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The path integral over the fermionic variables of the standard kinetic action functional for fermions (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.
For definiteness, we consider a sigma model quantum field theory on a worldvolume $\Sigma$ and pseudo-Riemannian target spacetime $X$ with fields
bosons: smooth functions $\phi : \Sigma \to X$
fermions (gravitino): $\psi \in \Gamma(S(\Sigma) \otimes \phi^* T M)$, sections of the tensor product of a spinor bundle on $\Sigma$ and the pullback of the cotangent bundle of $X$ along the given bosonic field $\phi$.
is the sum of the
bosonic action
fermionic action
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 $T^* X$ ).
One imagines than that the hypothetical path integral symboilically written as
can be computed in two steps
by first computing the integral over the fermions
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
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
makes the fermionic path integral into a genuine function
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
We are implicitly assuming that $dim \Sigma = 2$ or maybe $8 n + 2$ in the following. Needs to be generalized.
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
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 locally to a finite dimensional one in a consistent way, such that the above ordinary construction of Pfaffians applies.
In the above setup, write
for the space of spinor sections for given $\phi : \Sigma \to X$. Then the choral Dirac operators a maps
We also have a “quaternionic structure”
Define then an open cover of the space $C^\infty(\Sigma,X)$ of the space of bosonic fields with open sets $U_\mu$ for $(0 \leq \mu)$ given by
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 finite rank vector bundles
of eigenspaces of $D_\phi^2$ for eigenvalues bounded by $\mu$.
The Dirac operator that we are interested in is
This defines now a finite-dimensional matrix
whose Berezinian integral is the Pfaffian
One shows that these constructions for each $\mu$ glue together to define
a smooth line bundle $Pfaff \to C^\infty(\Sigma, X)$
with a smooth section $pfaff(D)$.
Moreover, there is canonically a hermitean metric? and a canonical unitary connection on a bundle (the Freed-Bismut connection?) on this bundle.
For the sigma model describing the 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.