Comment on the Surface Exponential for Tensor Fields

Starting from essentially commutative exponential map $E(B\mathrm{\mid }I)$ 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 $\mathcal{E}(B\mathrm{\mid }I\mathrm{\mid }{t}_{\gamma })$, 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).