On the Integrability Aspects of the Self Dual Membrane

TL;DR

This paper explores the quantum integrability of spherical membranes in flat spacetime, revealing connections to SU(infty) SDYM equations, Toda molecules, and infinite-dimensional algebras, advancing understanding of membrane quantum states.

Contribution

It demonstrates that spherical membranes admit integrable solutions linked to SU(infty) SDYM equations and constructs the associated algebraic structures and quantum states.

Findings

Integrable solutions linked to SU(infty) SDYM equations.

Explicit construction of the W(infty) algebra and its representations.

Development of quantum states including discrete states.

Abstract

The exact quantum integrability aspects of a sector of the membrane is investigated. It is found that spherical membranes moving in flat target spacetime backgrounds admit a class of integrable solutions linked to SU(infty) SDYM equations (dimensionally reduced to one temporal dimension). After a suitable ansatz, the SDYM equations can be recast in the form of the continuous Toda molecule equations whose symmetry algebra is the dimensional reduction of the W (infty} plus {\bar W}(infty} algebra. The latter algebra is explicitly constructed. Highest weight representations are built directly from the infinite number of defining relations among the highest weight states of W(\infty) algebras and the quantum states of the Toda molecule. Discrete states are also constructed. The full (dimensionaly reduced) quantum SU(infty) YM theory remains to be explored.