Black holes and large N complex saddles in 3D-3D correspondence

Abstract

We study the large N sign oscillation of the twisted indices for 3D theories of class ℛ obtained from M5-branes wrapped on a hyperbolic 3-manifold. Holographically, the oscillatory behavior can be understood from the imaginary part of on-shell actions for the two Euclidean supergravity solutions, Bolt± with p = 0, which are Wick rotation of magnetically charged AdS4 black holes. The two solutions have the same imaginary part with opposite sign. The imaginary part comes from the F ∧ F-term in the supergravity and the coefficient is proportional to the Chern-Simons invariant of 3-manifold. Combining the holographic computation with 3D-3D relation for twisted indices, we propose a non-trivial mathematical conjecture regarding the phase factor of a twisted Reidemeister-Ray-Singer torsion on hyperbolic 3-manifold.

Wrapped M5-branes and complex saddle points

“…The seven-dimensional internal space is an S 4 bundle over Σ 3 , and there is a consistent truncation of eleven-dimensional supergravity on this space that leads to four-dimensional minimal supergravity. 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]. Therefore, it is necessary to include a ϑ term when studying the gravity dual of the field theories obtained by wrapping M 5-branes on hyperbolic three-manifolds, which is a case much less studied than its previously considered counterpart [6,[17][18][19][20][21][22][23].…”

“…Instead, we work in four Euclidean dimensions, restricting ourselves to real metric and gauge field configurations. 2 Yet, 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.…”

“…This extends the results of [7] and shows the need of a finite counterterm for the gauge field. Then, in section 3, we consider a number of examples, both with an analytic expression for the fields and without it, including the example considered in [6].…”

Higher-derivative supergravity, AdS4 holography, and black holes

“…We will not study this complexified four-derivative Lagrangian further in this work. It will be interesting to understand how it can be used in a holographic setting to calculate the complexified partition function of the dual 3d SCFT on compact Euclidean manifolds, see [22] for a recent discussion on this in the two-derivative context.…”

“…At the two-derivative level, this model admits an embedding in maximal 4d supergravity and can be obtained as a consistent truncation from 11d supergravity on S 7 [110][111][112]. Part of our motivation to perform this analysis is to relate our supergravity results to the calculation of the subleading N 1 2 22 Here we assume that there are no additional subtleties involving counterterms arising from the full tower of higher-derivative terms.…”