Fermions, their path integrals and effective actions

Question

I just read the nice exposition Fermionic Path Integral on nLab and began to wonder about some details to which references appear to be lacking. Suppose we live on Euclidean space as in the Osterwalder-Schrader approach to QFT:

• Is there a deeper analogy between fermionic and bosonic integration? For bosons, we should consider e.g the space $\mathcal{S}'$ of Schwartz distributions as "path space". What is the corresponding - presumably non-commutative - fermionic space $\mathcal{F}'$?
• If such an analogy exists, what is the corresponding one between probability measures $\mu$ on $\mathcal{S}'$ and Berezin integrals $\nu$ on $\mathcal{F}'$?
• Is there an analogy to the spaces physicists would like to work on? i.e The space $\mathcal{S}$ of Schwartz functions would be nice and obliterate the need for regularisation/renormalisation, but unfortunately we have to work on $\mathcal{S}'$ instead. If such an analogy exists, what is $\mathcal{F}$?
• Finally, what features should an interaction $S^{\mathrm{int}}$ have in order to make \begin{equation} S^{\mathrm{eff}} \left( \phi \right) = - \ln \int_{\mathcal{F}'} \exp \left[ - S^{\mathrm{int}} \left( \phi, \psi \right) \right] \mathrm{d} \nu_{\mathrm{Berezin}} \left( \psi \right) \end{equation} well-defined? i.e How to integrate out a fermion?

EDIT: For the analogies, I was expecting something along the lines:

• For bosons, we encode $\mathbb{R}^4$ by a commutative ring of test functions on $\mathbb{R}^4$, e.g $\mathcal{S}$
• For fermions, I would expect something similar to give a non-commutative ring $\mathcal{F}$
• For bosons, consider the cylindrical measure of a free theory on $\mathcal{S}$ which extends to a measure on $\mathcal{S}'$
• For fermions, consider some weird (in some sense positive) linear functional on $C_b( \mathcal{F} )$ that is somehow compatible with the non-commutativity of $\mathcal{F}$, expose a failure of regularity and show that the failure disappears upon prolonging to $C_b( \mathcal{F}' )$ where $\mathcal{F}'$ is something even more untangible than $\mathcal{F}$ - but has a concrete description

Answer 1

The fermionic path integral is not an integral in the analytic sense, and that's good so because therefore we can evaluate it rigorously. So, this is quite unrelated to measures or distributions, and I think that there is no analogy like the one you are looking for.

Your last point can be answered completely, more or less along the lines of the nlab page you cite. The setting has be so the bosons $\phi$ parameterize a family of Dirac operators $D_\phi$ acting usually on a space of $L^2$-sections of a spinor bundle. Associated to this family of Dirac operators must be a Pfaffian line bundle $Pfaff(D)$ over the space of all bosons $\phi$.

It is not so that you would have any choice for the fermionic action functional, it only works for the standard functional $$ S^{int}(\phi,\psi) := \int \langle \psi,D_\phi \psi \rangle dvol_g, $$ where the integral is w.r.t. a volume form and orientation on the worldvolumes of fields. The reason is that the - a priori not well-defined - expression $$ \int_\psi e^{S^{int}(\phi,\psi)} d\psi $$ can then be interpreted as an element in $Pfaff(D)$ in the fibre over $\phi$. This has to do with the fact that $\langle -,D_\phi -\rangle$ is skew-hermitian (provided the setting is correctly set up), and Paffians are concerned with skew-hermitian operators. Under this interpretation, the map $$ \phi \mapsto \int_\psi e^{S^{int}(\phi,\psi)} d\psi $$ a well defined section (probably with zeros) of $Pfaff(D)$.

Further discussion ("anomaly cancellation") is then concerned with the question how the Paffian line bundle can be trivialized, so that this section becomes a complex-valued function on the space of bosons. Once this is achieved, one may multiply this function with any bosonic action functional; this gives the full integrand under the bosonic path integral.

For example, a spin structure on spacetime trivializes the Pfaffian line bundle in worldvolume dimension one, while a (geometric) string structure trivializes the Pfaffian line bundle in worldvolume dimension two.

Some reference for this are:

Freed, Daniel S.; Moore, Gregory W., Setting the quantum integrand of M-theory, Commun. Math. Phys. 263, No. 1, 89-132 (2006). ZBL1124.58011.

Freed, Daniel S., Determinants, torsion, and strings, Commun. Math. Phys. 107, 483-513 (1986). ZBL0606.58013.

Bunke, Ulrich, String structures and trivialisations of a Pfaffian line bundle, Commun. Math. Phys. 307, No. 3, 675-712 (2011). ZBL1238.58022.

I have also talked a lot about this, and have slides on my webpage, e.g. these.