spinᶜ Dirac operator in nLab
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Context
Index theory
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Definitions
Index theorems
Higher genera
Functional analysis
Overview diagrams
Basic concepts
Theorems
Topics in Functional Analysis
Higher spin geometry
spin geometry, string geometry, fivebrane geometry …
Ingredients
Spin geometry
String geometry
Fivebrane geometry
Ninebrane geometry
Contents
Idea
A type of Dirac operator defined on manifolds equipped with Spin^c structure.
Properties
Relation to the Dolbeault operator
Every almost complex manifold carries a canonical spin^c structure (as discussed there). If is a complex manifold, then under the identification
of the spinor bundle with that of holomorphic differential forms, the corresponding -Dirac operator is identified with the Dolbeault-Dirac operator
Genus
The A-hat genus for the -operator is the Todd genus (e.g. Kitada 75).
Applications
References
Original articles include
- Yasuhiko Kitada, Semi-free circle actions on -manifolds, Publ. RIMS, Kyoto Univ. 10(1975), 601-617 (pdf)
A quick statement of the definition is in
- Ulrich Krämer, -Dirac structures and Dirac operators (2009) (pdf)
Detailed accounts include
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J. J. Duistermaat, The Spin-c Dirac Operator, in The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator Modern Birkhäuser Classics, 2011, 41-51, DOI: 10.1007/978-0-8176-8247-7_5
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Eckhard Meinrenken, Symplectic Surgery and the Spin-C Dirac operator (arXiv:dg-ga/9504002)
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Charles Epstein, Subelliptic -Dirac operators, I (pdf)
Last revised on July 5, 2024 at 13:46:41. See the history of this page for a list of all contributions to it.