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In brief terms, the identification of

  1. $\mathfrak{sl}_2$ lowering operators "$f$" applied "in a conformal block" at the $i$th puncture $z_i$ in the Riemann sphere with

  2. the "hypergeometric" differential forms $ - \frac{m_i}{\kappa} d \log(z-z_i)$

for integral and "integrable" $\widehat{\mathfrak{sl}_2}$-weights $m_i$ at non-negative integer levels $k = \kappa - 2$

constitutes a natural linear transformation from:

  1. the corresponding conformal blocks to

  2. the Dolbeault cohomology of holomorphic differential forms on the configuration space of points in the punctured Riemann sphere, twisted by the holomorphic flat connection 1-form whose holonomy around $z_i$ is $e^{ 2 \pi \mathrm{i}\, m_i/\kappa }$.

This curious statement, which arises within the more general program of constructing solutions of the KZ-equation via "hypergeometric integrals"

  • V. Schechtman, A. Varchenko: Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 (doi:10.1007/BF00626523),

is "Main Theorem 3.4.1" & "Corollary 3.4.2" in

  • [FSV1] B. Feigin, V. Schechtman, A. Varchenko: On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 (doi:10.1007/BF02101739)

and as such is the special case $\widehat{\mathfrak{g}} = \widehat{\mathfrak{sl}_2}$ of a more general result in

  • [FSV2] B. Feigin, V. Schechtman, A. 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)

I am wondering what is known, in citable form, about the kernel and the cokernel of this linear transformation (from conformal blocks on the punctured sphere to twisted Dolbeault cohomology of configurations of points in the punctured sphere).

This is an (open?) issue highlighted already in [FSV2, p. 2]:

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 fact, [FSV1, final Rem. 3.4.3] claims that for $\widehat{\mathfrak{sl}_2}$ the mapping is injective. However, for proof, the reader is pointed to a preprint which seems not to be available. On request for a proof, I have been pointed broadly to the monograph:

  • [Var] A. 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)

Is there a proof of injectivity of the mapping in this book?

What I see is that the existence of the mapping itself (the subtle part of it) is considered around p. 289. Also I see that the twisted version of the Orlik-Solomon result, due to

  • H. Esnault, V. Schechtman, E. Viehweg: Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992): 557-561 (pdf)

is reviewed around [Var, Thm. 10.6.8]. I believe this result can be used (only) to show that the mapping in question is in fact an isomorphism away from the cases where the weights lead to "resonances", but these excluded cases are exactly where the subtlety lies that needs to be discussed for purposes of conformal blocks.

Kohno briefly touches on this issue in

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

where it briefly says on p. 107:

Furthermore, as is stated in [FSV1], it is shown in [Var], the induced map is injective.

So I guess I am missing something in [Var]. (I haven't fully absorbed yet the discussion in [Var] involving Hochschild cohomology; maybe that can be used to produce the desired injectivity proof? -- I'd be grateful for a hint if so, as I don't see it.)

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