On the Exact Quantum Integrability of the Membrane

TL;DR

This paper demonstrates that the spherical membrane in flat spacetime is exactly quantum integrable under specific conditions, linking it to advanced Toda theories and infinite-dimensional algebras, revealing both continuous and discrete energy spectra.

Contribution

It establishes the exact quantum integrability of the spherical membrane through a novel connection to $SU( olinebreak ext{infinity})$ Toda theory and $U_ olinebreak ext{infinity}$ algebra, providing explicit solutions and spectral analysis.

Findings

Identification of exact quantum integrability for spherical membranes.

Construction of the $U_ olinebreak ext{infinity}$ algebra from Toda theory solutions.

Discovery of both continuous and discrete energy levels in the spectrum.

Abstract

The exact quantum integrability problem of the membrane is investigated. It is found that the spherical membrane moving in flat target spacetime backgrounds is an exact quantum integrable system for a particular class of solutions of the light-cone gauge equations of motion : a dimensionally-reduced $SU(\infty)$ Yang-Mills theory to one temporal dimension. Crucial ingredients are the exact integrability property of the $3D~SU(\infty)$ continuous Toda theory and its associated dimensionally-reduced $SU(\infty)$ Toda $molecule$ equation whose symmetry algebra is the $U_\infty$ algebra obtained from a dimensional-reducion of the $W_\infty \oplus {\bar W}_\infty$ algebras that act naturally on the original $3D$ continuous Toda theory. The $U_\infty$ algebra is explicitly constructed in terms of exact quantum solutions of the quantized continuous Toda equation. Highest weight irreducible representations of the $W_\infty$ algebras are also studied in detail. Continuous and discrete energy levels are both found in the spectrum . Other relevant topics are discussed in the conclusion.