Extension of sine-Gordon field theory from generalized Clifford algebras

TL;DR

This paper extends the sine-Gordon field theory into multicomplex spaces derived from generalized Clifford algebras, connecting it with integrable models like Liouville and Toda in specific dimensions.

Contribution

It introduces multicomplex extensions of sine-Gordon theory using generalized Clifford algebras, broadening the mathematical framework of integrable field theories.

Findings

Identification of n=1,2,3,4 cases with Liouville, sine-Gordon, and deformed Toda models

Construction of multicomplex trigonometric functions

Extension of sine-Gordon theory to higher-dimensional multicomplex spaces

Abstract

Linearization of homogeneous polynomials of degree n and k variables leads to generalized Clifford algebras. Multicomplex numbers are then introduced in analogy to complex numbers with respect to usual Clifford algebra. In turn multicomplex extensions of trigonometric functions are constructed in terms of `compact' and `non-compact' variables. It gives rise to the natural extension of the d-dimensional sine-Gordon field theory in the n-dimensional multicomplex space. In dimension 2, the cases n=1,2,3,4 are identified as the quantum integrable Liouville, sine-Gordon and known deformed Toda models. The general case is discussed.