Geometrization of the TUY/WZW/KZ connection

Given a simple, simply connected, complex algebraic group G , a flat projective connection on the bundle of non-abelian theta functions on the moduli space of semistable parabolic G -bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier p...

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Bibliographic Details
Published in:Letters in mathematical physics 2024-06, Vol.114 (3), Article 85
Main Authors: Biswas, Indranil, Mukhopadhyay, Swarnava, Wentworth, Richard
Format: Article
Language:English
Subjects:
Complex Systems
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Physics
Physics and Astronomy
Tensors
Theoretical
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Summary:Given a simple, simply connected, complex algebraic group G , a flat projective connection on the bundle of non-abelian theta functions on the moduli space of semistable parabolic G -bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of non-abelian theta functions and the bundle of WZW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya–Ueno–Yamada. As an application, we give a geometric construction of the Knizhnik–Zamolodchikov connection on the trivial bundle over the configuration space of points in the projective line whose typical fiber is the space of invariants of tensor product of representations.
ISSN:1573-0530
0377-9017
1573-0530
DOI:10.1007/s11005-024-01834-8