There's something that's been confusing me about symplectic field theory, and I'm not sure if I can formulate it exactly, but I'll try asking an imprecise question.
Symplectic field theory, as described in the seminal paper (https://arxiv.org/abs/math/0010059) of Eliashberg-Givental-Hofer, appears superficially very similar to Gromov-Witten theory, in that the mathematical definitions of SFT correlators involve integration over moduli spaces of pseudoholomorphic curves. And in fact, SFT provides new ways of computing correlators in Gromov-Witten theory.
If you told me that you had some sort of "field-theoretic" construction similar to Gromov-Witten theory which counted holomorphic curves, I might guess that this thing had the structure of a 2d field theory, as Gromov-Witten theory does. (Gromov-Witten theory has the structure of a 2D "cohomological field theory," as described in http://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf. As one ought to expect, this is a 2-dimensional sort of structure structure, having to do with the moduli space of stable curves.) But SFT is not a 2-dimensional field theory; in fact, SFT is exactly what it sounds like: it is a functor from a certain category of $2n$-dimensional symplectic cobordisms. In other words, SFT is a $(2n)$-dimensional symplectic field theory, where we don't seem to have specified anywhere a choice of $n.$
This strikes me as really weird! My imprecise question is twofold: first, where did the "2-dimensional" aspect of the theory go? And second, how can it be that the constructions in SFT produce a $2n$-dimensional "field theory" for all $n$? From my experience with TFT and QFT, I expect that interesting theories live in a single specified dimension and have something to do with that definition. What makes SFT so different?
I have one guess, which is that in Gromov-Witten theory & related constructions, we specify a space $X$, and then to a surface $\Sigma$ the theory assigns a count of holomorphic maps $\Sigma\to X,$ and this has the locality properties on $\Sigma$ which are necessary to construct a topological field theory (or some related structure); but in SFT, somehow we are using a sort of locality property on the symplectic manifold $X$ which has something to do with contact hypersurfaces in $X$. If this is what's going on, then a slightly more precise version of my question is: what structure of a symplectic manifold allows us to do this? And why is SFT the only place where I've seen some version of this construction?