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For when the editing functionality is back, I’ll want to create an !include
-entry with references on the Schechtman-Varchenko construction of KZ-solutions/conformal blocks via hypergeometric functions realized as 1-twisted de Rham cohomology of configurations spaces of points in the plane – in order to easily include it into the References section.
Meanwhile, I’ll use this comment here to collect references (will be updating this comment here):
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Background results on twisted cohomology of complements of hyperplane arrangements:
Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56, 167–189 (1980) (doi:10.1007/BF01392549)
Hélène Esnault, Vadim Schechtman, Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992): 557-561 (pdf)
V. Schechtman, H. Terao, A. 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)
and on the Gauss-Manin connection:
Precursor constructions:
Vl. S. Dotsenko, V. A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B Volume 240, Issue 3, 15 October 1984, Pages 312-348 (doi:10.1016/0550-3213(84)90269-4)
P. Christe, R. 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 original SV-construction:
Vadim V. Schechtman, Alexander N. Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, August 1989, Preprint MPI/89- (cds:1044951)
Vadim V. Schechtman, Alexander N. Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 (doi:10.1007/BF00626523)
Vadim V. Schechtman, Alexander N. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 (dml:143938, pdf)
with an independent discussion for $\mathfrak{g} = \mathfrak{sl}(2,\mathbb{C})$ in:
Proof that for rational levels the construction yields WZW conformal blocks inside the KZ-solutions:
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)
Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170(1): 219-247 (1995) (arxiv:hep-th/9407010, euclid:cmp/1104272957)
See also:
Boris Feigin, Edward Frenkel, Nikolai Reshetikhin, Thm. 4 of: Gaudin Model, Bethe Ansatz and Critical Level, Commun. Math. Phys. 166 (1994) 27-62 (arXiv:hep-th/9402022, doi:10.1007/BF02099300)
R. Rimányi, V. Schechtman, A. Varchenko, Conformal blocks and equivariant cohomology, Moscow Mathematical Journal 11 3 (2010) (arXiv:1007.3155, mmj:vol11-3-2011)
P. Belkale, P. Brosnan, S. Mukhopadhyay, Hyperplane arrangements and invariant theory (pdf)
Vadim Schechtman, Alexander Varchenko, Rational differential forms on line and singular vectors in Verma modules over $\widehat{\mathfrak{sl}}_2$, Mosc. Math. J. 17 (2017), 787–80 (arXiv:1511.09014, mmj:2017-017-004/2017-017-004-011)
Alexey Slinkin, Alexander Varchenko, Twisted de Rham Complex on Line and Singular Vectors in sl2^ Verma Modules, SIGMA 15 (2019), 075 (arXiv:1812.09791, doi:10.3842/SIGMA.2019.075)
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)
P. I. Etingof, Igor Frenkel, Alexander A Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998)
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)
Toshiyake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39, 575–598 (2014). (doi:10.1007%2Fs40306-014-0088-6, pdf)
also
Discussion as braid representations and anyons:
Toshiyake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 (doi:10.5427/jsing.2012.5g, pdf)
Ivan G Todorov, L K Hadjiivanov, Monodromy Representations of the Braid Group, Phys. At. Nucl. 64 (2001) 2059-2068 (doi:10.1134/1.1432899, cds:480345)
Xia Gu, Babak Haghighat, Yihua Liu, Ising- and Fibonacci-Anyons from KZ-equations (arXiv:2112.07195)
As a public service, I’ll extract where in these references the holonomy group of the local system is restricted to be inside $\mathbb{Q}/\mathbb{Z} \xhookrightarrow{\;} \mathrm{U}(1)$ (though that’s never how these authors state it), corresponding to the case that the solution to the KZ-equation given by the hypergeometric integrals produces specifically the conformal blocks among more general solutions.
That this is the case is sort of highlighted in
P. I. Etingof, Igor Frenkel, Alexander A Kirillov, Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998)
namely on top of p. 106 and then in Sec. 13.4
but these comments don’t make it too clear whether they are concerned only with rationality of the level $\kappa$; and in the original articles the relevant condition/assumption is pretty hard to spot, among a long and ever-continuing list of “let this, put that”-declarations of notation and assumptions:
Namely, in the original announcement
the condition of rational phases is essentially coded in the last paragraph on p. 291 and the following second paragraph on p. 292, which states that the weights $\Lambda_m$ further below in (3.1) are to be “finite weights” and that this is meant to mean dominant integral weights. Since, moreover, the $\alpha_i$ denote simple roots (second sentence of the article) and $\theta$ the highest root, normalized as usual to $(\theta,\theta) = 2$ (third sentence), this implies, with $k - (\Lambda, \theta)$ being integer (bottom of first page) that $k$ is rational, and thus, with $\kappa \coloneqq k + g^{\vee}$ (top of third page) that, finally, the exponents $(-\alpha_i, \Lambda_m)/\kappa$ and $(-\alpha_i, -\alpha_{i'})/\kappa$, in (3.1), are rational.
(To see this from a standard textbook such as Hall 15: For the inner product of roots this follows by Def. 8.1 with Prop. 8.6 there – specifically Sec. 6.9.3 for the case $\mathfrak{sl}_n$ –; for the inner product of roots with integral weights this then follows with Def. 7.1/8.21. Finally, the dual Coxeter number $g^{\vee}$ is an integer, e.g. p. 3 in arXiv:hep-th/9407010, p. 2 in arXiv:0903.0398v1.)
In the followup
which is tailored towards the special case $\mathfrak{g} \,=\,\mathfrak{sl}_2$, the condition is declared in the first paragraph of Sec. 3.2, which demands that $m_i$ and $\kappa = k +2$ be integers and thus that the exponents ${2}/{\kappa}$ and $- {m_i}/{\kappa}$ (in the previous Sec. 3.1.2) are rational.
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The phenomenon is made more explicit in
Toshiyake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 (doi:10.5427/jsing.2012.5g, pdf)
namely on p. 106 (13 of 15)
and
Toshiyake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39, 575–598 (2014). (doi:10.1007%2Fs40306-014-0088-6, pdf)
namely on p. 19, which finally admits that the corresponding exponents are rational.
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It’s quite striking that $hol(\text{local system}) \,\subset\, \mathbb{Q}/\mathbb{Z} \xhookrightarrow{\;} \mathrm{U}(1)$ goes with $H_\bullet( \text{config space}, \, \text{local system} ) = \text{ConformalBlocks}$. I wonder if there is any author who would have highlighted this more explicitly?
On the issue of embedding the conformal blocks into the twisted cohomology of the configuration space:
in Part II
they left this open, but highlighted the relevance:
There are reasons to expect that the map is injective. It would be very interesting to define its image in topological terms; if the above expectation is true, we would have a topological description of the bundle of conformal blocks.
But in part I,
which is concerned with the special case $\mathfrak{g} = \mathfrak{sl}(2)$, this is claimed to be known: The final Remark 3.4.3 there asserts that the proof of injectivity in this case was given in a reference [V] that is cited as:
I haven’t yet found this preprint, or whatever became of it.
One would expect that this is discussed in Varchenko’s own monograph
and possibly some version of it is indeed stated in section 12 there, but I am not sure: I still keep getting stack overflow when unwinding the notation declarations in this book.
My impression is that Kohno was struggling with this same issue when he wrote (p. 107) in
that
Furthermore, as is stated in [6], it is shown in [22], the induced map is injective.
Since [6] is Part I above and [22] is the above Monograph, this sounds like Kohno didn’t find it in the latter either, but read the former as claiming it’s true. Strange though that Kohno seems to think that “Part I” claimed it in generality beyond the case $\mathfrak{g} = \mathfrak{sl}(2)$.
So I am little stuck on this point. Does anyone have a good reference for the statement that the Schechtman-Varchenko construction of hypergeometric integrals yields an injection from the space of conformal blocks to the twisted de Rham cohomology of the configuration space (i.e. the claim of Rem. 3.4.3 in Part I above)?
I am wondering now about the following:
The above SV-construction via hypergeometric integrals produces solutions of the KZ-equation in great generality of values of the level and the weights.
As recalled above, it is a classical result that if these values are integral (and satisfy some bounds) then the construction happens to give the conformal blocks of WZW models.
More recently, the algebra of the WZW model has been argued to make good sense also at fractional (i.e. rational) level (pointers here). This makes it natural to wonder:
For non-integral but rational levels, where the SV-construction still makes good sense, is it at all related to the WZW model at rational level?
The question in #3 above I have now forwarded to MathOverflow: MO:q/416486.
Now I am looking for references that would generalize the “hypergeometric” construction of conformal blocks referenced in #1 from the punctured sphere to the punctured torus.
This is what I found so far (is there more?):
M. Crivelli, G. Felder, C. Wieczerkowski, Generalized hypergeometric functions on the torus and the adjoint representation of $U_q(sl_2)$, Commun. Math. Phys. 154, 1–23 (1993) (doi:10.1007/BF02096829)
M. Crivelli, G. Felder & C. Wieczerkowski, Topological representations of $U_q(sl_2)$ on the torus and the mapping class group, Lett Math Phys, 30 (1994) 71–85 (doi:10.1007/BF00761424)
Review:
Oh, I see: Instead of starting from scratch, I should prove that sewing of sphere conformal blocks corresponds to excision on the twisted cohomology they come from. This should be fairly straightforward and will imply the statement for complex curves of any genus.
[edit: hm, maybe not all that straightforward, after all…]
With the editing functionality finally being back, I am now starting an entry on this material:
with the above list of references now stored in an !include
-entry at
On the off-chance that there is any discussion to be had on this, probably best to move to to the corresponding thread here
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