Toroidal p-branes, anharmonic oscillators and (hyper)elliptic solutions

TL;DR

This paper investigates the exact solvability of p-brane equations in higher-dimensional Minkowski space, revealing integrable cases and solutions expressed through elliptic and hyperelliptic functions, and explores their physical implications.

Contribution

It introduces a new invariant ansatz for p-brane solutions, reduces the problem to relativistic anharmonic oscillators, and finds explicit solutions in special cases using elliptic and hyperelliptic functions.

Findings

p-brane equations are integrable for degenerate p-torus with equal radii

Solutions are expressed in elliptic ($p=2$) or hyperelliptic ($p>2$) functions

Contracting p-brane solutions depend on p and energy density

Abstract

Exact solvability of brane equations is studied, and a new $U(1)\times U(1)\times... \times U(1)$ invariant anzats for the solution of $p$-brane equations in $D=(2p+1)$-dimensional Minkowski space is proposed. The reduction of the $p$-brane Hamiltonian to the Hamiltonian of $p$-dimensional relativistic anharmonic oscillator with the monomial potential of the degree equal to $2p$ is revealed. For the case of degenerate p-torus with equal radii it is shown that the $p$-brane equations are integrable and their solutions are expressed in terms of elliptic ($p=2$) or hyperelliptic ($p>2$) functions. The solution describes contracting $p$-brane with the contraction time depending on $p$ and the brane energy density. The toroidal brane elasticity is found to break down linear Hooke law as it takes place for the anharmonic elasticity of smectic liquid crystals.