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I know this is a very vague question, so I may restrict it to quantum theories for their more category theoretic setting. Even if the concept of crystalline cohomology is very abstract, it has made me wonder if there could be any applications of it to physical theories. This question arises from the growing use of higher category theory in many quantum theories and their mathematical analogs, for example homological mirror symmetry.

Question: Are there any applications of crystalline cohomology to physical theories?

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  • $\begingroup$ @Bertie I'm not a native english speaker, thanks for pointing out. $\endgroup$
    – Tatu
    Commented Apr 14, 2017 at 10:04
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    $\begingroup$ Is this the kind of thing you are looking for: arxiv.org/pdf/hep-th/0603106.pdf $\endgroup$
    – Jason Starr
    Commented Apr 14, 2017 at 10:05
  • $\begingroup$ @JasonStarr Somewhat yes, but it's not entirely crystalline cohomology, because there are other p-adic cohomology theories too. An interesting reference anyways. $\endgroup$
    – Tatu
    Commented Apr 14, 2017 at 10:13

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A talk that explores the physics connection to crystalline cohomology in the context of string theory is Motives and Strings by Jan Stienstra, with a challenge for each community:

Why look for a MOTIVE-STRING relation?

Computations in Type IIb string theory proceed by manipulating solutions of certain differential equations. During the computations there are many denominators. In the end these drop out and true integers remain. Many differential equations in Type IIb string can be recognized as Picard-Fuchs equations in De Rham cohomology of families of varieties. The integrality statements can be recognized as consequences of theorems about the crystalline cohomology of families of ordinary varieties.

Challenge for Motive people: Crystalline cohomology deals with only one prime $p$ at a time and puts out statements about $p$-adic integrality. What mechanism synchronizes the primes and leads to true integers?

Challenge for String people: Crystalline cohomology implies extra symmetries in the differential equations. Where are these extra symmetries in Nature?

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  • $\begingroup$ Are there any more references for this? The talk seems to be mostly about group theory, but gives a very interesting idea. $\endgroup$
    – Tatu
    Commented Apr 14, 2017 at 10:39

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