Chern–Weil forms and abstract homotopy theory
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- by Daniel S. Freed and Michael J. Hopkins PDF
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Abstract:
We prove that Chern–Weil forms are the only natural differential forms associated to a connection on a principal $G$-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebra as the de Rham complex of a specific simplicial sheaf, and similarly give a new interpretation of the Weil model in equivariant de Rham theory. There is an appendix proving a general theorem about set-theoretic transformations of polynomial functors.References
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Additional Information
- Daniel S. Freed
- Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
- Email: dafr@math.utexas.edu
- Michael J. Hopkins
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- Email: mjh@math.harvard.edu
- Received by editor(s): January 24, 2013
- Published electronically: April 17, 2013
- Additional Notes: The work of the first author was supported by the National Science Foundation under grants DMS-0603964, DMS-1207817, and DMS-1160461
The work of the second author was supported by the National Science Foundation under grants DMS-0906194, DMS-0757293, DMS-1158983 - © Copyright 2013 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 50 (2013), 431-468
- MSC (2010): Primary 58Axx; Secondary 53C05, 53C08, 55U35
- DOI: https://doi.org/10.1090/S0273-0979-2013-01415-0
- MathSciNet review: 3049871
Dedicated: In memory of Dan Quillen