Proving the equivalence of $c$-extremization and its gravitational dual for all toric quivers

Abstract

The gravitational dual of $c$-extremization for a class of $(0,2)$ two-dimensional theories obtained by twisted compactifications of D3-brane gauge theories living at a toric Calabi-Yau three-fold has been recently proposed. The equivalence of this construction with $c$-extremization has been checked in various examples and holds also off-shell. In this note we prove that such equivalence holds for an arbitrary toric Calabi-Yau. We do it by generalizing the proof of the equivalence between $a$-maximization and volume minimization for four-dimensional toric quivers. By an explicit parameterization of the R-charges we map the trial right-moving central charge $c_r$ into the off-shell functional to be extremized in gravity. We also observe that the similar construction for M2-branes on ${\mathbb{C}}^4$ is equivalent to the $\mathcal{I}$-extremization principle that leads to the microscopic counting for the entropy of magnetically charged black holes in AdS${}_4\times S^7$. Also this equivalence holds off-shell.

35 pages

References in corpus (10)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Sasaki-Einstein Manifolds and Volume Minimisation
• Dimer models and toric diagrams
• Four-Dimensional SCFTs from M5-Branes
• Exact two-dimensional superconformal R-symmetry and c-extremization
• Flow equations and attractors for black holes in N = 2 U(1) gauged supergravity
• Brane Tilings and M2 Branes
• Toric geometry and the dual of $c$-extremization
• Comments on a-maximization from gauged supergravity
• Comments on Anomalies and Charges of Toric-Quiver Duals

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• Localization and Attraction
• Evidence for a 5d F-theorem
• The joy of factorization at large $N$: five-dimensional indices and AdS black holes
• On Type IIA AdS${}_3$ solutions and massive GK geometries
• $\mathcal{I}$/$c$-Extremization in M/F-Duality
• The Near-Horizon Geometry of Supersymmetric Rotating AdS${}_4$ Black Holes in M-theory
• $N=(2,0)$ AdS${}_3$ Solutions of M-theory
• Uplifting dyonic AdS${}_4$ black holes on seven-dimensional Sasaki-Einstein manifolds
• $\mathcal{I}$-extremization with baryonic charges
• Constant maps in equivariant topological strings and geometric modeling of fluxes