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Preprint today by Yau et al., relating $p$-adic strings to the Riemann zeta function:
added more references. Should add some pointer to Bruhat-Tits trees. But no time now.
So this is developing the open bosonic corner you mention in the penultimate paragraph of your MO question?
Yes.
It sounds rather striking what they say about Riemann zeta zeros corresponding to the adelic string spectrum. But I haven’t had time yet to try to absorb it.
Yes, quite something if some aspect of the Riemann hypothesis emerges from one corner of a “more general number theoretic and homotopy-theoretic refinement of string scattering amplitudes”.
Is there anything deep in mathematics not touched by string/M-theory?
One thing I haven’t appreciated before is how a Bruhat-Tits building here serves as the disk-shaped worldsheet of the open string.
I have no idea how this relates to taking elliptic curves over arbitrary rings as closed string vacuum diagrams, as it happens in the construction of the string orientation of tmf.
To associate some fog with more fog, I wonder if topological Langlands is about here, relating arithmetic to homotopy theory. I see it gets a mention in Eric Peterson’s new book Formal Geometry and Bordism Operations footnote 18, p. 361.
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