Toric geometry and the dual of \(c\)-extremization. (English) Zbl 1409.81118

Summary: 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 \(\mathrm{AdS}_{3}\) 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.


81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
83E50 Supergravity
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
81T60 Supersymmetric field theories in quantum mechanics
53Z05 Applications of differential geometry to physics
Full Text: DOI arXiv


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