Abstract
We study the effects of the introduction of a ϑ term in minimal gauged supergravity in four dimensions. We show why this term is not present in supergravity duals of field theories arising on wrapped M2-branes, but is there in the case of M5-branes wrapping hyperbolic manifolds Σ3, and compute the higher-derivative corrections. Having proved that the on-shell supergravity action of any supersymmetric solution can be expressed in terms of data from the fixed points of a Killing vector, we show that it is proportional to a complex topological invariant of Σ3. This is consistent with the characteristics of the dual three-dimensional \U0001d4a9= 2 SCFT predicted by the 3d-3d correspondence, and we match the large N limit of its partition functions in the known cases.
Highlights
In a saddle point approximation in different regimes of the parameters.1 a saddle point approximation may receive leading contributions from complex on-shell actions
As pointed out in [6], it is possible to see a sign oscillation in some field theory partition functions. To match this in supergravity, we necessarily have to introduce an imaginary θ term for the Abelian gauge field, which by definition does not affect the equations of motion or the supersymmetry variations
We prove that in this reduction the eleven-dimensional topological term reduces to a four-dimensional θ term for the gauge field, confirming the statements in [6]
Summary
We consider four-dimensional Einstein-Maxwell theory with a cosmological constant and a θ term. Because of the results of the supersymmetry analysis, we focus on four-dimensional space-times solving the equations (2.2) that admit a U(1) symmetry generated by the vector field ξ preserving both metric and gauge field.4 This assumption means that we consider manifolds Y4 that have the structure of a circle fibration over an orbifold base B. where (Y4)0 is the subset of fixed points for the U(1) symmetry generated by ξ. We have introduced a local one-form φ with Lξφ = 0 and ξ φ = 0, a metric γ on B, and the square norm of the Killing vector V = ξ, ξ g It is a strictly positive global function on B, so V −1 is well-defined, and we use it to rescale η ≡ V −1ξ. It is easy to make sure that the expression for the bulk on-shell action is invariant under the gauge transformation (2.12): the real part in (2.13) shifts by a multiple of the equation of motion for φ, so on-shell is gauge-invariant, and the purely imaginary part shifts by a term that vanishes using the Bianchi identity
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