Abstract
We compute the on-shell action of static, BPS black holes in AdS4 from mathcal{N}=2 gauged supergravity coupled to vector multiplets and show that for a certain class it is equal to minus the entropy of the black hole. Holographic renormalization is used to demonstrate that with Neumann boundary conditions on the scalar fields, the divergent and finite contributions from the asymptotic boundary vanish. The entropy arises from the extrinsic curvature on Σg × S1 evaluated at the horizon, where Σg may have any genus g ≥ 0. This provides a clarification of the equivalence between the partition function of the twisted ABJM theory on Σg × S1 and the entropy of the dual black hole solutions. It also demonstrates that the complete entropy resides on the AdS2 × Σg horizon geometry, implying the absence of hair for these gravity solutions.
Highlights
Vector multiplets and show that the on-shell action coincides with the entropy
We compute the on-shell action of static, BPS black holes in AdS4 from N = 2 gauged supergravity coupled to vector multiplets and show that for a certain class it is equal to minus the entropy of the black hole
This comes from the extrinsic curvature on Σg × S1 evaluated at the horizon3 and we show this to be precisely equal to the entropy of the black hole
Summary
We review some facts about quarter-BPS black holes in AdS4 with boundary Σg×S1. The reader who is already familiar with this literature may wish to skip to section 3. The bulk action of four dimensional N = 2 FI-gauged supergravity with nv vector mulitplets is. 2see the interesting work [9] for work on holographic renormalization of finite temperature black holes in AdS4. The equation of motion for the metric which follows from (2.1) will be utilized later so we give it explicitly here 1 − Rμν − 2 gμν. The equations of motion for the scalar fields and Maxwell’s equation will not be needed in this work
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