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 Σ 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 Σ g . We furthermore conjecture that the trial central charge 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. † On leave at the Galileo Galilei Institute,

Twisted $$ \mathcal{N} $$ = 1 SCFTs and their AdS3 duals

“…Identifying the 2d SCFTs dual to the AdS 3 solutions with a < 0 constructed in this paper remains an open problem. This same phenomenon has been noted previously in [16,17,39] for solutions with internal space T 2 × Y p,q , T 2 × L a,b,c and a specific fibration of Y p,q over S 2 . In the Y p,q cases the field theory is characterized by integers p > q > 0, however the naive dual supergravity solutions, when known are instead characterized by integers q > p > 0.…”

“…With these values for the mixing parameters one finds the right-moving central charge 17) and the R-charges of the baryonic operators are given by…”

“…In this section we utilize the geometric dual of c-extremization [16,17] to compute the central charges and R-charges of the L a,b,c quiver theories reduced on a Riemann surface Σ g with constant curvature. We apply geometric c-extremization to geometries of the form AdS 3 × Y 7 , where Y 7 is a fibration of L a,b,c over Σ g .…”

“…For the latter case of Y 5 = Y p,q , local supergravity solutions have been found in [11] but their global regularity has not been studied so far, see also [14,15]. However, even if explicit solutions are not known, one can still employ the recently found geometric dual of c-extremization [16][17][18][19] 2 . For this technique one has to assume the existence of a solution and 1 One can make progress in understanding the gravity theory if one knows the topological data of the solution [2, 3].2 The closely related geometric dual of I-extremization [20,21] was also put forward in [16] and further studied in [22][23][24].…”

“…On the field theory side we compute their central charges and R-charges using c-extremization. On the geometry side we compute these same charges, first by employing geometric c-extremization which has been developed for toric Y 5 in [17]. However, as mentioned above, to use this technique we must assume the existence of the explicit solution.…”