Comment on the Surface Exponential for Tensor Fields

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 $\mathcal{E}(B|I|t_{\gamma })$, 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).

6 pages

References in corpus (6)

• On Background Independent Open-String Field Theory
• WDVV-like equations in N=2 SUSY Yang-Mills Theory
• Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
• Free-Field Representation of Group Element for Simple Quantum Group
• Bogolubov's Recursion and Integrability of Effective Actions
• Hopf algebra of ribbon graphs and renormalization

Cited by in corpus (4)

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