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\begin{document}

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\section*{fermionic path integral}

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

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{setup}{Setup}\dotfill \pageref*{setup} \linebreak
\noindent\hyperlink{pfaffian_bundles}{Pfaffian bundles}\dotfill \pageref*{pfaffian_bundles} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak


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

The [[path integral]] over the [[fermion]]ic variables of the standard kinetic [[action functional]] for [[fermion]]s (see for instance [[spinors in Yang-Mills theory]]) has a well-defined meaning as a [[section]] of the [[Pfaffian line bundle]] of the corresponding [[Dirac operator]].

\hypertarget{setup}{}\subsection*{{Setup}}\label{setup}

For definiteness, we consider a [[sigma model]] [[quantum field theory]] on a worldvolume $\Sigma$ and [[pseudo-Riemannian manifold|pseudo-Riemannian]] target [[spacetime]] $X$ with fields

\begin{itemize}%
\item [[boson]]s: [[smooth function]]s $\phi : \Sigma \to X$


\item [[fermion]]s ([[gravitino]]): $\psi \in \Gamma(S(\Sigma) \otimes \phi^* T M)$, [[section]]s of the [[tensor product]] of a [[spinor bundle]] on $\Sigma$ and the [[pullback]] of the [[cotangent bundle]] of $X$ along the given bosonic field $\phi$.



\end{itemize}
The [[action functional]]

\begin{displaymath}
S : (\phi, \psi) \mapsto S^{bos}(\phi) + S^{ferm}_{\phi}(\psi)
\end{displaymath}
is the sum of the

\begin{itemize}%
\item bosonic action

\begin{displaymath}
S^{bos})(\phi) 
   = 
  \int_\Sigma \langle d \phi \wedge \star d \phi\rangle
\end{displaymath}

\item fermionic action

\begin{displaymath}
S^{ferm}_\phi(\psi)
   = \int_\Sigma \langle \psi, D_\phi \psi\rangle
\end{displaymath}
where $D_\phi$ is a [[Dirac operator]] on $S \otimes \phi^* T M$ (the Dirac operator on $S$ twisted by the pullback of the [[Levi-Civita connection]] on $T^* X$ ).



\end{itemize}
One imagines than that the hypothetical [[path integral]] symboilically written as

\begin{displaymath}
\int [d \phi] [d \psi] \exp(S(\psi)(\phi,\psi))
\end{displaymath}
can be computed in two steps

\begin{displaymath}
\cdots = \int [d \phi] \exp(S^{bos}(\phi)) 
   \left(
     \int [d \psi]
     \exp(S^{ferm}_\phi(\psi))
   \right)
\end{displaymath}
by first computing the integral over the fermions

\begin{displaymath}
pfaff(\phi)
   := 
  \int [d \psi]
     \exp(S^{ferm}_\phi(\psi))
\end{displaymath}
and then inserting this into the remaining bosonic integral. Now, as opposed to the bosonic integral, this fermionic integral can be given well-defined sense by interpreting it as an infinite-dimensional [[Berezinian integral]].

However, while this makes the expression well defined, the result is not quite a function of $\phi$, but is instead a [[section]] $pfaff$ of a [[Pfaffian line bundle]]

\begin{displaymath}
\itexarray{
    && Pfaff 
     \\
     {}^{pfaff := Z_{eff}^{ferm}}\nearrow & \downarrow 
    \\
   C^{\infty}(\Sigma, X)
 &= &  C^{\infty}(\Sigma, X)
  }
\end{displaymath}
over the space of bosonic field configurations.

If $Pfaff$ is not isomorphic to the trivial line bundle, we say the system has a fermionic [[quantum anomaly]]. If instead $Pfaff$ is trivializable, any choice of trivialization

\begin{displaymath}
t : Paff \stackrel{\simeq}{\to} C^\infty(\Sigma, X) \times \mathbb{C}
\end{displaymath}
makes the fermionic path integral into a genuine function

\begin{displaymath}
Z_{eff}^{ferm}
   : = 
 (t \circ pfaff) :
  C^\infty(\Sigma, X) \to \mathbb{C}
  \,.
\end{displaymath}
Any such choice of $t$ is called a choice of [[quantum integrand]].

With this one can then try to enter the remaining bosonic path integral

\begin{displaymath}
\int [d \phi] \exp(S^{bos}(\phi)) Z_{eff}^{ferm}(\phi)
\end{displaymath}
\hypertarget{pfaffian_bundles}{}\subsection*{{Pfaffian bundles}}\label{pfaffian_bundles}

\begin{quote}%
We are implicitly assuming that $dim \Sigma = 2$ or maybe $8 n + 2$ in the following. Needs to be generalized.


\end{quote}
For $n \in \mathbb{N}$, there the [[square root]] of a skew symmetric $(n\times n)$-[[matrix]] $D$ -- the [[Pfaffian]] of the matric -- can be understood as the [[Berezinian integral]]

\begin{displaymath}
pfaff(D) 
    =  
   \int [d \vec \theta]
   \exp( \langle \theta , D \theta\rangle)
  \in
  det \mathbb{R}^n
\end{displaymath}
over the [[Grassmann algebra]] elements $\theta_i$. Written this way this is an element of the [[determinant line]] of $\mathbb{R}^n$: its identification with a number depends on the choice of basis for $\mathbb{R}^n$. For this case this is unproblematic, since there is a canonical choice of basis for the single vector space $\mathbb{R}^n$, but when $D$ instead depends on a parameter $\phi$, then in general its Pfaffian can at best be a section of a [[determinant line bundle]].

We now generalize this to the case that $D$ is not a finite-dimensional matrix, but a [[Dirac operator]] acting on spaces of sections of a [[spinor bundle]]. We discuss that we can reduce this ``infinite-dimensional matrix'' in a sense \emph{locally} to a finite dimensional one in a consistent way, such that the above ordinary construction of Pfaffians applies.

In the above setup, write

\begin{displaymath}
\mathcal{H}_\phi^{\pm} := \Gamma(S^{\pm} \otimes \phi^* T^* X)
\end{displaymath}
for the space of spinor sections for given $\phi : \Sigma \to X$. Then the choral [[Dirac operator]]s a maps

\begin{displaymath}
D_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi
  \,.
\end{displaymath}
We also have a ``quaternionic structure''

\begin{displaymath}
J_\phi^{\pm} : \mathcal{H}_\phi^{\pm} \to \mathcal{H}^{\mp}_\phi
\end{displaymath}
Define then an [[open cover]] of the space $C^\infty(\Sigma,X)$ of the space of bosonic fields with [[open set]]s $U_\mu$ for $(0 \leq \mu)$ given by

\begin{displaymath}
U_\mu := \{ \phi \in C^\infty(\Sigma,X)  | \mu \nin Spec D_\phi^2\}
  \,,
\end{displaymath}
hence the collection of bosonic field configurations such that $\mu$ is not in the [[operator spectrum]] of the squared [[Dirac operator]].

Over these open subsets we have the \emph{finite} [[rank]] [[vector bundle]]s

\begin{displaymath}
\mathcal{H}_\phi^{\mu \pm} 
  :=
   \oplus_{0 \leq \epsilon \leq \mu}
  Eig(D_\phi^2, \epsilon)
\end{displaymath}
of [[eigenspace]]s of $D_\phi^2$ for [[eigenvalue]]s bounded by $\mu$.

The Dirac operator that we are interested in is

\begin{displaymath}
D_\phi^\mu
  :=
  J_\phi^- \circ D_\phi^+ : \mathcal{H}_\phi^{\mu,+} \to \mathcal{H}_\phi^{\mu,+}
 \,.
\end{displaymath}
This defines now a finite-dimensional [[matrix]]

\begin{displaymath}
\langle -, D_\phi^\mu -\rangle
\end{displaymath}
whose [[Berezinian integral]] is the [[Pfaffian]]

\begin{displaymath}
\int [d \psi] \exp(\langle \psi , D^\mu_\phi \phi \rangle )
  =
  pfaff(D^\mu_\phi) \in det \mathcal{H}^{\mu \pi}_\phi
  \,.
\end{displaymath}
One shows that these constructions for each $\mu$ glue together to define

\begin{itemize}%
\item a smooth [[line bundle]] $Pfaff \to C^\infty(\Sigma, X)$


\item with a smooth [[section]] $pfaff(D)$.



\end{itemize}
Moreover, there is canonically a [[hermitean metric]] and a canonical unitary [[connection on a bundle]] (the [[Freed-Bismut connection]]) on this bundle.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

For the [[sigma model]] describing the [[brane|heterotic superstring]] propagating on a [[pseudo-Riemannian manifold]] $X$, the trivialization of the Pfaffian line bundle, hence the cancellation of its fermionic [[quantum anomaly]] is related to the existence of a (twisted) [[differential string structure]] on $X$. See there for more details.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[path integral]]


\item [[spinors in Yang-Mills theory]]



\end{itemize}
[[!redirects fermionic path integrals]]



\end{document}