Correlation functions in ω-deformed N = 6 $$ \mathcal{N}=6 $$ supergravity
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Abstract
Gauged $$ \mathcal{N}=8 $$ supergravity in four dimensions is now known to admit a deformation characterized by a real parameter ω lying in the interval 0 ≤ ω ≤ π/8. We analyse the fluctuations about its anti-de Sitter vacuum, and show that the full $$ \mathcal{N}=8 $$ supersymmetry can be maintained by the boundary conditions only for ω = 0. For non-vanishing ω, and requiring that there be no propagating spin s > 1 fields on the boundary, we show that $$ \mathcal{N}=3 $$ is the maximum degree of supersymmetry that can be preserved by the boundary conditions. We then construct in detail the consistent truncation of the $$ \mathcal{N}=8 $$ theory to give ω-deformed SO(6) gauged $$ \mathcal{N}=6 $$ supergravity, again with ω in the range 0 ≤ ω ≤ π/8. We show that this theory admits fully $$ \mathcal{N}=6 $$ supersymmetry-preserving boundary conditions not only for ω = 0, but also for ω = π/8. These two theories are related by a U(1) electric-magnetic duality. We observe that the only three-point functions that depend on ω involve the coupling of an SO(6) gauge field with the U(1) gauge field and a scalar or pseudo-scalar field. We compute these correlation functions and compare them with those of the undeformed $$ \mathcal{N}=6 $$ theory. We find that the correlation functions in the ω=π/8 theory holographically correspond to amplitudes in the U(N) k ×U(N)−k ABJM model in which the U(1) Noether current is replaced by a dynamical U(1) gauge field. We also show that the ω-deformed $$ \mathcal{N}=6 $$ gauged supergravities can be obtained via consistent reductions from the eleven-dimensional or ten-dimensional type IIA supergravities.