Higher-form symmetries in supergravity, scalar charges and black-hole thermodynamics

TL;DR

The work develops a unified treatment of higher-form and ordinary symmetries in minimal 5d supergravity and the T^3 model in 4d, clarifying how generalized Komar charges encode black-hole thermodynamics across dimensions. By exploiting p-form gauge invariance, GCTs and global higher-form symmetries, it derives on-shell closed currents and charges, including scalar charges, and constructs Smarr formulas and first laws through a generalized Komar framework. Dimensional reduction reveals precise mappings between 5d and 4d charges, currents, and potentials, and exposes a KK-origin of new 4d structures while highlighting obstructions to closure that relate to horizon topology and possible solitonic, horizonless solutions. The results extend to Einstein–Maxwell–type theories with CS terms, indicating a broad relevance to string/M-theory compactifications (e.g., 11d supergravity) and illuminating the role of higher-form symmetries in black-object thermodynamics and no-hair properties.

Abstract

Minimal 5-dimensional supergravity compactified on a circle gives the T$^{3}$ model of $\mathcal{N}=2,d=4$ supergravity, whose duality group is SL$(2,\mathbb{R})$. We study exhaustively the relations between all the local and global symmetries of both theories and between the corresponding conserved currents and charges, including the on-shell closed generalized Komar charges associated to isometries. We find that the 2-dimensional subgroup of SL$(2,\mathbb{R})$ that does not include electric-magnetic transformations is realized as a higher-form symmetry group that acts on the 5-dimensional metric and vector field. Using the generalized Komar charges we compute the Smarr formulas for black holes, showing that they are identical once the relations between all the 5- and 4-dimensional thermodynamical quantities are taken into account, which is only possible if certain constraints on the fields are satisfied. We notice that on-shell closed 5-dimensional 3-form charges give, upon dimensional reduction, on-shell closed 3-form currents and 2-form charges. The dimensional reduction of the 5-dimensional generalized Komar 3-form charge associated to a Killing vector gives a new 4-dimensional on-shell closed 3-form current which must be associated to a new global symmetry of the theory when it admits that Killing vector. Some of the results that we have derived are valid for theories of Einstein--Maxwell-like theories of $(p+1)$-forms with Chern--Simons terms, which includes 11-dimensional supergravity as a particular example.