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.
Highlights
Correspond to the complex conjugate pair of the Euclidean solutions, which are the Wick rotated version of the electrically charged rotating AdS5 BPS black hole solutions [12, 13]
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
In [17], using the 3D-3D correspondence, the large N limit of the topologically twisted index Ig on Riemann surface Σg of genus g > 1 was computed and its large N exponential growth was shown to agree with the Bekenstein-Hawking entropy of magnetic BPS black holes in the AdS4 dual with AdS2 × Σg near-horizon geometry [18, 19]
Summary
One may want to interpret the above large N complex saddle points and the phase of the entropy in the dual gravity side. The Bekenstein-Hawking entropy is just given by the horizon area of the black hole, and the chemical potentials are the electrostatic potentials and the angular velocities at the horizon. There are always a complex conjugate pair of Euclidean solutions corresponding to one rotating Lorentzian BPS black hole solution. This conjugate pair of the Euclidean solutions naturally corresponds to the conjugate pair of the Legendre transform saddle points of the CFT index in the large N limit. The extremized chemical potentials, free energy, and the entropy, which are complex, are precisely matched between each solution of the conjugate pair [13]
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