I was reading some references on the slave-particle approach to the Kondo problem and Anderson model. It is known that the slave-particle is introduced in the large Hubbard $U$ limit of the system so that the Hilbert space for the localized orbital (e.g., $f$ electron) on a single site only include: $\{ | 0 \rangle, |\uparrow \rangle, |\downarrow\rangle \}$.
The constraint $n_f \le 1$ then becomes $$b^\dagger b + \sum_{\sigma} s^\dagger_\sigma s_\sigma =1$$ (here $b$ is boson and $s_\sigma$ is fermion). In the path integral, people say the measure of the integration becomes:
$$\int D(s^\dagger,s)D(b^\dagger,b)\delta(b^\dagger b + \sum_{\sigma} s^\dagger_\sigma s_\sigma-1)$$
and a Lagrange multiplier is then introduced which replace the Dirac delta function above by: $$\int D\lambda \ e^{i\lambda(b^\dagger b + \sum_{\sigma} s^\dagger_\sigma s_\sigma-1)}.$$
Here I would like to ask how does the imaginary time dependence entered into the constraint? Because originally we just have a constraint of the particle number operators, which can be transformed into a delta function at a specific time, say, at $\tau = 0$.