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The mirror map in mirror symmetry is the change-of-variables between the natural coordinatizations on the two mirror sides and is typically a highly-complicated transcendental function (indeed, should be something like a ratio of periods). Nonetheless, it has been noticed that the Taylor coefficients of the expansion of this function about the large complex-structure point often have surprising integrality properties (Lian-Yau, Zudilin, and now more recently I've found Krattenthaler-Rivoal and Delaygue). I have some questions about this phenomenon.

Question 1. Do we know any examples where the mirror map does not seem to have integral Taylor coefficients?

Next, it seems that many of the cases where we do know (either conjecturally by computation or by proof) integrality is in the case that the periods satisfy a Gelfand-Kapranov-Zelevinsky (GKZ) system. I'm not sure exactly when this happen -- I think for complete-intersection Calabi-Yaus in toric varieties, or on the physics side for (abelian?) gauged linear \sigma-models (GLSMs). This case often seems to be suspiciously nicer and I'm wondering if we only know of integrality in this case.

Question 2. Do we know of any examples of integrality of mirror map Taylor coefficients outside of GLSMs/examples arising from a GKZ system?

I'm mostly interested in the cases of compact Calabi-Yau threefolds, but I'm sure I'll find anything related to be of interest. Speculation, further references, pure straight knowledge, corrections to my understanding above, or general philosophizing are all appreciated!

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  • $\begingroup$ See arxiv.org/abs/1110.4439 and references (and follow-ups) there for the case of toric Calabi-Yau varieties. The fact that the natural algebraic parameters on the complex side are expressed as power series in the exponentiated Kähler parameters on the symplectic side with coefficients given by counts of holomorphic disks with boundary on SYZ fibers should be general, even if I am not sure if a precise statement is written somewhere. The idea going back to SYZ is that the mirror is a moduli space of A-branes and so receives corrections from open worldsheet instantons. $\endgroup$
    – user25309
    May 23, 2017 at 22:47
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    $\begingroup$ For complete intersections in toric varieties, one has standard coordinates for the complex moduli space of the mirror, arising from toric geometry - e.g. the mirror quintic is described as an explicit family over $\mathbb{P}^1$. The mirror map is the base change from these coordinates to the canonical coordinates - the ones that have intrinsic mirror symmetric meaning. But outside such a context, I'm not sure what integrality of the mirror map would mean, because I don't know which coordinates on the moduli space, near the LCSL, one would want to compare to the canonical coordinates. $\endgroup$
    – Tim Perutz
    May 24, 2017 at 2:55

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