# Does such kind of generalization of Feynman diagram exist?

We draw FD diagram on a paper which is essentially a $$2$$-$$d$$ space and then use perturbative Feynman rules to write the algebraic expression of the perturbative term. I want to ask if drawing it on a cylinder, a torus(problem of CTC), sphere, or any other curved surface will make some change in interpreting the FD.

I understand the rules to interpret FD arises by expanding order by order $$\frac{\langle0|T \Big\{ \phi_0(x_1)...\phi_0(x_n) e^{i\int d^4x\mathcal{L}_{i}[\phi_0]}\Big\}|0\rangle}{\langle0|T \Big\{e^{i\int d^4x\mathcal{L}_{i}[\phi_0]} \Big\}|0\rangle}$$

But my question arises from a statement about FD in Zee QFT book (chapter $$1$$ or $$2$$) "...Later Feynman diagrams acquire a life of their own..." he was probably referring to kinks and solitons (chapter $$18$$ most probably).

TL;DR So my question is whether taking an FD drawn/embedded on a curved surface and taking it as an independent entity rather than a rule invented to write the perturbative term compactly and succinctly; ever used in the literature? And if so how?

Zee's also probably thinking of large $$N_c$$ theories and planar diagrams. There's also the topological expansion of diagrams in string theory.

Perhaps more to your line of thought, I like to think of amplitudes squared as living on a cylinder with both a final state cut across the lines that match the final state particles from the amplitude and its complex conjugate (which is common to think of) but also with an initial state cut across the lines that match the initial state particles from the amplitude and its complex conjugate (which I think of as on the back side of the cylinder).