pointer

- Anton Alekseev, Florian Naef, Muze Ren.
*Generalized Pentagon Equations*(2024). (arXiv:2402.19138).

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]

added pointer to:

- Ralph Blumenhagen, Erik Plauschinn, §3.5 of:
*Introduction to Conformal Field Theory – With Applications to String Theory*, Lecture Notes in Physics**779**, Springer (2009) [doi:10.1007/978-3-642-00450-6]

and this one:

- Xia Gu, Babak Haghighat, Yihua Liu,
*Ising- and Fibonacci-Anyons from KZ-equations*[arXiv:2112.07195]

also pointer to:

- Ivan Todorov, Ludmil Hadjiivanov,
*Monodromy Representations of the Braid Group*, Phys. Atom. Nucl.**64**(2001) 2059-2068; Yad.Fiz.**64**(2001) 2149-2158 [arXiv:hep-th/0012099, doi:10.1134/1.1432899]

added pointer to:

- Ivan Marin,
*Sur les représentations de Krammer génériques*, Annales de l’Institut Fourier,**57**6 (2007) 1883-1925 [numdam:AIF_2007__57_6_1883_0]

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.

]]>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 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 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.

]]>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$]$

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$]$

?

]]>[ removed, sorry for the noise]

]]>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

added pointer to:

- Ivan Marin,
*Knizhnik-Zamolodchikov bundles are topologically trivial*(arXiv:0809.3590)

finally added the actual definition, `!include`

-ed from *Knizhnik-Zamolodchikov-Kontsevich construction – definition* (as per the discussion here)