references/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+KZ-solutions+--+references/11">v11</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references">current</a>

   added pointer to:

   • 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$]$

   diff, v11, current

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * [[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](https://doi.org/10.1007/BF02102089)$]$ <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+KZ-solutions+--+references/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+KZ-solutions+--+references/13">v13</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references">current</a>

    added pointer to:

    • 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$]$

    diff, v13, current

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2023
    • (edited Oct 22nd 2023)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * [[Peter Orlik]], *Hypergeometric integrals and arrangements*, Journal of Computational and Applied Mathematics **105** (1999) 417–424 $[$<a href="https://doi.org/10.1016/S0377-0427(99)00036-9">doi:10.1016/S0377-0427(99)00036-9</a>, [pdf](https://core.ac.uk/download/pdf/82631708.pdf)$]$ <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+KZ-solutions+--+references/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+KZ-solutions+--+references/18">v18</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+KZ-solutions+--+references">current</a>

    added pointer to:

    • 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$]$

    diff, v18, current