nLab Dirac operator

Idea

General

For $S \to X$ a spinor bundle over a Riemannian manifold $(X,g)$, a Dirac operator on $S$ is an differential operator on (sections of) $S$ whose principal symbol is that of $c \circ d$, where $d$ is the exterior derivative and $c$ is the symbol map.

More abstractly, for $D$ a Dirac operator, its normalization $D(1+ D^2)^{-1/2}$ 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 $n$-dimensional manifold $M$ is an $SO(n)$-bundle. Orientation means that the first Stiefel-Whitney class $w_1(M)$ is zero. If $w_2(M)$ is zero than the $SO(n)$ bundle can be lifted to a $Spin(n)$-bundle. A choice of connection on such a $Spin(n)$-bundle is a $Spin$-structure on $M$. There is a standard $2^{[n/2]}$-dimensional representation of $Spin(n)$-group, so called Spin representation. If $n$ is odd it is irreducible, and if $n$ is even it decomposes into the sum of two irreducible representations of equal dimension $S_+$ and $S_-$. Thus we can associate associated bundles to the original $Spin(n)$ bundle $P$ with respect to these representations. Thus we get the spinor bundles $E_\pm := P\times_{Spin(n)} S_\pm\to M$ and $E = E_+\oplus E_-$.

Gamma matrices, which are the representations of the Clifford algebra

$$\gamma_a \gamma_b + \gamma_b \gamma_a = -2\delta_{ab} I$$

$$\gamma_5 = i^{n(n+1)/2}\gamma_1\cdots\gamma_n, \,\,\,\,\gamma^2_5 = I$$

thus act on such a space; certain combinations of products of gamma matrices with partial derivatives define a first order Dirac operator $\Gamma(E)\to\Gamma(E_-)$; there are several versions, in mathematics is pretty important the chiral Dirac operator

$$\Gamma(M,E_+)\to \Gamma(M,E_-)$$

given by local formula

$$\sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}$$

where $e^\mu_a(x)$ are orthonormal frames of tangent vectors and $\nabla_\mu$ is the covariant derivative with respect to the Levi-Civita spin connection. The expression $\frac{1+\gamma_5}{2}$ 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 $D$ expresses the functional determinant of its Laplace operator $H = D^2$.

Index and partition function

Proposition

Let $(X,g)$ be a compact Riemannian manifold and $\mathcal{E}$ a smooth super vector bundle and indeed a Clifford module bundle over $X$. Consider a Dirac operator

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

with components (with respect to the $\mathbb{Z}_2$-grading) to be denoted

$$D = \left[ \array{ 0 & D^- \\ D^+ & 0 } \right] \,,$$

where $D^- = (D^+)^\ast$. Then $D^+$ is a Fredholm operator and its index is the supertrace of the kernel of $D$, as well as of the heat kernel of $D^2$:

$$\begin{aligned} ind(D^+) & \coloneqq dim(ker(D^+)) - dim(coker(D^+)) \\ & = dim(ker(D^+)) - dim(ker(D^-)) \\ & = sTr(ker(D)) \\ & = sTr( \exp(-t \, D^2) ) \;\;\; \forall t \gt 0 \end{aligned} \,.$$

This appears as (Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50), based on (MacKean-Singer 67).

Remark

If one thinks of $D^2$ as the time-evolution Hamiltonian of a system of supersymmetric quantum mechanics with $D$ the supercharge on the worldline, then $ker(D)$ is the space of supersymmetric quantum states, $\exp(-t \, D^2)$ 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

• Kähler-Dirac operator

• Dolbeault-Dirac operator

• Spin^c Dirac operator

Related concepts

• Dirac spinor, Dirac field

• Dirac equation

• spin^c Dirac operator

• index of a Dirac operator

• Fredholm module, K-homology, KK-theory

• Lichnerowicz formula

• Dirac-Ramond operator, Dirac operator on smooth loop space

• Feynman slash notation

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

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-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) [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.