Holonomy, twisting cochains and characteristic classes

G. Sharygin[1]

  • [1] Institute of Theoretical and Experimental Physics 25 ul. B. Cheremushkinskaya, Moscow, 117259 Russia.

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 2, page 295-366
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Sharygin, G.. "Holonomy, twisting cochains and characteristic classes." Annales de la faculté des sciences de Toulouse Mathématiques 20.2 (2011): 295-366. <http://eudml.org/doc/219763>.

@article{Sharygin2011,
abstract = {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.},
affiliation = {Institute of Theoretical and Experimental Physics 25 ul. B. Cheremushkinskaya, Moscow, 117259 Russia.},
author = {Sharygin, G.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {index; Dirac operator; vector bundle; monodromy; gauge; characteristic class; free loop space; cyclic Chern character; Bismut's class; twisting cochain; principal bundle; Getzler-Jones-Petrack map; Getzler-Jones classes},
language = {eng},
month = {4},
number = {2},
pages = {295-366},
publisher = {Université Paul Sabatier, Toulouse},
title = {Holonomy, twisting cochains and characteristic classes},
url = {http://eudml.org/doc/219763},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Sharygin, G.
TI - Holonomy, twisting cochains and characteristic classes
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/4//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 2
SP - 295
EP - 366
AB - 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.
LA - eng
KW - index; Dirac operator; vector bundle; monodromy; gauge; characteristic class; free loop space; cyclic Chern character; Bismut's class; twisting cochain; principal bundle; Getzler-Jones-Petrack map; Getzler-Jones classes
UR - http://eudml.org/doc/219763
ER -

References

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