Vacuum stability, string density of states and the Riemann zeta function
We study the distribution of graded degrees of freedom in classically stable oriented closed string vacua and use the Rankin-Selberg transform to link it to the finite one-loop vacuum energy. In particular, we find that the spectrum of physical excitations not only must enjoy asymptotic supersymmetry but actually, at very large mass, bosonic and fermionic states must follow a universal oscillating pattern, whose frequencies are related to the zeros of the Riemann ζ-function. Moreover, the convergence rate of the overall number of the graded degrees of freedom to the value of the vacuum energy is determined by the Riemann hypothesis. We discuss also attempts to obtain constraints in the case of tachyon-free open-string theories.
KeywordsSuperstrings and Heterotic Strings Superstring Vacua
- D. Zagier, Eisenstein series and the Riemann Zeta-function, in Automorphic forms, representation theory and arithmetic: papers, presented at the Bombay Colloquium 1979, Springer-Verlag (1981), pg. 303–355.Google Scholar
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.