Numbers and Strings

Abstract

Compact Riemann surfaces figure prominently in string theory where they appear as string world sheets. Compact Riemann surfaces are “algebraic curves” (curves, in that their complex dimension is one while their real dimension is two), and their algebraic geometry has strong arithmetic (number-theoretic) flavor. Some of this arithmetic geometry has already surfaced in discussions of higher loop diagrams in string theory. It is well known that in algebraic geometry it pays to be flexible concerning the nature of the number-field over which the algebraic curves (or varieties) are defined. Could one be equally flexible in physics, and envisage strings over the still locally compact but non-archimedean field of p-adic numbers? Could such non-archimedean strings shed light on the physically interesting archimedean string over the field R of real numbers? As we shall see ^1),2) this is indeed the case and the extremely simple non-archimedean strings can be viewed, essentially, as the basic building blocks of the much richer physical (archimedean) string. Before constructing such non-archimedean strings one has to confront three crucial decisions.

The actual talk I gave at the Boulder Workshop under the title “Non-archimedean Strings” presented only the work¹⁾ on closed non-archimedean strings by M. Olson and myself. The progress registered since, has made me opt here for a presentation centered on the simpler open non-archimedean string based on work by E. Witten and myself,²⁾ and including the adelic string constructed in reference 2). This work was started after the Boulder Workshop and causality prevented its presentation there. The present work was supported in part by the NSF Grant No. PHY 85-21588.