Contents

# Contents

## Idea

What is called $p$-adic string theory is the study of a variant of scattering amplitudes in string theory (hence string scattering amplitudes) where the worldsheet of the string is regarded not as a Riemann surface but as an object in p-adic geometry.

### Open string amplitudes

The original observation is that of (Volovich 87, reviewed in VVZ 95, section XIV) that the integral expression for the Veneziano amplitude of the open bosonic string naturally generalizes from an integral over the real numbers (which in this case parameterize the boundary of the open string worldsheet) to the p-adic numbers by adelic integration.

The standard Veneziano amplitude has the expression

$\int_{\mathbb{R}} {\vert x\vert}^{a-1} \cdot {\vert 1-x\vert}^{b-1} d x$

where the norm involved is the usual absolute value, and the proposed $p$-adic version is hence

$\int_{\mathbb{Q}_p} {\vert x\vert}_p^{a-1} \cdot {\vert 1-x\vert}_p^{b-1} d x$

The main result here is (Freund-Witten 87) that the ordinary Veneziano amplitude equals the inverse of the product of its $p$-adic versions, for all primes $p$, apparently a version of the idelic product formula.

With due regularization this result carries over to other string scattering amplitudes, too. When forming these products one also speaks of adelic string theory.

Since the Veneziano amplitude concerns the bosonic string tachyon state, p-adic string theory has been discussed a lot in the context of tachyon condensation and Sen's conjecture (Cottrell 02).

Traditionally literature on $p$-adic string theory asserts that the generalization of this from the open string to the closed string remains unclear (e.g CMZ 89, section 4, Cottrell 02, section 5), since it is unclear which adic version of the complex numbers to use. However, in other parts of the literature adic versions of closed strings are common, this we discuss below.

### Closed string 1-loop vacuum amplitudes (topological Witten genus)

Generally, the development of string theory has shown that its worldsheet is usefully regarded as an object in algebraic geometry (see also at number theory and physics) and mathematically the generalization from algebraic varieties over the complex numbers to more general algebraic varieties (or schemes) is often natural, if not compelling.

For instance when the Witten genus (essentially the partition function of the superstring) is refined to the string orientation of tmf then the elliptic curves over the complex numbers which serve as the toroidal worldsheets over the complex numbers are generalized to elliptic curves over general rings and by the fracture theorems the computations in tmf in fact typically proceed (see here) by decomposing the general problem into that of ellitpic curves over the rational numbers and over the p-adic integers. The result refines the Witten genus

$\Omega^String_{\bullet} \longrightarrow MF_\bullet$

(being a ring homomorphism) from the String structure cobordism ring to that of modular forms to one of E-∞ rings

$M String \longrightarrow tmf$

from the String structure Thom spectrum to tmf. Notice that $M String$ here classifies String-cobordism and hence parameterizes ordinary (not $p$-adic) target spacetime manifolds, while $tmf$ on the right does regard the genus-1 worldsheet as a general elliptic curve, hence in particular possibly as an elliptic curve over the p-adic integers.

The original articles include

The early history of the subject is recalled in

• Paul H. Frampton, Particle Theory at Chicago in Late Sixties and p-Adic Strings (arXiv:2001.10915)

That the ordinary Veneziano amplitude is the inverse product of all its $p$-adic versions is due to

also

A detailed discussion of $p$-adic open string scattering amplitudes is in

• L. Chekhov, A. Mironov, A. Zabrodin, Multiloop calculations in $p$-adic String theory and Bruhat-Tits Trees, Comm. Math. Phys. 125, 675-711 (1989) (Euclid)

A review of this is in

• Lee Brekke and Peter Freund, $p$-Adic numbers in physics, Phys. Rep. 233, 1 (1993) (web record)

and with an eye towards AdS-CFT duality in

Suggestion that the disk worldsheet of the open p-adic string is to be identified with the Bruhat-Tits tree $T_p$:

Discussion of tachyon condensation in $p$-adic string theory includes

• William Cottrell, $p$-adic Strings and Tachyon Condensation, 2002 (pdf)