The Hodge Operator Revisited
L. Castellani, R. Catenacci, and P.A. Grassi
TL;DR
This paper introduces a novel Fourier-integral-based construction of the Hodge operator on differential manifolds, providing a simple formula for the Hodge dual of wedge products, with extensions to supergeometry and non-commutative geometry.
Contribution
It presents a new Fourier (Berezin)-integral approach to defining the Hodge operator, simplifying calculations and extending applicability to advanced geometric frameworks.
Findings
Derived a simple formula for the Hodge dual of wedge products.
Extended the analysis to supergeometry and non-commutative geometry.
Provided a Fourier-integral representation for the Hodge operator.
Abstract
We present a new construction for the Hodge operator for differential manifolds based on a Fourier (Berezin)-integral representation. We find a simple formula for the Hodge dual of the wedge product of differential forms, using the (Berezin)-convolution. The present analysis is easily extended to supergeometry and to non-commutative geometry.
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Taxonomy
Topicsadvanced mathematical theories · Algebraic and Geometric Analysis