Hodge Duals in Spherical Compactifications

TL;DR

This paper develops a systematic method for computing Hodge duals of differential forms constrained to spheres, crucial for Kaluza-Klein compactifications in supergravity and string theory.

Contribution

It introduces a general formalism for Hodge duals under spherical constraints, validated through examples, aiding compactification analyses.

Findings

Derived explicit formulas for constrained Hodge duals

Validated the formalism with examples in three dimensions

Facilitates consistent truncations in supergravity

Abstract

We present a general formalism for computing the Hodge dual of differential forms in arbitrary dimensions subject to a spherical constraint. This problem arises naturally in Kaluza-Klein compactifications, where sphere reductions demand careful treatment of differential forms constrained to lie on embedded submanifolds. We derive an explicit expression for the Hodge dual of a $p$-form in the presence of such constraints and validate our general ansatz through illustrative examples in three dimensions, including both flat and diagonal metric backgrounds. The resulting framework offers a systematic and practical tool for handling constrained Hodge duals, with direct applications to consistent truncations in supergravity and string theory compactifications.