It is commonly believed that a unitary supersymmetric quantum field theory (QFT) involving graviton and gravitino fields on fixed 4-dimensional de Sitter spacetime (dS4) cannot exist due to known challenges associated with supersymmetry (SUSY) in dS4. In this paper, we contradict this expectation by presenting a new unitary supersymmetric QFT on dS4: the free supersymmetric theory of the chiral graviton and chiral gravitino fields. By chiral we mean that the corresponding field strengths are anti-self-dual, and the gauge potentials are complex, each carrying a single complex propagating degree of freedom. The global SUSY transformations are generated by the standard Dirac Killing spinors of dS4. The theory overcomes the known obstacles to unitary global SUSY on dS4 by closing the commutator between two SUSY transformations on so(4, 2) ⊕ u(1) rather than the de Sitter algebra so(4, 1). Crucially, the so(4, 2) symmetry is realised through unconventional conformal-like transformations. This free theory cannot become interacting while preserving SUSY in a way that makes the spin-2 sector the true graviton sector of General Relativity, as the three-graviton coupling cannot be u(1)-invariant.We establish the unitarity of the free supersymmetric theory in two complementary ways. First, by studying the action of the superalgebra generators on the space of physical gravitino and graviton mode solutions. In particular, we introduce positive-definite, invariant inner products and demonstrate that the SUSY representation is unitary, forming a direct sum of two unitary irreducible representations — one with negative-helicity modes and the other with positive-helicity modes. Second, by quantising the fields and explicitly constructing the complex quantum supercharges QA and QA†, we show that the trace <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\sum }_{A}\left\{{Q}_{A},{Q}^{A \dagger}\right\}\)</EquationSource> </InlineEquation> is positive-definite.Before constructing the supersymmetric theory, we examine the free graviton and gravitino fields on dS4, where the gravitino is known to have an imaginary mass parameter. We introduce a Hermitian, gauge-invariant, and local Lagrangian for the free gravitino field and explain why the requirement of unitarity forces the field to be chiral, removing half of the propagating helicity states.

Unitary rigid supersymmetry for the chiral graviton and chiral gravitino in de Sitter spacetime

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2025年12月12日

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Atsushi Higuchi,
Vasileios A. Letsios

摘要

It is commonly believed that a unitary supersymmetric quantum field theory (QFT) involving graviton and gravitino fields on fixed 4-dimensional de Sitter spacetime (dS₄) cannot exist due to known challenges associated with supersymmetry (SUSY) in dS₄. In this paper, we contradict this expectation by presenting a new unitary supersymmetric QFT on dS₄: the free supersymmetric theory of the chiral graviton and chiral gravitino fields. By chiral we mean that the corresponding field strengths are anti-self-dual, and the gauge potentials are complex, each carrying a single complex propagating degree of freedom. The global SUSY transformations are generated by the standard Dirac Killing spinors of dS₄. The theory overcomes the known obstacles to unitary global SUSY on dS₄ by closing the commutator between two SUSY transformations on so(4, 2) ⊕ u(1) rather than the de Sitter algebra so(4, 1). Crucially, the so(4, 2) symmetry is realised through unconventional conformal-like transformations. This free theory cannot become interacting while preserving SUSY in a way that makes the spin-2 sector the true graviton sector of General Relativity, as the three-graviton coupling cannot be u(1)-invariant.

We establish the unitarity of the free supersymmetric theory in two complementary ways. First, by studying the action of the superalgebra generators on the space of physical gravitino and graviton mode solutions. In particular, we introduce positive-definite, invariant inner products and demonstrate that the SUSY representation is unitary, forming a direct sum of two unitary irreducible representations — one with negative-helicity modes and the other with positive-helicity modes. Second, by quantising the fields and explicitly constructing the complex quantum supercharges Q_A and Q^A†, we show that the trace \({\sum }_{A}\left\{{Q}_{A},{Q}^{A \dagger}\right\}\) is positive-definite.

Before constructing the supersymmetric theory, we examine the free graviton and gravitino fields on dS₄, where the gravitino is known to have an imaginary mass parameter. We introduce a Hermitian, gauge-invariant, and local Lagrangian for the free gravitino field and explain why the requirement of unitarity forces the field to be chiral, removing half of the propagating helicity states.