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Mirror symmetry is, very roughly, a relation between symplectic geometry on one side and complex/algebraic geometry on the other side. For example, counts of pseudoholomorphic spheres in a closed symplectic manifold (Gromov-Witten invariants) are related to variations of the complex structure on a "mirror" complex manifold.

A natural generalization of the study of pseudoholomorphic curves in closed symplectic manifolds is given by the study of pseudoholomorphic curves in symplectic manifold with boundaries. From the point of view of symplectic geometry, it is natural to assume that these boundaries are contact manifolds and natural boundaries for pseudoholomorphic curves are provided by Reeb orbits in contact manifolds. Rather than to only consider closed symplectic manifolds and to associate numbers to them, it is therefore more natural to consider the category whose objects are contact manifolds and whose morphisms are symplectic cobordisms, and to associate vector spaces to contact manifolds ("contact homology") and to recover numbers associated to closed symplectic manifolds via cutting along contact submanifolds and gluing formulas, in accordance with the general philosophy of field theory. This program is called symplectic field theory (SFT) and was initiated in https://arxiv.org/abs/math/0010059 .

My question is: what is the mirror of symplectic field theory?

For example, what is the mirror of a contact manifold? what is the mirror of the operation of cutting a symplectic manifold along a contact submanifold? what is the mirror of contact homology?

These questions are maybe too naive and certainly too general (even in the "classical case", I don't know what mirror symmetry is (or even shoud be) in general). Any insights and/or concrete examples would be appreciated.

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A partial answer is as follows.

In order to make use of the theory of holomorphic curves (with boundaries or asymptotics, whatever), we should restrict ourselves to symplectic manifolds with convex boundary, or more restrictively, Liouville domains or Weinstein domains. These domains then admit completions into Liouville/Weinstein manifolds by attaching cylindrical ends. (Strictly speaking, the notion of a Liouville manifold should mean something more general, and even symplectic manifolds which are of infinite topological type.)

In the case of Weinstein domains, the relation between SFT and wrapped Fukaya categories is well-known. Explicitly, let $(M,\theta,\phi)$ be a Weinstein domain, with $\theta$ a contact form on $\partial M$ and $\phi:M\rightarrow\mathbb{R}$ the Morse function which determines the structure of $M$. Then $M$ can be realized as a handle body decomposition

$M=M_0\cup_{i=1}^k H_i,$

where $M_0=N^{2n-2}\times D^2$ (with its corners rounded off) is a subcritical Weinstein domain (with $N$ a $2n-2$-dimensional manifold with boundary), and $H_i$ are critical handles (in this case, $n$-handles).

For every $H_i$, there is a Legendrian attaching sphere $\Lambda_i$, so all together we get a Legendrian link $\Lambda=\cup_{i=1}^k\Lambda_i$. For $\Lambda$, there is a well defined $\mathbb{Z}$-graded DG-algebra over a semisimple ring $\mathbf{k}=\oplus_{i=1}^k\mathbb{K}$, namely the Chekanov-Eliashberg DG-algebra $\mathit{LHA}^\ast(\Lambda)$. One important result that can be extracted from the paper Effect of Legendrian surgery due to Bourgeois-Ekholm-Eliashberg is that $\mathit{LHA}^\ast(\Lambda)$ is quasi-isomorphic, as an $A_\infty$ algebra over $\mathbf{k}$, to the wrapped Fukaya category $\mathscr{W}(M)$ of the Weinstein domain $M$.

This quasi-isomorphism can be interpreted in another way, in the sense of Abouzaid's geometric generation criterion. Namely the Lagrangian cocores $L_1,\cdot\cdot\cdot,L_k\subset M$ associated to the handles $H_1,\cdot\cdot\cdot,H_k$ together generate the wrapped Fukaya category $\mathscr{W}(M)$.

Now this fact has a number of applications. For example, in the case when $M$ carries an exact symplectic Lefschetz fibration $\pi:M\rightarrow D^2$ (in fact, by a theorem of Giroux-Pardon, this is always the case for Weinstein manifolds), it's natural to take $M_0$ to be $F\times D^2$, where $F$ is a smooth fiber of $\pi$, then the above result can be applied to verify that $\mathscr{W}(M)$ is the localization of $\mathscr{F}(\pi)$ along a natural transformation $T:\mu\rightarrow\mathrm{id}$. Here $\mathscr{F}(\pi)$ is the directed $A_\infty$ category associated to the Lefschetz fibration $\pi$, $\mu$ is the autoequivalence on $\mathscr{F}(\pi)$ induced by the global monodromy of $\pi$.

In the context of homological mirror symmetry, $\pi:M\rightarrow D^2$ is usually mirror to a complete algebraic variety $X$ over a specified field $\mathbb{K}$ together with a (usually singular) divisor $D\subset X$. Denote by $Y$ the non-compact algebraic variety $X\setminus D$, which should be the putative mirror of $M$. The localization result above then reduces the verification of the triangulated equivalence

$D^\pi\mathscr{W}(M)\cong D^b\mathit{Coh}(Y)$

to two (usually much easier to verify) triangulated equivalences

$D^b\mathscr{F}(\pi)\cong D^b\mathit{Coh}(X)$

and

$D^\pi\mathscr{F}(F)\cong\mathit{Perf}(D)$,

where $\mathit{Perf}(D)\subset D^b\mathit{Coh}(D)$ is the triangulated category of perfect complexes. As $D$ is usually singular, it's usually not equivalent to $D^b\mathit{Coh}(D)$.

Conclusion. What I want to say is that most versions of mirror symmetry makes sense only in the homological setting (with the Gross-Siebert program as an exception), namely you should deal with invariants which come from Lagrangian Floer or SFT rather than working on the geometric level directly. Once you group these invariants together, those come from Floer theory and SFT are actually compatible with each other, and in many cases, they are just different ways of understanding the same thing, and combining these two viewpoints together do give you something interesting. On the other hand, if you think directly on the geometric level (say asking what is the mirror of certain surgeries in the symplectic or contact category) is kind of wild or unprofessional, and I don't think this is the correct way of doing things. What you should really look at is the effect of these surgeries on invariants, and try to figure out what kind of operations on the mirror have similar effect, then you would probably get some kind of surgery or operation which would conjecturally fit into the mirror correspondence. There are considerations of this flavor due to Katzarkov, Kerr, etc., they considered what happens to a Landau-Ginzburg model as its mirror goes through a birational cobordism.

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