Toric geometry and the dual of $c$-extremization

Abstract

We consider D3-brane gauge theories at an arbitrary toric Calabi-Yau 3-fold cone singularity that are then further compactified on a Riemann surface ${\Sigma }_g$, with an arbitrary partial topological twist for the global $U(1)$ symmetries. This constitutes a rich, infinite class of two-dimensional $(0,2)$ theories. Under the assumption that such a theory flows to a SCFT, we show that the supergravity formulas for the central charge and $R$-charges of BPS baryonic operators of the dual AdS${}_3$ solution may be computed using only the toric data of the Calabi-Yau 3-fold and the topological twist parameters. We exemplify the procedure for both the $Y^{p,q}$ and $X^{p,q}$ 3-fold singularities, along with their associated dual quiver gauge theories, showing that the new supergravity results perfectly match the field theory results obtained using $c$-extremization, for arbitrary twist over ${\Sigma }_g$. We furthermore conjecture that the trial central charge $\mathcal{Z}$, which we define in gravity, matches the field theory trial $c$-function off-shell, and show this holds in non-trivial examples. Finally, we check our general geometric formulae against a number of explicitly known supergravity solutions.

62 pages. Typos corrected, published version

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• Sasaki-Einstein Manifolds and Volume Minimisation
• Exact two-dimensional superconformal R-symmetry and c-extremization
• The Master Space of N=1 Gauge Theories
• Obstructions to the Existence of Sasaki-Einstein Metrics
• Supersymmetric AdS_3 solutions of type IIB supergravity
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Cited by in corpus (13)

• Supersymmetric spindles
• Equivariant localization and holography
• Proving the equivalence of $c$-extremization and its gravitational dual for all toric quivers
• AdS${}_2$ duals to ADHM quivers with Wilson lines
• Fibred GK geometry and supersymmetric $AdS$ solutions
• Localization and Attraction
• On Type IIA AdS${}_3$ solutions and massive GK geometries
• $\mathcal{I}$/$c$-Extremization in M/F-Duality
• $N=(2,0)$ AdS${}_3$ Solutions of M-theory
• Uplifting dyonic AdS${}_4$ black holes on seven-dimensional Sasaki-Einstein manifolds
• Constant maps in equivariant topological strings and geometric modeling of fluxes
• $\mathcal{I}$-extremization with baryonic charges
• Giant Gravitons and Volume Minimisation