Noncommutative self-dual supersymmetric Yang–Mills theory

Hitoshi Nishino,Subhash Rajpoot

doi:10.1016/j.physletb.2003.07.086

Abstract

We formulate noncommutative self-dual N=4 supersymmetric Yang–Mills theory in D=2+2 dimensions. As in the corresponding commutative case, this theory can serve as the possible master theory of all the noncommutative supersymmetric integrable models in lower dimensions. As a by-product, noncommutative self-dual N=2 supersymmetric Yang–Mills theory is obtained in D=2+2. We also perform a dimensional reduction of the N=2 theory further into N=(2,2) in D=1+1, as a basis for more general future applications. As a typical example, we show how noncommutative integrable matrix N=(1,0) supersymmetric KdV equations in D=1+1 arise from this theory, via the Yang–Mills gauge groups GL(n,R) or SL(2n,R).

Highlights

Noncommutative geometry has attracted attention nowadays, after the discovery of its importance in terms of noncommutative gauge theories [1] associated with M-theory and/or superstring theory.Based on a completely different motivation, there has been a long-standing conjecture [2] that all of the integrable systems in lower dimensions, such as KdV equations, KP hierarchies, Liouville equations, or Toda theories, are generated by four-dimensional (4D) self-dual Yang-Mills (SDYM) theory3 [3], which serves as a ‘master theory’ of lower dimensional integrable models
We have presented the formulation of noncommutative N = 4 supersymmetric self-dual Yang-Mills (SSDYM) in D = 2 + 2 for the first time
This may well serve as the ‘master theory’ of all the lower-dimensional noncommutative supersymmetric integrable models, as the corresponding commutative case [4][5][6] can do

Summary

Introduction

Noncommutative geometry has attracted attention nowadays, after the discovery of its importance in terms of noncommutative gauge theories [1] associated with M-theory and/or superstring theory. Other supersymmetric integrable models, such as supersymmetric KP systems are shown to be generated from SSDYM in 4D [7] Motivated by these two different developments, there have been works combining noncommutative gauge theories and integrable models [8]. A formulation of noncommutative SDYM has been established, with dimensional reductions to chiral field model and Hitchin equations [9]. Considering these developments, it is a natural step to seek a possible noncommutative version of ‘master theory’ generating all the integrable supersymmetric systems in lowerdimensions. For a compact gauge group, all the generators τI are anti-hermitian, and all the fields such as AμI are hermitian.

The complete set of field equations in our system is

Concluding Remarks