The complete action for $\mathcal{N}=2$ de Sitter pure supergravity

Technical Summary: The Complete Action for N = 2 de Sitter Pure Supergravity

Problem Statement
The paper addresses the long-standing challenge of constructing a consistent, real Lagrangian for pure supergravity in four-dimensional de Sitter (dS$_4$) spacetime. While supergravity theories naturally favor negative cosmological constants (anti-de Sitter), the observed universe possesses a positive cosmological constant. Previous attempts, most notably the seminal work by Pilch, van Nieuwenhuizen, and Sohnius (PvN) [10], identified that a consistent $N=2$ supergravity in dS$_4$ might exist but required a "ghostly" graviphoton (a vector field with a wrong-sign kinetic term). However, the PvN paper left the theory incomplete, providing an action invariant under local supersymmetry only up to terms quartic in fermions. Furthermore, the issue of unitarity remained ambiguous; while the ghostly graviphoton was known to introduce negative-norm states, the status of the gravitino sector in de Sitter space was not fully clarified in the context of the full theory.

Methodology
The authors employ a rigorous field-theoretic approach combining modern cohomological techniques with the antifield formalism to reconstruct the theory from first principles:

1. Free Theory Analysis: They begin by analyzing the free theory on a dS$_4$ background containing a Dirac massless spin-3/2 field (gravitino), a real spin-2 field (graviton), and a real spin-1 field (graviphoton). They utilize recent results [29, 34, 35] to demonstrate that the quantization of the free Dirac gravitino on Lorentzian dS$_4$ leads to indefinite-norm states, a feature previously overlooked in the context of the full interacting theory.
2. Cubic Interactions via Noether Method: Using the cohomological reformulation of the Noether method [32, 48], the authors classify all possible local, cubic interactions among the fields. They derive the necessary deformations to gauge transformations and the gauge algebra.
3. Algebraic Constraints: By enforcing the Jacobi identities for the gauge algebra, they establish strict constraints on the coupling constants. This analysis determines the relative signs of the kinetic terms for the spin-1 and spin-2 fields without requiring prior knowledge of the quartic fermion terms.
4. Construction of the Full Action: Inspired by the construction of $N=2$ supergravity in anti-de Sitter space [57, 58], the authors propose a complete action. They verify that this action is invariant under local supersymmetry to all orders in the fermionic fields, explicitly constructing the previously missing quartic terms.

Key Contributions and Results

• Existence and Uniqueness: The paper rigorously proves the existence and uniqueness of the simplest possible supergravity theory in dS$_4$. The theory is identified as $N=2$ pure supergravity.
• Complete Lagrangian: The authors provide the unique, complete Lagrangian invariant under $N=2$ local supersymmetry. Unlike the work of PvN [10], this action is valid to all orders in the coupling constant (Newton's constant) and, crucially, includes all terms quartic in fermions.
• Clarification of Non-Unitarity: The study confirms that the graviphoton necessarily possesses a wrong-sign kinetic term, leading to negative-norm states. However, a significant new finding is that the gravitino sector also contributes to non-unitarity. The authors clarify that the massless gravitino on Lorentzian dS$_4$ inherently leads to indefinite-norm states upon quantization, meaning the graviphoton is not the sole source of non-unitarity in the theory.
• Superalgebra Structure: The work explicitly defines the $N=2$ de Sitter Lie superalgebra, with a bosonic subalgebra isomorphic to $\mathfrak{so}(4,1) \oplus \mathfrak{u}(1)$, and constructs the corresponding gauge fields and curvatures.

Significance and Outlook
The paper claims that its primary significance lies in completing the theoretical framework for $N=2$ de Sitter supergravity, which was left open for decades. By providing the full action, the authors set the necessary foundation for studying the theory in the Euclidean approach to quantum gravity.

The authors argue that while the Lorentzian theory suffers from non-unitarity (indefinite norms), these issues may be "harmless" or reinterpreted in the Euclidean signature. Citing recent results on the 1-loop sphere partition function [38], they suggest that in the Euclidean path integral, complex actions for the gravitino can yield real-valued results expressed in terms of unitary characters of the de Sitter group. Consequently, the complete action derived in this paper is presented as a prerequisite for investigating the consistency of pure $N=2$ de Sitter supergravity in Euclidean signature and for computing higher-point cosmological correlators for fermions. The work does not claim to resolve the non-renormalizability of gravity or the embedding of de Sitter vacua in String/M theory, but rather provides the specific, complete field-theoretic object required to further explore quantum gravity in de Sitter space.