Noncommutative Self-Dual Supersymmetric Yang-Mills Theory

TL;DR

This paper formulates a noncommutative self-dual supersymmetric Yang-Mills theory in four dimensions, serving as a foundational framework for various lower-dimensional noncommutative integrable models, including supersymmetric KdV equations.

Contribution

It introduces a new noncommutative self-dual N=4 supersymmetric Yang-Mills theory in D=2+2 dimensions and explores its dimensional reductions to generate integrable models.

Findings

Derived noncommutative self-dual N=2 supersymmetric Yang-Mills theory in D=2+2.

Reduced the theory to N=(2,2) in D=1+1 dimensions.

Showed how noncommutative supersymmetric KdV equations emerge from the theory.

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).