spin geometry, string geometry, fivebrane geometry …
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Hirzebruch signature theorem?
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.
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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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 $n/2$-dimensional representation of $Spin(n)$-group, so called Spin representation, which is depending, if $n$ is odd 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
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
given by local formula
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.
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
with components (with respect to the $\mathbb{Z}_2$-grading) to be denoted
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 $D^2$:
This appears as (Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50), based on (MacKean-Singer 67).
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
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
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
based on original articles such as
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.