Extended Complex Trigonometry in Relation to Integrable 2D-Quantum Field Theories and Duality

Multicomplex numbers of order n have an associated trigonometry (multisine functions with (n-1) parameters) leading to a natural extension of the sine-Gordon model. The parameters are constrained from the requirement of local current conservation. In two dimensions for n < 6 known integrable models (deformed Toda and non-linear sigma, pure affine Toda…) with dual counterparts are obtained in this way from the multicomplex space MC itself and from the natural embedding ${\text{MC}}_n\subset {\text{MMC}}_m,n<m$. For $n\ge 6$ a generic constraint on the space of parametersis obtained from current conservation at first order in the interaction Lagragien.