I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be a close relation between p-adic string theory and the refinement of the superstring partition function to what in mathematics is known as the String orientation of tmf. But I provide some more background first.

The very last words of the seminal article

  • Edward Witten, Elliptic Genera And Quantum Field Theory, Commun.Math.Phys. 109 525 (1987) (Euclid)

on the partition function of the superstring -- now called the Witten genus -- were the following:

A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means.

One of the breakthrough results in pure mathematics originally motivated by this was the construction of tmf and of the string orientation of tmf by Michael Hopkins and collaborators:

where the original Witten genus is a homomorphism

$$ Z_{superstring} : \Omega^{String}_\bullet \longrightarrow MF_\bullet $$

from the cobordism ring of spacetime manifolds with String structure (meaning: Green-Schwarz anomaly-free spacetimes) to the ring of modular forms (which are the possible 1-loop correlators, modular invariant up to the relevant anomaly), the string orientation of tmf (or "topological Witten genus") is a homomorphism of coherently homotopy-commutative ring spectra

$$ \sigma : M String \to tmf $$

from the String-structure Thom spectrum of cobordism cohomology theory to topological modular forms.

This is a homotopy-theoretic refinement of the Witten genus, the latter is the "decategorification" of $\sigma$ in that it is reproduced on homotopy groups:

$$ Z_{superstring} = \pi_\bullet(\sigma) \,, $$

a result due to

  • Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf).

This is maybe noteworthy in view of the quote above, since $tmf$ and its String-orientation have a fairly "god-given" origin right at the foundations of stable homotopy theory ("chromatic stable homotopy theory").

By some magic (rigorously proven magic, though) this fundamental abstract math knows the modular invariant of the heterotic superstring including its Green-Schwarz anomaly cancellation. And it knows something more, since the Witten genus is only the shadow of this on homotopy groups.

Now to come to my question, one thing that is maybe noteworthy here is that where the Witten genus $Z_{superstring}$ is built from ordinary string worldsheets being ordinary genus-1 Riemann surfaces, hence elliptic curves over the complex numbers, its homotopy-theoretic refinement $\sigma$ is built from all elliptic curves in the sense of algebraic geometry, hence elliptic curves over general base rings.

Moreover, in the standard construction of tmf via arithmetic fracture squares it is explicitly built from a piece over the rational numbers and one piece which for each prime number $p$ receives a contribution from p-adic elliptic curves.

Here an elliptic curve over the p-adic integers $\mathbb{Z}_p$ is in a precise sense a genus-1 closed string worldsheet, but not over the complex numbers, but over the $p$-adic integers.

This of course now reminds one of what is called p-adic string theory. This is a field driven by the observation that the integrals over the real numbers which give the scattering amplitudes of the open string -- where the real line parameterize its boundary -- still make sense and still are very interesting when one replaces the real numbers by the p-adic numbers $\mathbb{Q}_p$.

This means to regard the boundary of the open string as an object in p-adic geometry. In traditional literature on $p$-adic string theory it is usually stated that the generalization of the $p$-adic theory to closed strings remains unclear, since it remains unclear which adic version of the complex numbers (parameterizing the interior of the string worldsheet) one should use.

But now by the above, in algebraic geometry there is in fact an obvious concept at least of adic genus-1 closed string worldsheets: these are just the elliptic curves over the $p$-adics. And moreover, since the Witten genus, being the partition function, is a particular case of a string scattering amplitude, it follows by the above story of the String-orientation of $tmf$ that making this identification indeed does relate closed $p$-adic string worldsheets to actual physical closed string scattering (at least to the 1-loop vacuum amplitude).

In conclusion, it seems that there should be a close interrelation between p-adic string theory and the refinement of the superstring partition function to the string orientation of tmf.

My question, finally, is: has anyone made this close interrelation more explicit? Is there anything in the literature that makes this connection? Or else, if unpublished, has anyone seriously thought about this?

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    $\begingroup$ If I remember correctly (from Zabrodin's 2nd paper in CMP on the subject, with Chekhov?), the $p$-adic string theory literature so far only applies to Mumford curves which can be "uniformized". I don't know if there is a $p$-adic string theory for more general curves over $\mathbb{Q}_p$. $\endgroup$ Mar 21, 2021 at 16:48


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