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!
This post imported from StackExchange MathOverflow at 2023-09-02 11:17 (UTC), posted by SE-user Arnav Tripathy