Noncommutative Sine-gordon Model Extremizing the Sine-gordon Action

As I briefly review, the sine-Gordon model may be obtained by dimensional and algebraic reduction from 2+2 dimensional self-dual U(2) Yang-Mills through a 2+1 dimensional integrable U(2) sigma model. I argue that the noncommutative (Moyal) deformation of this procedure should relax the algebraic reduction from U(2) → U(1) to U(2) → U(1)×U(1). The result are novel noncommutative sine-Gordon equations for a pair of scalar fields. The dressing method is outlined for constructing its multi-soliton solutions. Finally, I look at tree-level amplitudes to demonstrate that this model possesses a factorizable and causal S-matrix in spite of its time-space noncommutativity. 1 Classical sine-Gordon model . . . Extremizing the sine-Gordon action S = 1 2 ∫ dt dy [ (∂tφ) 2 − (∂yφ) + 8α(cosφ− 1) ] (1) for a scalar field φ(t, y) on R with mass = 2α yields the sine-Gordon equation, (∂ t − ∂ y)φ + 4α sin φ = 0 . (2) This famous equation has many remarkable features, such as a Lax-pair or zero-curvature representation, infinitely many conserved local charges, a factorizable S-matrix without particle production, as well as soliton and breather solutions. The simplest soliton configuration (with velocity v) is kink-like, φkink(t, y) = 4 arctan e −2αη with η = y−vt √ 1−v2 . (3) For later use I introduce light-cone coordinates u := 1 2 (t+ y) , v := 1 2 (t− y) =⇒ ∂u = ∂t + ∂y , ∂v = ∂t − ∂y . (4) 1Talk presented at the XIIIth International Colloquium Integrable Systems and Quantum Groups in Prague 17-19 June 2004 and at the 37th International Symposium Ahrenshoop on the Theory of Elementary Particles in Berlin-Schmöckwitz 23-27 August 2004.