I am interested in (typically topological) field theories arising from Lagrangians of the form.
$f(\Phi) \lambda$,
where $\lambda$ is a Lagrange multiplier field not appearing in $f(\Phi)$. Perturbatively, the partition function just counts the solutions to $f(\Phi)=0$. What about nonperturbatively? Are there exponentially suppressed kinks that affect the partition function?
Has anyone considered these?
For a path integral, the integral
$$ \int e^{i\int \lambda(x) F(x)} d\lambda = \prod_x (2\pi) \delta(F(x)) $$
Where the product is over all points x (imagine a lattice). So that the result is not just a saddle point identity (saddle points and perturbative approximations are not the same, perturbative is by expanding the exponential in a series), it is an exact path integration identity.
While the result is not fully rigorous, that is only because of the issue of path integration definition, the limit of continuous space as the lattice gets finer. Ignoring this mathematical point, there is no way this identity is avoided in any proper definition of path integration, since it holds without any regulator ambiguity.
