$$p$$-Adic Fractal Strings of Arbitrary Rational Dimensions and Cantor Strings

The Geometry of p-Adic Fractal Strings: A Comparative Survey

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the tubular neighborhood of a p-adic fractal string Lp, expressed in terms of th...

Contemporary MathematicsComplex Dimensions of Fractal Strings

We put the theory of Dirichlet series and integrals in the geometric setting of`fractal strings' (one-dimensional drums with fractal boundary). The poles of a Dirichlet series thus acquire the geometric meaning of`complex dimensions' of the associated fractal string, and they describe the geometric and spectral oscillations of this string by means of anèxplicit formula'. We deene thèspectral op...

Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation

On the relation between p-adic and ordinary strings

The amplitudes for the tree-level scattering of the open string tachyons, generalised to the field of p-adic numbers, define the p-adic string theory. There is empirical evidence of its relation to the ordinary string theory in the p → 1 limit. We revisit this limit from a worldsheet perspective and argue that it is naturally thought of as a continuum limit in the sense of the renormalisation g...

Exact Noncommutative Solitons in p -Adic Strings and BSFT

The tachyon field of p-adic string theory is made noncommutative by replacing ordinary products with noncommutative products in its exact effective action. The same is done for the boundary string field theory, treated as the p → 1 limit of the p-adic string. Solitonic lumps corresponding to D-branes are obtained for all values of the noncommutative parameter θ. This is in contrast to usual sca...