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\begin{document}

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\section*{Dirac operator}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{spin_geometry}{}\paragraph*{{Spin geometry}}\label{spin_geometry}

[[!include higher spin geometry - contents]]

\hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory}

[[!include index theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{origin_and_role_in_physics}{Origin and role in Physics}\dotfill \pageref*{origin_and_role_in_physics} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{eta_invariant_and_functional_determinant}{Eta invariant and functional determinant}\dotfill \pageref*{eta_invariant_and_functional_determinant} \linebreak
\noindent\hyperlink{index_and_partition_function}{Index and partition function}\dotfill \pageref*{index_and_partition_function} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

For $S \to X$ a [[spinor bundle]] over a [[Riemannian manifold]] $(X,g)$, a \emph{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]].

\hypertarget{origin_and_role_in_physics}{}\subsubsection*{{Origin and role in Physics}}\label{origin_and_role_in_physics}

The first relativistic Schr\"o{}dinger 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).

(\ldots{})

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{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 \textbf{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]]

\begin{displaymath}
\gamma_a \gamma_b + \gamma_b \gamma_a = -2\delta_{ab} I
\end{displaymath}
\begin{displaymath}
\gamma_5 = i^{n(n+1)/2}\gamma_1\cdots\gamma_n, \,\,\,\,\gamma^2_5 = I
\end{displaymath}
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

\begin{displaymath}
\Gamma(M,E_+)\to \Gamma(M,E_-)
\end{displaymath}
given by local formula

\begin{displaymath}
\sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}
\end{displaymath}
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]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{eta_invariant_and_functional_determinant}{}\subsubsection*{{Eta invariant and functional determinant}}\label{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]] $H = D^2$.

\hypertarget{index_and_partition_function}{}\subsubsection*{{Index and partition function}}\label{index_and_partition_function}

\begin{prop}
\label{}\hypertarget{}{}
Let $(X,g)$ be a [[compact topological space|compact]] [[Riemannian manifold]] and $\mathcal{E}$ a smooth [[super vector bundle]] and indeed a [[Clifford module bundle]] over $X$. Consider a Dirac operator

\begin{displaymath}
D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E})
\end{displaymath}
with components (with respect to the $\mathbb{Z}_2$-[[graded vector space|grading]]) to be denoted

\begin{displaymath}
D = 
  \left[
    \itexarray{
       0 & D^-
       \\
       D^+ & 0
    }
  \right]
  \,,
\end{displaymath}
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{displaymath}
\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}
  \,.
\end{displaymath}
\end{prop}
This appears as (\href{BerlineGetzlerVergne04}{Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50}), based on (\hyperlink{MacKeanSinger67}{MacKean-Singer 67}).

\begin{remark}
\label{}\hypertarget{}{}
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

\begin{itemize}%
\item index = partition function .

\end{itemize}
\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Kähler-Dirac operator]]


\item [[Dolbeault-Dirac operator]]


\item [[Spin{\tt \symbol{94}}c Dirac operator]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Dirac spinor]], [[Dirac field]]


\item [[Dirac equation]]


\item [[spin{\tt \symbol{94}}c Dirac operator]]


\item [[index of a Dirac operator]]


\item [[Fredholm module]], [[K-homology]], [[KK-theory]]


\item [[Lichnerowicz formula]]


\item [[Dirac-Ramond operator]], [[Dirac operator on smooth loop space]]


\item [[Feynman slash notation]]



\end{itemize}
[[!include genera and partition functions - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Textbooks include

\begin{itemize}%
\item [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], \emph{[[Spin geometry]]}, Princeton University Press (1989)


\item [[Thomas Friedrich]], \emph{Dirac operators in Riemannian geometry}, Graduate studies in mathematics 25, AMS (1997)



\end{itemize}
The relation to [[index theory]] is discussed in

\begin{itemize}%
\item [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], \emph{Heat Kernels and Dirac Operators}, Springer Verlag Berlin (2004)

\end{itemize}
based on original articles such as

\begin{itemize}%
\item H. MacKean, [[Isadore Singer]], \emph{Curvature and eigenvalues of the Laplacian}, J. Diff. Geom. 1 (1967)

\end{itemize}
\begin{itemize}%
\item [[Michael Atiyah]], [[Raoul Bott]], V. K. Patodi, \emph{On the heat equation and the index theorem}, Invent. Math. 19 (1973), 279--330.

\end{itemize}
See also

\begin{itemize}%
\item [[Dan Freed]], \emph{Geometry of Dirac operators} (\href{http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item C. Nash, \emph{Differential topology and quantum field theory}, Acad. Press 1991.


\item [[Eckhard Meinrenken]], \emph{Clifford algebras and Lie groups}, Lecture Notes, University of Toronto, Fall 2009.


\item Jing-Song Huang, Pavle Pandi, J.-S. Huang, P. Pandzic, Dirac Operators in Representation Theory,. Birkh\"a{}user, Boston, 2006, 199 pages; short version \emph{Dirac operators in representation theory}, 48 pp. \href{http://www.ims.nus.edu.sg/Programs/liegroups/files/singnotes.pdf}{pdf}


\item J.-S. Huang, Pavle Pandi, \emph{Dirac cohomology, unitary representations and a proof of a conjecture of Vogan}, J. Amer. Math. Soc. \textbf{15} (2002), 185---202.


\item R. Parthasarathy, \emph{Dirac operator and the discrete series}, Ann. of Math. \textbf{96} (1972), 1-30.



\end{itemize}
[[!redirects Dirac operators]]



\end{document}