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Hidden Kac-Moody structures in the fermionic sector of five-dimensional supergravity

par Damour, Thibault;Spindel, Philippe
Référence Physical Review D, 105, 12, 125006
Publication Publié, 2022-06

Article révisé par les pairs

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Résumé :

We study the supersymmetric quantum dynamics of the cosmological models obtained by reducing D=5 supergravity to one timelike dimension. This consistent truncation has fourteen bosonic degrees of freedom, while the quantization of the homogeneous gravitino field leads to a 216-dimensional fermionic Hilbert space. We construct a consistent quantization of the model in which the wave function of the Universe is a 216-component spinor depending on fourteen continuous coordinates, which satisfies eight Dirac-like wave equations (supersymmetry constraints) and one Klein-Gordon-like equation (Hamiltonian constraint). The fermionic part of the quantum Hamiltonian is built from operators that generate a 216-dimensional representation of the (infinite-dimensional) maximally compact subalgebra K(G2++) of the rank-4 hyperbolic Kac-Moody algebra G2++. The (quartic-in-fermions) squared-mass term μ^2 entering the Klein-Gordon-like equation has several remarkable properties: (i) it commutes with the generators of K(G2++); and (ii) it is a quadratic polynomial in the fermion number NF∼ψ¯ψ, and a symplectic fermion bilinear CF∼ψCψ. Some aspects of the structure of the solutions of our model are discussed, and notably the Kac-Moody meaning of the operators describing the reflection of the wave function on the fermion-dependent potential walls ("quantum fermionic Kac-Moody billiard").