Wrapped M5-branes and complex saddle points

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 Σ ₃, 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 Σ ₃. This is consistent with the characteristics of the dual three-dimensional N= 2 SCFT predicted by the 3d-3d correspondence, and we match the large N limit of its partition functions in the known cases.