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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 $\mathbb{C}$.

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I'm going to follow you around reading your questions from now on, Kevin. Always informative and entertaining! – GS Jan 14 '10 at 10:10
Upvoted for the punctuation in the title, although the question deserves it anyway. – Harrison Brown Jan 14 '10 at 21:58
Note that I'm not asking about p-adic physics. But I welcome someone else to ask a question about it... – Kevin H. Lin Jan 28 '10 at 3:44
up vote 8 down vote accepted

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.

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By "Batyrev's combinatorial mirror symmetry", do you mean the Hodge diamond mirror symmetry? – Kevin H. Lin Jan 31 '10 at 22:58
Well, it says quite generally that the family of CY hypersurfaces (or c.i. as per Batyrev-Borisov) in the first toric variety is the mirror for that in the second one. Then you can apply this to all kinds of things. For example, degenerations of the CYs in the first family correspond to deformations of CYs in the second family. Etc. Yes, that includes the duality between the Hodge and Betti numbers. There are so many "mirror symmetries": combinatorial, homological... Some of them are more char-p friendly than others. – VA. Jan 31 '10 at 23:11
Yeah, I'm aware of the different "mirror symmetries", but as I say in the original question, what I'm most interested in is whether the "mirror conjecture" of Candelas et. al. is at all characteristic $p$ friendly. Do you know anything about this? – Kevin H. Lin Feb 1 '10 at 22:08
In the work of Candelas-de la Ossa-Green-Parkes, the mirror symmetry is between the complex deformations on one side and the deformations of the (complexified) Kahler parameter of its mirror. The first one you can replace by algebraic deformations; char-p is OK. But the second one is essentially non-algebraic and needs C. Here, I don't know what to do, sorry. Perhaps mine is not the best answer here (personally, I upvoted Ben-Zvi's answer). – VA. Feb 2 '10 at 0:56

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.

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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, sets $k = \mathbf{Z}/2\mathbf{Z}$.

For a quartic surface, 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?

Caveat emptor: I have no idea, but I think it would be interesting to find out.

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You seem to be suggesting that various notions from symplectic geometry, such as Fukaya categories, can be defined for ... $p$-adic manifolds? – Kevin H. Lin Feb 1 '10 at 1:13
All the manifolds are still regular symplectic manifolds. We just change the coefficient field used in Floer homology. All we change is the field that is the receptacle for the counts of the numbers of pseudo-holomorphic polygons. For the elliptic curve case, we compare the Fukaya category, with coefficients in a $p$-adic field, of a real 2-torus to the bounded derived category of coherent sheaves of some other elliptic curve defined over that same $p$-adic field. – Matthew Ballard Feb 1 '10 at 1:46
Ok, I see, thanks. – Kevin H. Lin Feb 1 '10 at 2:33

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.

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In re your second paragraph: Where does arithmetic come in? I don't see any arithmetic in Atiyah-Bott nor in Narasimhan-Seshadri. – Kevin H. Lin Jan 14 '10 at 6:00
Forgive me in advance if this is an impertinent comment (I know literally nothing about string theory), but you do realize that the $p$-adics have characteristic $0$, right? Can you reduce string theory modulo an ideal?? – Pete L. Clark Jan 14 '10 at 6:55
@Kevin: A-B use Morse theory to compute cohomology of the space of flat connections mod gauge, which by N-S is the moduli space of stable bundles, whose Betti numbers (for general rank) had been found by Harder-Narasimhan by char p methods (Weil conjectures). (BTW, the gauge theory proof of N-S itself is Donaldson's rather than Atiyah-Bott's.) – Tim Perutz Jan 14 '10 at 14:01
@Pete: you're completely right, of course. I took "mod p" not literally as finite field, but as arithmetic in general. This is what happens when write a post in a rush before going on a trip :( – José Figueroa-O'Farrill Jan 14 '10 at 20:54
@Tim: Thanks. Sounds a lot like the number theory <-> curves over F_q <-> Riemann surfaces <-> physics "dictionary" which is explained here – Kevin H. Lin Jan 15 '10 at 1:32

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."

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In re the Kato quote: It doesn't seem clear to me whether, e.g., zeta values directly have something to do with physics. Many arithmetic things are probably analogous to things in physics (see Whether they actually are physics is, I think, a much stronger claim that needs much more evidence. – Kevin H. Lin Jan 14 '10 at 9:46
Hazewinkel is quoting Manin from 'Reflections on Arithmetical Physics' in 'Mathematics as Metaphor'. – David Corfield Jan 14 '10 at 11:52
Thanks, David! (BTW, do such google-book links work, or are they only temporary links?) – Thomas Riepe Jan 14 '10 at 12:40
The link works fine for me – Jonas Morrissey Jan 14 '10 at 17:38

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