Return to Answer

The "2-dimensional" aspect concerns the 2-dimensional curves and their punctures/nodes, and this went into the SFT Hamiltonian/potential.

SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and that's what the "n-dimensional field theory" plays with. For GW we're analyze the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory.

So in each case (SFT versus GW) the field theory structure is about different things. This algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, "Symplectic field theory and its applications".

You also see this in Embedded Contact Homology, concerning contact 3-manifolds and symplectic 4-cobordisms. It recovers Taubes' Gromov invariant, and there is a forthcoming paper of Hutchings on "ECH as a field theory" (but you can find a sketch in his blog posts). This is expected, because ECH was inspired by SFT and Taubes' Gromov invariant.

The "2-dimensional" aspect concerns the 2-dimensional curves and their punctures/nodes, and this went into the SFT Hamiltonian/potential.

SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and that's what the "n-dimensional field theory" plays with (for each n). For GW we're analyzing the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory (independent of n). So in each case (SFT versus GW) the field theory structure is about different things. The algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, "Symplectic field theory and its applications".

You also see this in Embedded Contact Homology, concerning contact 3-manifolds and symplectic 4-cobordisms. It recovers Taubes' Gromov invariant, and there is a forthcoming paper of Hutchings on "ECH as a field theory" (you can find a sketch now in his blog posts). This is expected, because ECH was inspired by SFT and Taubes' Gromov invariant.

The "2-dimensional" aspect went into the SFT Hamiltonian/potential.

SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and that's what the "n-dimensional field theory" plays with. For GW we analyze the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory.

This algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, "Symplectic field theory and its applications".

You also see this in Embedded Contact Homology, concerning contact 3-manifolds and symplectic 4-manifolds. It can recover Taubes' Gromov invariant! Michael Hutchings will be uploading a paper soon about "ECH as a field theory" (but you can find a sketch in his blog posts). This is satisfying, because ECH was inspired by SFT and Taubes' Gromov invariant (and, unrelated to the point of this question, its relation to Seiberg-Witten theory). There is also a blog post of Hutchings concerning the possibility of extracting ECH from SFT.

The "2-dimensional" aspect concerns the 2-dimensional curves and their punctures/nodes, and this went into the SFT Hamiltonian/potential.

SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and that's what the "n-dimensional field theory" plays with. For GW we're analyze the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory.

So in each case (SFT versus GW) the field theory structure is about different things. This algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, "Symplectic field theory and its applications".

You also see this in Embedded Contact Homology, concerning contact 3-manifolds and symplectic 4-cobordisms. It recovers Taubes' Gromov invariant, and there is a forthcoming paper of Hutchings on "ECH as a field theory" (but you can find a sketch in his blog posts). This is expected, because ECH was inspired by SFT and Taubes' Gromov invariant.

The "2-dimensional" aspect went into the SFT Hamiltonian/potential.

SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and that's what the "n-dimensional field theory" plays with. For GW we analyze the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory.

This algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, "Symplectic field theory and its applications".

You also see this in Embedded Contact Homology, between contact 3-manifolds and symplectic 4-manifolds. It can recover Taubes' Gromov invariant! Michael Hutchings will be uploading a paper soon about "ECH as a field theory" (but you can find a sketch in his blog posts). This is satisfying, because ECH was inspired by SFT and Taubes' Gromov invariant (and, unrelated to the point of this question, its relation to Seiberg-Witten theory). There is also a blog post of Hutchings concerning the possibility of extracting ECH from SFT.

The "2-dimensional" aspect went into the SFT Hamiltonian/potential.

SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and that's what the "n-dimensional field theory" plays with. For GW we analyze the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory.

This algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, "Symplectic field theory and its applications".

You also see this in Embedded Contact Homology, concerning contact 3-manifolds and symplectic 4-manifolds. It can recover Taubes' Gromov invariant! Michael Hutchings will be uploading a paper soon about "ECH as a field theory" (but you can find a sketch in his blog posts). This is satisfying, because ECH was inspired by SFT and Taubes' Gromov invariant (and, unrelated to the point of this question, its relation to Seiberg-Witten theory). There is also a blog post of Hutchings concerning the possibility of extracting ECH from SFT.