ag.algebraic geometry - Mirror symmetry mod p?! ... Physics mod p?!

For fixed integers $g,n$, any projective scheme $X$ over a field $k$, and a linear map $beta:operatorname{Pic}(X)tomathbb Z$, 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 $overline{M}_{g,n}(X,beta) times_R mathbb Z/pmathbb Z$ is a Deligne-Mumford stack for almost all primes $p$. For such $p$, $overline{M}_{g,n}(X,beta) times_R mathbb Z/pmathbb Z$ 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.

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