Integrable deformations of the Dirac--sinh-Gordon system

Abstract

We construct a two-dimensional family of integrable coupled Dirac--scalar field theories in $1+1$ dimensions, parameterized by $(\thz,α)\in[0,π/2]^2$, whose Lax connection takes values in $\slC$ throughout. The family arises as the orbit of the Dirac--sinh-Gordon system under the $U(1)\times U(1)$ maximal torus of $\mathrm{Inn}(\slC)\cong PSL(2,\CC)$: the $\thz$-$U(1)$ rotates the constant phase of the Dirac mass; the $\hat{I}\pm$-$U(1)$ rotates its field-dependent phase via $β\to|β|\ee^{\imα}$. Integrability throughout the parameter space follows from a single principle: any automorphism of the Lax algebra preserves the zero-curvature condition, since the condition depends only on the Lie bracket of $\slC$.

22 pages, no figures