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finally added the actual definition, !include
-ed from Knizhnik-Zamolodchikov-Kontsevich construction – definition (as per the discussion here)
added pointer to:
I am finally adding a section with references on the “hypergeometric” construction of conformal blocks/KZ-solutions, via twisted de Rham cohomology of configuration spaces of points.
A start is now here, but I will put this into a stand-alone entry, to be !include
-ed here and in other related entries
[ removed, sorry for the noise]
The entry used to claim (am removing it for the moment) that:
The interpretation of $[$ the KZ-equation $]$ in terms of a flat connection on the moduli space of conformal structures is due to:
- Graeme Segal, Conformal field theory, Oxford preprint and lecture at the IAMP Congress, Swansea July 1988.
Is that really the case?
(I forget who added this reference. There is a good chance that it was me, my apologies.)
If so, where exactly inside the following three items should we be pointing for the KZ-equation:
Graeme Segal, The definition of conformal field theory, in: K. Bleuler, M. Werner (eds.), Differential geometrical methods in theoretical physics (Proceedings of Research Workshop, Como 1987), NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. 250 Kluwer Acad. Publ., Dordrecht (1988) 165-171 $[$doi:10.1007/978-94-015-7809-7$]$
Graeme Segal, Two-dimensional conformal field theories and modular functors, in: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, 1988, Hilger, Bristol (1989) 22-37.
Graeme Segal, The definition of conformal field theory, in: Ulrike Tillmann (ed.), Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser. 308, Cambridge University Press (2004) 421-577 $[$doi:10.1017/CBO9780511526398.019, pdf$]$
?
added pointer to these surveys:
Ivan Cherednik, Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 $[$doi:10.2969/msjmemoirs/00101C010$]$
Toshitake Kohno, Section1 1.5 and 2.1 in: Conformal field theory and topology, transl. from the 1998 Japanese original by the author. Translations of Mathematical Monographs 210. Iwanami Series in Modern Mathematics. Amer. Math. Soc. 2002 $[$AMS:mmono-210$]$
I have come to think that the main part of the main theorem in the hypergeometric-integral construction of KZ solutions becomes a triviality when looked at from a HoTT point of view:
Namely, the main theorem says that the twisted cohomology groups of $Conf_{n+N}(\mathbb{C})\vert_{N}$ for fixed positions of $N$ of the points form a local system over $Conf_N(\mathbb{C})$.
But since the twisted cohomology depends only on the shape of $Conf_{n+N}(\mathbb{C})$, and since it is represented by a classifying space, the system of cohomology groups is given by an internal hom out of $ʃ Conf_{n + N}(\mathbb{C})$ in the slice over $ʃ Conf_N(\mathbb{C})$. Such a slice hom is again a fibration over $ʃ Conf_N(\mathbb{C})$, and its fibers are the desired fiberwise cohomology groups. Upon fiberwise truncation, this is the statement of that main theorem.
I should add that this proof uses that fiberwise 0-truncation preserves fiber products (which it does) combined with the assumption that any point inclusion into the base type is already 0-truncated. So this works over configuration spaces of points (since these are $K(G,1)$s) as needed here for the KZ-equation, but not for Gauss-Manin connections over higher truncated base spaces.
I have now typed out the argument in point-set model presentation. Unsure where this should go, for the moment I put it into the entry on Gauss-Manin connections: here.
I have strengthened statement and proof (here) to say that on a locally trivial fibration, the local system of cohomology groups has a compatible local trivialization.
This will serve to prove that, when applied to fibrations of configuration spaces, this abstract argument really reproduces the hypergeometric solutions to the KZ-equation.
The only further lemma for this conclusion is that the statement also works for fiberwise twisted cohomology. Will type this out next.
added pointer to:
also pointer to:
and this one:
added pointer to:
added these pointers on
KZ-equations controlling codimension$=2$ defects in D=4 super Yang-Mills theory:
Nikita Nekrasov, BPS/CFT correspondence V: BPZ and KZ equations from $q q$-characters [arXiv:1711.11582]
Nikita Nekrasov, Alexander Tsymbaliuk, Surface defects in gauge theory and KZ equation, Letters in Mathematical Physics 112 28 (2022) [arXiv:2103.12611, doi:10.1007/s11005-022-01511-8]
Saebyeok Jeong, Norton Lee, Nikita Nekrasov, Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations, J. High Energ. Phys. 2021 120 (2021) [arXiv:2103.17186, doi:10.1007/JHEP10(2021)120]
pointer
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