Dirac operator in nLab
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Context
Spin geometry
spin geometry, string geometry, fivebrane geometry …
Ingredients
Spin geometry
String geometry
Fivebrane geometry
Ninebrane geometry
Index theory
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Definitions
Index theorems
Higher genera
Contents
Idea
General
For a spinor bundle over a Riemannian manifold , a Dirac operator on is an differential operator on (sections of) whose principal symbol is that of , where is the exterior derivative and is the symbol map.
More abstractly, for a Dirac operator, its normalization is a Fredholm operator, hence defines an element in K-homology.
Origin and role in Physics
The first relativistic Schrödinger type equation found was Klein-Gordon. At first it did not look that K-G equation could be interpreted physically because of negative energy states and other paradoxes. Paul Dirac proposed to take a square root of Laplace operator within the matrix-valued differential operators and obtained a Dirac equation; matrix valued generators involved representations of a Clifford algebra. It also had negative energy solutions, but with half-integer spin interpretation which was appropriate the Pauli exclusion principle together with the Dirac sea picture came at rescue (Klein-Gordon is now also useful with more modern formalisms).
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Definition
In components
The tangent bundle of an oriented Riemannian -dimensional manifold is an -bundle. Orientation means that the first Stiefel-Whitney class is zero. If is zero than the bundle can be lifted to a -bundle. A choice of connection on such a -bundle is a -structure on . There is a standard -dimensional representation of -group, so called Spin representation. If is odd it is irreducible, and if is even it decomposes into the sum of two irreducible representations of equal dimension and . Thus we can associate associated bundles to the original bundle with respect to these representations. Thus we get the spinor bundles and .
Gamma matrices, which are the representations of the Clifford algebra
thus act on such a space; certain combinations of products of gamma matrices with partial derivatives define a first order Dirac operator ; there are several versions, in mathematics is pretty important the chiral Dirac operator
given by local formula
where are orthonormal frames of tangent vectors and is the covariant derivative with respect to the Levi-Civita spin connection. The expression is the chirality operator.
In Euclidean space the Dirac operator is elliptic, but not in Minkowski space.
The Dirac operator is involved in approaches to the Atiyah-Singer index theorem about the index of an elliptic operator: namely the index can be easier calculated for Dirac operator and the deformation to the Dirac operator does not change the index. An appropriate version of a Dirac operator is a part of a concept of the spectral triple in noncommutative geometry a la Alain Connes.
Properties
Eta invariant and functional determinant
The eta function (see there for more) of a Dirac operator expresses the functional determinant of its Laplace operator .
Index and partition function
Proposition
Let be a compact Riemannian manifold and a smooth super vector bundle and indeed a Clifford module bundle over . Consider a Dirac operator
with components (with respect to the -grading) to be denoted
where . Then is a Fredholm operator and its index is the supertrace of the kernel of , as well as of the heat kernel of :
This appears as (Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50), based on (MacKean-Singer 67).
Examples
References
Textbooks include
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H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)
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Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate studies in mathematics 25, AMS (1997)
The relation to index theory is discussed in
- Nicole Berline, Ezra Getzler, Michèle Vergne, Heat Kernels and Dirac Operators, Springer Verlag Berlin (2004)
based on original articles such as
- H. MacKean, Isadore Singer, Curvature and eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967)
- Michael Atiyah, Raoul Bott, V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330.
See also
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Daniel Freed, Geometry of Dirac operators, (1987) [pdf, pdf]
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C. Nash, Differential topology and quantum field theory, Acad. Press 1991.
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Eckhard Meinrenken, Clifford algebras and Lie groups, Lecture Notes, University of Toronto, Fall 2009.
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Jing-Song Huang, Pavle Pandžić, J.-S. Huang, P. Pandzic, Dirac Operators in Representation Theory,. Birkhäuser, Boston, 2006, 199 pages; short version Dirac operators in representation theory, 48 pp. pdf
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J.-S. Huang, Pavle Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185—202.
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R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1-30.
Last revised on August 14, 2022 at 15:55:44. See the history of this page for a list of all contributions to it.