Nambu-Poisson Bracket on Superspace

TL;DR

This paper extends the Nambu-Poisson bracket to superspaces using superdeterminants, proving its properties and analyzing its structure, including a new hi-bracket component, in superspaces R^{n|1} and R^{n|2}.

Contribution

It introduces a novel extension of n-ary Nambu-Poisson brackets to superspaces via superdeterminants, establishing their fundamental properties and structural decomposition.

Findings

The extended bracket satisfies skew-symmetry, Leibniz rule, and Filippov-Jacobi identity.

In R^{n|2}, the bracket decomposes into a standard Nambu-Poisson and a new hi-bracket.

The construction depends on invertible transformations of Grassmann coordinates.

Abstract

We propose an extension of n-ary Nambu-Poisson bracket to superspace R^{n|m} and construct by means of superdeterminant a family of Nambu-Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace R^{n|m}. We prove in the case of the superspaces R^{n|1} and R^{n|2} that our n-ary bracket, defined with the help of superdeterminant, satisfies the conditions for n-ary Nambu-Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov-Jacobi identity (fundamental identity). We study the structure of n-ary bracket defined with the help of superdeterminant in the case of superspace R^{n|2} and show that it is the sum of usual n-ary Nambu-Poisson bracket and a new n-ary bracket, which we call \chi-bracket, where \chi is the product of two odd degree smooth functions.