Mirror symmetry mod p?! ... Physics mod p?! In his answer to this question, Scott Carnahan mentions "mirror symmetry mod p". What is that? 
(Some kind of) Gromov-Witten invariants can be defined for varieties over fields other than $\mathbb{C}$. Moreover other things that come up in mirror symmetry, like variation of Hodge structure, and derived categories of coherent sheaves, also make sense. (Though I can't imagine that it's possible to talk about Fukaya categories...) Can we formulate any sort of sensible mirror symmetry statement, similar to say that of Candelas-de la Ossa-Green-Parkes relating Gromov-Witten invariants of a quintic threefold to variation of Hodge structure of the mirror variety, when the varieties are over some field other than $\mathbb{C}$? In particular, can we do anything like this for fields of positive characteristic?
I googled "arithmetic mirror symmetry" and "mirror symmetry mod p", and I found some stuff about the relationship between the arithmetic of mirror varieties, but nothing about Gromov-Witten invariants. I did find notes from the Candelas lectures that Scott referred to, but I wasn't able to figure out what was going on in them.
More generally, there are many examples of mathematical statements about complex algebraic varieties which come from physics/quantum field theory/string theory. Some of these statements (maybe with some modification) can still make sense if we replace "variety over $\mathbb{C}$ with "variety over $k$", where $k$ is some arbitrary field, or a field of positive characteristic, or whatever. Are there any such statements which have been proven?
Edit: I'm getting some answers, and they are all sound very interesting, but I'm still especially curious about whether anybody has done anything regarding Gromov-Witten invariants over fields other than $\mathbb{C}$.
 A: For fixed integers $g,n$, any projective scheme $X$ over a field $k$, and a linear map $\beta:\operatorname{Pic}(X)\to\mathbb Z$, the space $\overline{M}_{g,n}(X,\beta)$ of stable maps is well defined as an Artin stack with finite stabilizer, no matter the characteristic of $k$. You can even replace $k$ by $\mathbb Z$ if you like.
Now if $X$ is a smooth projective scheme over $R=\mathbb Z[1/N]$ for some integer $N$, then $\overline{M}_{g,n}(X,\beta) \times_R \mathbb Z/p\mathbb Z$ is a Deligne-Mumford stack for almost all primes $p$. For such $p$,  $\overline{M}_{g,n}(X,\beta) \times_R \mathbb Z/p\mathbb Z$ has a virtual fundamental cycle, and so you have well-defined Gromov-Witten invariants. This holds for all but finitely many $p$. Nothing about $\mathbb C$ here, that is my point, the construction is purely algebraic and very general.
It is when you say "Hodge structures" then you better work over $\mathbb C$, unless you mean $p$-Hodge structures. 
As far as mirror symmetry in characteristic $p$, much of it is again characteristic-free. For example Batyrev's combinatorial mirror symmetry for Calabi-Yau hypersurfaces in toric varieties is simply the duality between reflexive polytopes. You can do that in any characteristic, indeed over $\mathbb Z$ if you like.
A: To me, it seems that the operation mirror to the changing of the base field is the changing of coefficients in Floer homology. Let me give you some examples.
For the case when $k$ is any field, we have the following example:
Take $\mathbf{P}^2_k$ as our variety and its mirror $W: \left(\mathbf{C}^{\times}\right)^2 \rightarrow \mathbf{C}$, $W(x,y) = 1 + x + y - 1/xy$. For the Fukaya-Seidel category of vanishing cycles of $W$ take the coefficients in $k$ and forget about weighting by exponentials of the areas of holomorphic polygons. Then, the bounded derived category of coherent sheaves on $\mathbf{P}^2_k$ is equivalent the idempotent-completed derived Fukaya-Seidel category of $W$ with coefficients in $k$. In fact, the first statement of this result in writing, in Seidel's More on vanishing cycles and mutation (https://doi.org/10.1142/9789812799821_0012), sets $k = \mathbf{Z}/2\mathbf{Z}$.
For a quartic surface (Paul Seidel, Homological mirror symmetry for the quartic surface, published as https://doi.org/10.1090/memo/1116), we know one side of mirror symmetry holds when $k$ is the rational Novikov field over $\mathbf{C}$, $\Lambda_{\mathbf{Q}}$. Precisely, we have an equivalence between the idempotent-completed derived Fukaya category of a smooth quartic surface over $\mathbf{C}$, with coefficients in $\Lambda_{\mathbf{Q}}$, and the bounded derived category of the mirror of a smooth quartic surface over $\Lambda_{\mathbf{Q}}$. Here it seems perfectly plausible to replace $\mathbf{C}$ by $k$ again. However, there is a significant difference with the previous example. For $\mathbf{P}^2$, we never had to worry about convergence of the power series defining the products in the Fukaya category thanks to the exactness of everything in sight. But, here a lot of important questions are over $\mathbf{C}$ and depend on convergence. So it would make the most sense to take something like a $p$-adic field for $k$.
Changing coefficients may not seem very sexy and it probably will not have much to say about GW invariants of varieties over finite fields, but it may nonetheless provide interesting results. The first case to investigate: try mirror symmetry for an elliptic curve over a $p$-adic field. As first step, can one reproduce a statement like that of Polishchuk and Zaslow in Categorical Mirror Symmetry: The Elliptic Curve (https://dx.doi.org/10.4310/ATMP.1998.v2.n2.a9)?
Caveat emptor: I have no idea, but I think it would be interesting to find out.
A: Although not an answer to your question, strictly speaking, there has been some "physics mod p"  in the past.  In the 1980s there was some work on p-adic string theory.  If you google that you will find a number of articles on the subject.  People like Frampton and Volovich (father) have worked on this subject.  Even outside the realm of string theory, there has been some work on p-adic physics, seriously entertaining the notion of non-archimedian completions of the rationals, the rationale (no pun intended!) being that we are only reaslly able to measure positive rationals in the lab.
More generally still, the work of Atiyah and Bott on The Yang-Mills equations over Riemann surfaces, where they rederive using gauge-theoretic methods an earlier result of Narasimhan and Seshadri on the topology of the moduli space of holonomorphic bundles over a Riemann surface, suggests very strongly -- at least to Atiyah! -- a very strong relation between Physics and Arithmetic, which is still to be elucidated.
A: Hazewinkel quotes Y.I. Manin's "Reflections on Arithmetical Physics" as "main conjecture" in his "Niceness Theorems"
:
"On the fundamental level our world is neither real, nor p-adic; it is adèlic. For
some reasons reflecting the physical nature of our kind of living matter (e.g., the fact that we are built of massive particles), we tend to project the adèlic picture onto its real side. We can equally well spiritually project it upon its non-Archimedean side and calculate most important things arithmetically."
and gives examples and bibl. infos (not copied here): "There are applications of this idea to the Polyakov measure (Polyakov partition function), string theory, Yang-Mills theory, and much more. Add to this that the p-adic versions are often easier to handle and one finds some good justification for the discipline of p-adic physics."
Kazuya Kato writes in his lectures on Iwasawa theory: "Mysterious           properties           of   zeta      values     seem     to    tell         us    (in  a
not    so    loud     voice)       that     our    universe         has    the    same     properties:               The
universe         is   not     explained         just      by   real     numbers.       It    has     p-adic
properties           (as    is   claimed        by   some     people       in   physics)        and        it    is
related        to   profound        objects        which     we   calI     for    simplicity            the      crane,
the    galaxy       train,       and    the     homeland        of    zeta     values.    We  o u r s e l v e s         may
have     the     same     properties.
 Are     there      physical        meanings         of    zeta     elements?"
Edit: A new article on the arxiv on "arithmetic properties of field theories. In particular, we study the vacuum structure of supersymmetric gauge theories as algebraic varieties over number fields of finite characteristic." 
A: The most interesting answer I know to this question is the recent work of Albert Schwarz with Vadim Vologodsky, Ilya Shaprio and Maxim Kontsevich, in which for example they use properties of the Frobenius action on p-adic cohomology to establish properties of the mirror map, see e.g.
here,
here, or
here.
In another interesting direction there are the papers of Philip Candelas, Xenia de la Ossa and Fernando Rodriguez Villegas on Calabi-Yau manifolds over finite fields and "Dwork theory for physicists"
here,
here, and
here.
