Deformation Quantization of sdiff($\Sigma_{2}$) SDYM Equation

TL;DR

This paper explores the deformation quantization of the SDYM equation related to area-preserving diffeomorphisms, leading to a master equation with conserved charges, Lax pairs, hidden symmetries, and a twistor construction.

Contribution

It introduces the deformation quantization of the SDYM equation for sdiff(Σ₂), deriving the master equation with conserved charges, Lax pairs, and twistor methods.

Findings

Identification of conserved charges for the master equation

Construction of Lax pairs and hidden symmetries

Development of twistor construction for the deformed equation

Abstract

Deformation quantization (the Moyal deformation) of SDYM equation for the algebra of the area preserving diffeomorphisms of a 2-surface $\Sigma_{2}$, sdiff($\Sigma_{2}$), is studied. Deformed equation we call the master equation (ME) as it can be reduced to many integrable nonlinear equations in mathematical physics. Two sets of concerved charges for ME are found. Then the linear systems for ME (the Lax pairs) associated with the conserved charges are given. We obtain the dressing operators and the infinite algebra of hidden symmetries of ME. Twistor construction is also done.