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Geometrization of the TUY/WZW/KZ connection
Algebraic Geometry · Submitted 1 Oct 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada.
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The abstract, sentence by sentence
Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian 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 nonabelian theta functions and the bundle of WZNW 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian 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. | background 0.92 | 0.85 | 0.06 | asserted 1.00 |
| s1 | Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. | result 0.90 | 0.93 | 0.69 | asserted 1.00 |
| s2 | 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. | result 0.69 | 0.94 | 0.29 | asserted 1.00 |
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The paper
Original abstract
Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian 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 nonabelian theta functions and the bundle of WZNW 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.
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How the literature uses it
3 arXiv papers cite this in 5 sentences
Papers citing it · 4 sentences without a confident label
Builds on it 1 paper
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Recently this has been extended to the case of marked points, see [ref] and references therein. the main result of [ref] is that one can also do quantisation in the nongeneric case, namely quantising the system of [ref] , generalising all the above.
Local wild mapping class groups and cabled braids · Some remarks about quantisation · 0.83
What it cites · 16 sentences without a confident label
Uses the method or tool of 1 paper
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This was used in [ref] to construct a Hitchin type connection for parabolic bundles and it will be crucial here as well.
A Hitchin connection on non abelian theta functions for parabolic G-bundles · Parabolic bundles as equivariant bundles · 0.94
Compares with 1 paper
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In this section, we give a proof of the main theorem in this article by comparing the Sugawara tensor and the parabolic analog of the heat operator with a given symbol constructed by the authors in [ref] .
A Hitchin connection on non abelian theta functions for parabolic G-bundles · Proof of Theorem [ref] · 0.93
Cites as background 2 papers
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In this section we first briefly recall the construction of the Hitchin connection for the moduli space of parabolic bundles $M^par,rs_G,τ$ obtained in [ref] .
A Hitchin connection on non abelian theta functions for parabolic G-bundles · Parabolic Hitchin symbol as in Biswas-Mukhopadhyay-Wentworth · 0.93
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We also refer to the several complementary approaches of Andersen [ref] , Axelrod-Witten-della Pietra [ref] , Baier-Bolognesi-Martens-Pauly [ref] , Faltings [ref] , Ginzburg [ref] , Ran [ref] , Ramadas [ref] , Sun-Tsai [ref] and for generalizations to reductive groups, Belkale [ref] .
The Hitchin connection in arbitrary characteristic · Introduction · 0.99
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