Democratic Formulation of Supergravity

• The democratic formulation of supergravity is a framework that treats all form fields and their duals equally, avoiding any a priori selection of fundamental variables.
• It employs methodologies like generalized geometry, Free Differential Algebras, and Lₙ-infinity algebras to encode gauge symmetries and duality constraints.
• The approach unifies superspace formulations, BV formalism, and pseudoactions across dimensions, providing deep insights into duality, extended symmetries, and M-theory structures.

The democratic formulation of supergravity is a paradigm in which all form fields and their duals are treated on an equal footing, both at the level of the Lagrangian and in the underlying algebraic structure, often implemented through geometric and cohomological frameworks such as generalised geometry, Free Differential Algebras (FDA), and their mathematical avatars, the $L_\infty$-algebras. This approach eschews any a priori selection of "fundamental" variables (electric versus magnetic potentials, metric versus vielbein, etc.), and instead emphasizes formulations in which all relevant degrees of freedom and their duals, together with the entire hierarchy of gauge symmetries and constraints, are captured simultaneously by the action principle, the Batalin–Vilkovisky (BV) formalism, or generalized action functionals.

1. Geometric and Cohomological Foundations

The geometric approach forms the backbone of the democratic formulation, assembling all physical fields—including the graviton, gravitino, and higher-rank antisymmetric tensors—into differential forms on (super)group manifolds or superspace (Andrianopoli et al., 22 Apr 2024). The theory's dynamics are recast in terms of generalised Maurer–Cartan (MC) equations, or their extension as FDA: $$d\,\Theta^{A(p)} + \frac{1}{n!}\,C^{A(p)}_{\,B_1(p_1)\cdots B_n(p_n)}\;\Theta^{B_1(p_1)}\wedge\cdots\wedge \Theta^{B_n(p_n)} = 0$$ This framework bypasses the need for a metric or Hodge dual in constructing the action; duality is incorporated algebraically in the structure of the FDA itself. Notably, in $D = 11$ supergravity, the FDA encodes the graviton, gravitino, a 3-form $A^{(3)}$, and their duals (e.g., a 6-form $B^{(6)}$), while Bianchi identities and constraints are handled through Chevalley-Eilenberg cocycles and their Hodge duals (Grassi, 2023).

Mathematically, FDAs are equivalent to strong homotopy Lie ($L_\infty$) algebras, where the generalized MC equations implement the nilpotency constraints (strong homotopy Jacobi identities) of such algebras (Andrianopoli et al., 22 Apr 2024).

2. Superspace, Integral Forms, and Democratic Actions

In superspace formulations, the democratic principle is implemented by encoding all supergravity fields (physical and auxiliary) and their duals as superforms. The construction of actions proceeds via the ectoplasm (or superform) method: supersymmetric invariants and equations are written as integrals of closed superforms of appropriate degree. For instance, in $D=10$, $N=1$ supergravity with Lorentz Chern–Simons form, the construction in superspace involves imposing rigid torsion constraints and solving the Bianchi identities exactly to all orders in $\alpha'$ (0802.3869).

Integral forms and picture-changing operators (PCOs) furnish a coordinate-free scheme to define superspace actions on supermanifolds, interpolating naturally between the superfield and component (spacetime) formulations (Castellani et al., 2016, Grassi, 2023). In this context, the action takes the form: $$S = \int_{M^{(d|n)}} \mathcal{L}^{(d|0)} \wedge Y^{(0|n)}$$ where $Y^{(0|n)}$ is a PCO localizing integration along odd directions. This formalism ensures full "democratic" treatment of fields by distributing superspace integration evenly between all degrees.

3. Democratic Pseudoactions and Duality Constraints

Modern treatments, especially for type II supergravities, construct universal democratic Lagrangians for the bosonic sector by incorporating all Ramond–Ramond (RR) field strengths as inhomogeneous forms: $$F_{IIA} = F_2 + F_4 + F_6 + F_8 + F_{10}, \quad F_{IIB} = F_1 + F_3 + F_5 + F_7 + F_9$$ and organizing their dynamics through a pseudoaction (Mkrtchyan et al., 2022): $$S_{R}(A, R, a) = \frac{1}{2} \int_M \Big( (F + a Q, \star (F + a Q) ) + 2 (F, a Q) \Big)$$ with $F = DA$ and $Q = DR$, and $\star^2 = 1$ (duality operator). The auxiliary scalar $a$ encodes self-duality constraints (notably for $F_5$ in type IIB). Manifest duality invariance is present, and in type IIB, the formulation can make $SL(2, \mathbb{R})$ symmetry explicit. Demanding the self-duality only at the level of equations of motion ensures no overcounting of degrees of freedom.

The democratic pseudoaction for scalars—relevant in four-dimensional maximal and half-maximal supergravities—is constructed by doubling the variables, including both scalars and their dual $(d-2)$-form potentials, with gauge-invariant kinetic terms parameterized by a matrix built from the target space Killing vectors (Fernandez-Melgarejo et al., 2023): $$S = S_{\text{grav}} + \frac{1}{2} \int d\phi^x \wedge \star d\phi^y + \frac{1}{4} \int M_{AB} G^A \wedge \star G^B + \ldots,$$ ensuring $G$-invariance without mismatch of true propagating degrees.

4. Democratic Formulation in Supergravity BV and Generalized Geometry

The Batalin–Vilkovisky (BV) formalism is applied in full generality to the democratic formulation, as recently demonstrated for $\mathcal{N}=1$ supergravity in $D=10$ (Kupka et al., 8 Aug 2025). Here, the field space is organized as a fibration: bosonic fields (generalized metrics, half-densities) form the base, while fermionic fields (gravitinos, dilatinos) are the fibers. The use of generalized geometry, notably Courant algebroids, enables encoding all bosonic fields—including the metric, B-field, and gauge fields—within a unified geometric object.

Symmetry transformations, including supersymmetry and generalized diffeomorphisms, act as vector fields on the field space. The supersymmetry algebra closes only on-shell in the fermionic sector, but the inclusion of quadratic antifield terms (in the BV action) ensures the classical master equation is satisfied. The proper connection on the field space is necessary for comparing field-dependent spinors, and curvature terms generated thereby are crucial for the correct structure of the supersymmetry algebra. This yields an efficient BV action with no need for local Lorentz ghost fields; all degrees of freedom and gauge redundancies are treated in a fully democratic way.

5. Physical Implications: Cohomology, Duality, and Emergent Symmetry

The democratic approach clarifies the relationship between field content, cohomology, and duality symmetries. Chevalley–Eilenberg cocycles in the relevant FDA encode physical field strengths (e.g., 4- and 7-cocycles for $D=11$ supergravity), while their super Hodge duals and cohomological constraints select unique representatives relevant for the action (Grassi, 2023). The trivializability of certain cocycles through introduction of further higher-degree forms is directly linked to the presence of extended objects (e.g., membranes, 5-branes) and to hidden/super/graded Lie algebras underlying "M-algebra" structure (Andrianopoli et al., 22 Apr 2024).

By formulating supergravity entirely in terms of differential forms and their free differential algebra, the democratic approach naturally unifies electric and magnetic sectors, revealing hidden symmetries and providing a geometric rationale for duality and central extensions in supersymmetry algebras.

When applied to scalar sectors parametrizing symmetric spaces $G/H$ (with $G$ global symmetry), democratic dualization further enables the construction of pseudoactions with manifest $G$-covariance; an explicit instance is the SL(2,$\mathbb{R}$)-invariant pseudoaction for type IIB supergravity containing all dual 0-, 2-, 4-, 6-, and 8-forms (Fernandez-Melgarejo et al., 2023).

6. Extensions and Special Models

The democratic formulation extends beyond higher-dimensional supergravity to lower-dimensional examples and to different contraction limits:

• In two-dimensional $N=2$ Carroll dilaton supergravity, both "democratic" and "despotic" versions arise from different contraction procedures of the Lorentzian superalgebra. The democratic version treats both supercharges symmetrically and retains a clean BF/gPSM structure (Grumiller et al., 26 Sep 2024).
• The dilatonic subsector of $D=10$ supergravity, with only the dilaton and dilatino dynamical, provides a simplified but structurally faithful arena. The theory remains invariant under generalized diffeomorphisms and local supersymmetry; its full BV extension encodes the democratic algebraic structure and serves as a laboratory for understanding the full theory (Kupka et al., 26 Aug 2024).

7. Historical and Mathematical Context

The identification of the geometric approach with a hidden graded Lie algebra (or $L_\infty$-algebra) has profound implications. For $D=11$ supergravity, the FDA underlying the theory is shown to be a deformation of the $DF$-algebra, embedding p-brane charges in the algebra of supersymmetry generators (Andrianopoli et al., 22 Apr 2024). This suggests a deep and inseparable connection between the structure of democratic formulations and the algebraic backbone of supergravity, providing a rationale for emergent symmetries and dualities in M-theory and related frameworks.

The democratic paradigm is not only a modern technical device but a natural mathematical outcome of formulating supergravity and its extensions in the most general and geometrically transparent terms.

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In summary, the democratic formulation of supergravity provides a universal formalism where all fields and their duals are incorporated symmetrically, implemented through geometric, cohomological, and $L_\infty$-algebraic structures. This approach naturally encodes the full set of gauge symmetries, unifies the treatment of physical and dual degrees of freedom, and serves as an essential bridge between algebraic, geometric, and physical insights in supergravity and string/M-theory (0802.3869, Mkrtchyan et al., 2022, Grassi, 2023, Fernandez-Melgarejo et al., 2023, Andrianopoli et al., 22 Apr 2024, Kupka et al., 8 Aug 2025).