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

TL;DR

This paper extends complex trigonometry using multicomplex numbers to develop new integrable 2D quantum field models, revealing dualities and constraints in parameter spaces for models with n<6 and n≥6.

Contribution

It introduces a multicomplex-based extension of sine-Gordon models, establishing new integrable models and dualities in 2D quantum field theories with parameter constraints.

Findings

Known integrable models for n<6 are derived from multicomplex space.

Dual models are identified through natural embeddings of multicomplex spaces.

Parameter constraints are established for models with n≥6 based on current conservation.

Abstract

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 $\MC_n \subset \MMC_m, n < m$. For $ n \ge 6$ a generic constraint on the space of parametersis obtained from current conservation at first order in the interaction Lagragien.