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nLab > Latest Changes: Knizhnik-Zamolodchikov equation

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1 to 16 of 16

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2020
    • (edited Jan 13th 2020)

    finally added the actual definition, !include-ed from Knizhnik-Zamolodchikov-Kontsevich construction – definition (as per the discussion here)

    diff, v13, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 22nd 2021
    • (edited Dec 22nd 2021)

    added pointer to:

    diff, v17, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2022

    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

    diff, v18, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2022
    • (edited Jun 6th 2022)

    [ removed, sorry for the noise]

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2022

    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]]

    ?

    diff, v23, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2022

    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]]

    diff, v23, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2022

    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(ℂ)| NConf_{n+N}(\mathbb{C})\vert_{N} for fixed positions of NN of the points form a local system over Conf N(ℂ)Conf_N(\mathbb{C}).

    But since the twisted cohomology depends only on the shape of Conf n+N(ℂ)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 ʃConf n+N(ℂ)ʃ Conf_{n + N}(\mathbb{C}) in the slice over ʃConf N(ℂ)ʃ Conf_N(\mathbb{C}). Such a slice hom is again a fibration over ʃConf N(ℂ)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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2022
    • (edited Jun 20th 2022)

    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)K(G,1)s) as needed here for the KZ-equation, but not for Gauss-Manin connections over higher truncated base spaces.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2022

    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.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2022

    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.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2022

    added pointer to:

    diff, v29, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2022

    also pointer to:

    diff, v30, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2022

    and this one:

    diff, v30, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2022

    added pointer to:

    diff, v32, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2024

    added these pointers on

    KZ-equations controlling codimension=2=2 defects in D=4 super Yang-Mills theory:

    diff, v37, current

    • CommentRowNumber16.
    • CommentAuthorperezl.alonso
    • CommentTimeMar 1st 2024

    pointer

    diff, v38, current

1 to 16 of 16

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