Dirac operator (changes) in nLab
Showing changes from revision #20 to #21: Added | Removed | Changed
Context
Spin geometry
spin geometry, string geometry, fivebrane geometry …
Ingredients
Spin geometry
rotation groups in low dimensions:
see also
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 S→XS \to X a spinor bundle over a Riemannian manifold (X,g)(X,g), a Dirac operator on SS is an differential operator on (sections of) SS whose principal symbol is that of c∘dc \circ d, where dd is the exterior derivative and cc is the symbol map.
More abstractly, for DD a Dirac operator, its normalization D(1+D 2) −1/2D(1+ D^2)^{-1/2} 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).
(…)
Definition
In components
The tangent bundle of an oriented Riemannian nn-dimensional manifold MM is an SO(n)SO(n)-bundle. Orientation means that the first Stiefel-Whitney class w 1(M)w_1(M) is zero. If w 2(M)w_2(M) is zero than the SO(n)SO(n) bundle can be lifted to a Spin(n)Spin(n)-bundle. A choice of connection on such a Spin(n)Spin(n)-bundle is a SpinSpin-structure on MM. There is a standard 2 [n/2]2^{[n/2]}-dimensional representation of Spin(n)Spin(n)-group, so called Spin representation. If nn is odd it is irreducible, and if nn is even it decomposes into the sum of two irreducible representations of equal dimension S +S_+ and S −S_-. Thus we can associate associated bundles to the original Spin(n)Spin(n) bundle PP with respect to these representations. Thus we get the spinor bundles E ±:=P× Spin(n)S ±→ME_\pm := P\times_{Spin(n)} S_\pm\to M and E=E +⊕E −E = E_+\oplus E_-.
Gamma matrices, which are the representations of the Clifford algebra
γ aγ b+γ bγ a=−2δ abI \gamma_a \gamma_b + \gamma_b \gamma_a = -2\delta_{ab} I
γ 5=i n(n+1)/2γ 1⋯γ n,γ 5 2=I \gamma_5 = i^{n(n+1)/2}\gamma_1\cdots\gamma_n, \,\,\,\,\gamma^2_5 = I
thus act on such a space; certain combinations of products of gamma matrices with partial derivatives define a first order Dirac operator Γ(E)→Γ(E −)\Gamma(E)\to\Gamma(E_-); there are several versions, in mathematics is pretty important the chiral Dirac operator
Γ(M,E +)→Γ(M,E −) \Gamma(M,E_+)\to \Gamma(M,E_-)
given by local formula
∑ aγ ae a μ(x)∇ μ1+γ 52 \sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}
where e a μ(x)e^\mu_a(x) are orthonormal frames of tangent vectors and ∇ μ\nabla_\mu is the covariant derivative with respect to the Levi-Civita spin connection. The expression 1+γ 52\frac{1+\gamma_5}{2} 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 DD expresses the functional determinant of its Laplace operator H=D 2H = D^2.
Index and partition function
Proposition
Let (X,g)(X,g) be a compact Riemannian manifold and ℰ\mathcal{E} a smooth super vector bundle and indeed a Clifford module bundle over XX. Consider a Dirac operator
D:Γ(X,ℰ)→Γ(X,ℰ) D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E})
with components (with respect to the ℤ 2\mathbb{Z}_2-grading) to be denoted
D=[0 D − D + 0], D = \left[ \array{ 0 & D^- \\ D^+ & 0 } \right] \,,
where D −=(D +) *D^- = (D^+)^\ast. Then D +D^+ is a Fredholm operator and its index is the supertrace of the kernel of DD, as well as of the heat kernel of D 2D^2:
ind(D +) ≔dim(ker(D +))−dim(coker(D +)) =dim(ker(D +))−dim(ker(D −)) =sTr(ker(D)) =sTr(exp(−tD 2))∀t>0. \begin{aligned} ind(D^+) & \coloneqq dim(ker(D^+)) - dim(coker(D^+)) \\ & = dim(ker(D^+)) - dim(ker(D^-)) \\ & = sTr(ker(D)) \\ & = sTr( \exp(-t \, D^2) ) \;\;\; \forall t \gt 0 \end{aligned} \,.
This appears as (Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50), based on (MacKean-Singer 67).
Examples
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
References
Textbooks include
-
H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)
-
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
-
Daniel Freed, Geometry of Dirac operators , 1987 (1987) ( [[pdf](http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf),pdf, pdf ) ]
-
C. Nash, Differential topology and quantum field theory, Acad. Press 1991.
-
Eckhard Meinrenken, Clifford algebras and Lie groups, Lecture Notes, University of Toronto, Fall 2009.
-
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
-
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.
-
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.