Suppose you had some infinite dimensional path integral, and you want to get some finite dimensional approximation. Physicist told you to get some effective theory with a cut-off, but with interaction coefficients depending on the cut-off scale.

Vivek says, can we do the samething. When ever you see the action $S$, you do Fukaya category with $S: (Fields) \to \C$. Can you compare this with $S_{eff}: (low modes fields) \to \C$?