On the generalized Komar charge of Kaluza-Klein theories and higher-form symmetries

TL;DR

This work addresses how to define a conserved mass-charge in five-dimensional gravity with Kaluza-Klein boundary conditions by incorporating a higher-form global symmetry. The authors construct a one-parameter family of generalized Komar charges, showing that with the choice $\alpha = \tfrac{1}{2}$ the KK-related scalar contamination cancels and the KK circle integration yields the four-dimensional generalized Komar charge, including KK monopole contributions and electric-magnetic duality. They also show how to remove residual infinity-charge contributions by shifting the time-translation generator $l$ by $-\Phi_{\infty} k$, ensuring the infinity integral reproduces the 4D mass $M$. The results clarify the role of higher-form symmetries in KK theories and provide a framework for deriving consistent Smarr relations and first laws in higher dimensions, with possible extensions to matter couplings.

Abstract

The generalized Komar $(d-2)$-form charge can be modified by the addition of any other on-shell closed (conserved) $(d-2)$-form charge. We show that, with Kaluza--Klein boundary conditions, one has to add a charge related to the higher-form symmetry identified in Ref.~\cite{Gomez-Fayren:2024cpl} to the naive Komar charge of pure 5-dimensional gravity to obtain a conserved charge charge whose integral at spatial infinity gives the mass. The new term also contains the contribution of the Kaluza--Klein monopole charge leading to electric-magnetic duality invariance.