nLab
Dirac operator

Contents

Idea

General

For SXS \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 cdc \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).

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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 n/2n/2-dimensional representation of Spin(n)Spin(n)-group, so called Spin representation, which is depending, if nn is odd 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 1γ 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 \mathcal{E} a smooth super vector bundle and indeed a Clifford module bundle over XX. Consider a Dirac operator

D:Γ(X,)Γ(X,) D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E})

with components (with respect to the 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).

Remark

If one thinks of D 2D^2 as the time-evolution Hamiltonian of a system of supersymmetric quantum mechanics with DD the supercharge on the worldline, then ker(D)ker(D) is the space of supersymmetric quantum states, exp(tD 2)\exp(-t \, D^2) is the Euclidean time evolution operator and its supertrace is the partition function of the system. Hence we have the translation

  • index = partition function .

Examples

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

References

Textbooks include

The relation to index theory is discussed in

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

  • 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.

Revised on August 25, 2014 03:21:26 by Urs Schreiber (82.113.121.153)