Massless $p2$-brane modes and the critical line

TL;DR

This paper models a p2-brane using a Bruhat-Tits tree framework, revealing that unitarity constrains spectral parameters to special loci including the critical line, with implications for interpreting Riemann zeta zeros as massless particles.

Contribution

It introduces a p2-brane model on a Bruhat-Tits tree, linking spectral localization to unitarity and connecting Riemann zeta zeros with massless particles in a novel theoretical setting.

Findings

Spectral parameters localize at special loci including the critical line.

Excitations resemble those of the bosonic string with conserved Poincaré algebra.

Riemann zeta zeros correspond to massless photon and graviton.

Abstract

We consider a $p2$-brane model as a theory of maps from the vertices of the Bruhat-Tits tree times $\mathbb{Z}$ into $\mathbb{R}^d$. We show that in order for the worldsheet time evolution to be unitary, a certain spectral parameter of the model must localize at special loci in the complex plane, which include the (0,1) interval and the critical line of real part $1/2$. The excitations of the model are supported at these loci, with commutation relations closely resembling those of the usual bosonic string. We show that the usual Hamiltonian, momentum, and angular momentum are conserved quantities, and the Poincar\'e algebra is obeyed. Assuming an Euler product relation for the spectrum, the Riemann zeta zeros on the critical line have spectral interpretation as the massless photon and graviton in the Archimedean theory.