Comment on the Surface Exponential for Tensor Fields

TL;DR

This paper proposes a multi-parametric exponential map for tensor-valued 2-forms, extending the non-commutative exponential to higher-rank tensor fields, with potential applications in multi-directional evolution operators.

Contribution

It introduces a novel multi-parametric exponential for tensor fields, generalizing the P-exponent to higher ranks and emphasizing commutative algebra backgrounds.

Findings

Defines a multi-parametric exponential for tensor-valued 2-forms.

Restricts construction to backgrounds with commutative algebra structure constants.

Provides a framework for multi-directional evolution operators in tensor field theory.

Abstract

Starting from essentially commutative exponential map $E(B|I)$ for generic tensor-valued 2-forms $B$, introduced in \cite{Akh} as direct generalization of the ordinary non-commutative $P$-exponent for 1-forms with values in matrices (i.e. in tensors of rank 2), we suggest a non-trivial but multi-parametric exponential ${\cal E}(B|I|t_\gamma)$, which can serve as an interesting multi-directional evolution operator in the case of higher ranks. To emphasize the most important aspects of the story, construction is restricted to backgrounds $I_{ijk}$, associated with the structure constants of {\it commutative} associative algebras, what makes it unsensitive to topology of the 2d surface. Boundary effects are also eliminated (straightfoward generalization is needed to incorporate them).