Holonomy, twisting cochains and characteristic classes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 2, p. 295-366

This paper contains a description of various geometric constructions associated with fibre bundles, given in terms of important algebraic object, the “twisting cochain". Our examples include the Chern-Weil classes, the holonomy representation and the so-called cyclic Chern character of Bismut and others (see [2, 11, 27]), also called the Bismut’s class. The later example is the principal one for us, since we are motivated by the attempt to find an algebraic approach to the Witten’s index formula. We also give several examples of the twisting cochain associated with a given principal bundle. In particular, our approach allows us to obtain explicit formulas for the Chern classes and for an analogue of the cyclic Chern character in the terms of the glueing functions of the principal bundle. We discuss few modifications of this construction. We hope that this approach can turn fruitful for the investigations of the Witten index formula.

Nous proposons diverses constructions naturelles que l’on peut associer à un espace fibré. Nous les décrivons à l’aide d’un objet algébrique appelé « la cochaîne tordue » (« twisting cochains »). Nous considérons les classes caractéristiques de Chern-Weil, la représentation d’holonomie et le caractère de Chern cyclique défini par Bismut. Nous cherchons une approche algébrique de la formule de l’index de Witten. En outre, nous donnons quelques constructions explicites de la « cochaîne tordue » associée au fibré principal donné. En particulier, nos méthodes permettent d’obtenir des formules explicites pour les classes de Chern et pour un analogue du caractère de Chern cyclique en fonction du cocycle noncommutatif qui définit ce fibré principal. Nous discutons aussi certaines versions modifiées de cette construction. On espère que ces idées peuvent être utile pour l’étude de formule d’index de Witten.

@article{AFST_2011_6_20_2_295_0,
     author = {Sharygin, G.},
     title = {Holonomy, twisting cochains and characteristic classes},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 20},
     number = {2},
     year = {2011},
     pages = {295-366},
     doi = {10.5802/afst.1294},
     mrnumber = {2847886},
     zbl = {1276.57028},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2011_6_20_2_295_0}
}
Sharygin, G. Holonomy, twisting cochains and characteristic classes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 2, pp. 295-366. doi : 10.5802/afst.1294. http://www.numdam.org/item/AFST_2011_6_20_2_295_0/

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