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

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.

18 pages, extended version accepted in Nucl. Phys. B; discussions on integrability and geometric approach added; correspondence between anharmonic elasticity in toroidal branes and smectic liquid crystals revealed; references and acknowledgements updated; typos corrected

References in corpus (3)

• Exact solutions for U(1) globally invariant membranes
• U(1)-invariant membranes: the geometric formulation, Abel and pendulum differential equations
• Extended Particles and the Exterior Calculus

Cited by in corpus (5)

• Integrability cases for the anharmonic oscillator equation
• Laplace-Beltrami operator and exact solutions for branes
• On brane symmetries
• $p$-Branes with $Ad{S}_{p+1}$ vacuum as models of $R^2$ gravity
• Branes as solutions of gauge theories in gravitational field