Wrapped $M5$-branes and complex saddle points

TL;DR

This paper investigates the role of a $ heta$ term in four-dimensional minimal gauged supergravity, especially for M5-branes on hyperbolic manifolds, linking supergravity actions to topological invariants and dual field theories.

Contribution

It demonstrates the presence of a $ heta$ term in M5-brane scenarios, relates supergravity actions to topological invariants, and matches large N limits with dual SCFT predictions.

Findings

The $ heta$ term appears in M5-brane wrapping hyperbolic manifolds.

Supergravity actions are proportional to topological invariants of $ ext{Σ}_3$.

Large N partition functions match dual 3d $ ext{SCFT}$ predictions.

Abstract

We study the effects of the introduction of a $\vartheta$ 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 $\Sigma_3$, 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 $\Sigma_3$. This is consistent with the characteristics of the dual three-dimensional $\mathcal{N}=2$ SCFT predicted by the $3d$-$3d$ correspondence, and we match the large $N$ limit of its partition functions in the known cases.