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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?

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    $\begingroup$ For an analogy: consider the worldline formalism and an ordinary QFT. $\endgroup$ – AHusain Aug 30 '16 at 21:40
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    $\begingroup$ One can take a slightly broader view of what SFT is, giving rise to different algebraic structure. For example for a fixed completed Liouville domain, you can define a version of symplectic cohomology (but with holomorphic curves instead of Floer solutions) as a non-equivariant version of linearized contact homology. This has many of the same algebraic/TQFT structures as those defined using Gromov-Witten invariants (pair of pants product, homotopy S^1 action etc...). In the equivariant setting, you have arxiv.org/pdf/0706.3284.pdf and homotopy refinements by the same authors + Fukaya. $\endgroup$ – Daniel Pomerleano Aug 31 '16 at 1:20
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What follows is a guess. I don't know enough about Symplectic Field Theory to be sure.

In perturbative string theory, one describes physics in the target spacetime $X$ by summing over maps from Riemann surfaces into $X$. These sums have an interpretation in terms of a 2d CFT living on the Riemann surfaces, but for physicists, the 2d constructions are auxiliary. They're primarily interested in physics in the target space (and indeed, there are constructions of string theory, such as Matrix theory and AdS/CFT, which make no mention of worldsheets).

The physics on the worldsheet is local on the worldsheet, as you point out. The physics on the target spacetime $X$ isn't local in general -- that doesn't seem to be compatible with what we know of gravity -- but it does reduce to local quantum field theory on $X$ in some limits and special situations.

  • The most obvious example is the low-energy limit, where the target space string theory can be approximated by a gauge theory on $X$ or some compactification of $X$.

  • N=4 super Yang-Mills theory describes life in the 10d string theory spacetime as perceived by a 3-dimensional D3-brane embedded in the spacetime. (The 6 scalar fields on the 4d worldvolume parametrize the transverse fluctuations of the brane inside the 10d spacetime.)

  • Perhaps more interesting to mathematicians: Witten's realization of Chern-Simons theory on $M$ as the string theory whose worldsheet CFT is the open-string topological A-model with target space $X = T^*M$ (and appropriate boundary conditions!). This is physically degenerate, but mathematically rather nice: The target space physics is not just local, but actually topological!

You find this sort of construction everywhere in the string theory literature. (Since real string theory calculations are rather hard, most string theory 'consistency checks' involve finding two quantities that can be computed in QFT limits and checking that they match.)

I suspect that symplectic field theory is an example of the third sort: A topologically twisted A-model CFT with appropriate boundary conditions gives rise to a target space string theory which is actually a TQFT, namely symplectic field theory. I'd be more certain of this if I knew exactly which boundary conditions were involved.

The topological twist in the worldsheet CFT also explains the lack of dependence on the target dimension. For untwisted theories, there's a strong constraint on target space dimension. The 'conformal anomalies' of the 2d sigma model and 2d gravity have to sum to zero. In untwisted theories, the anomaly of the 2d sigma model is 3/2 $d$, with $d$ the dimension of the spacetime and the anomaly of 2d gravity is $-15$, leading to $d=10$. In topologically twisted theories, both anomalies vanish, so there's no constraint implied on target space dimension.

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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.

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  • $\begingroup$ Can you please explain the downvote? $\endgroup$ – Chris Gerig Aug 30 '16 at 23:57
  • $\begingroup$ I'm also curious. This answer seems like a reasonable response. $\endgroup$ – user1504 Aug 31 '16 at 0:14

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