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Toric Geometry, Sasaki-Einstein Manifolds and a New Infinite Class of AdS/CFT Duals

Dario Martelli, James Sparks

TL;DR

This work constructs and analyzes an infinite family of AdS/CFT duals arising from Sasaki–Einstein manifolds $Y^{p,q}$ on $S^2\times S^3$ and their toric Calabi–Yau cones $C(Y^{p,q})$. Using toric geometry and Delzant-type constructions, the cones are realized as GLSM vacua with charges $(p,p,-p+q,-p-q)$, and their toric diagrams are embedded in $\mathbb{C}^3/\mathbb{Z}_{p+1}\times\mathbb{Z}_{p+1}$; this yields dual $\,\mathcal{N}=1$ toric quiver gauge theories with gauge group $SU(N)^{2p}$. The paper highlights $Y^{2,1}$ as the horizon of the complex cone over the first del Pezzo surface $\mathbb{F}_1$, showing the IR central charge obtained from geometry matches the $a$-maximisation result in the dual field theory. Collectively, these results provide a concrete and scalable framework to generate and test an infinite class of AdS/CFT duals, with central charges typically quadratic irrational and accessible via toric methods.

Abstract

Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on S^2 x S^3 has been discovered, where p and q are two coprime positive integers, with q<p. These give rise to a corresponding family of Calabi-Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kahler quotients C^4//U(1), namely the vacua of gauged linear sigma models with charges (p,p,-p+q,-p-q), thereby generalising the conifold, which is p=1,q=0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold C^3/Z_{p+1}xZ_{p+1} for all q<p with fixed p. We hence find that the Y^{p,q} manifolds are AdS/CFT dual to an infinite class of N=1 superconformal field theories arising as IR fixed points of toric quiver gauge theories with gauge group SU(N)^{2p}. As a non-trivial example, we show that Y^{2,1} is an explicit irregular Sasaki-Einstein metric on the horizon of the complex cone over the first del Pezzo surface. The dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its IR fixed point using the AdS/CFT correspondence. The value we obtain is a quadratic irrational number and, remarkably, agrees with a recent purely field theoretic calculation using a-maximisation.

Toric Geometry, Sasaki-Einstein Manifolds and a New Infinite Class of AdS/CFT Duals

TL;DR

This work constructs and analyzes an infinite family of AdS/CFT duals arising from Sasaki–Einstein manifolds on and their toric Calabi–Yau cones . Using toric geometry and Delzant-type constructions, the cones are realized as GLSM vacua with charges , and their toric diagrams are embedded in ; this yields dual toric quiver gauge theories with gauge group . The paper highlights as the horizon of the complex cone over the first del Pezzo surface , showing the IR central charge obtained from geometry matches the -maximisation result in the dual field theory. Collectively, these results provide a concrete and scalable framework to generate and test an infinite class of AdS/CFT duals, with central charges typically quadratic irrational and accessible via toric methods.

Abstract

Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on S^2 x S^3 has been discovered, where p and q are two coprime positive integers, with q<p. These give rise to a corresponding family of Calabi-Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kahler quotients C^4//U(1), namely the vacua of gauged linear sigma models with charges (p,p,-p+q,-p-q), thereby generalising the conifold, which is p=1,q=0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold C^3/Z_{p+1}xZ_{p+1} for all q<p with fixed p. We hence find that the Y^{p,q} manifolds are AdS/CFT dual to an infinite class of N=1 superconformal field theories arising as IR fixed points of toric quiver gauge theories with gauge group SU(N)^{2p}. As a non-trivial example, we show that Y^{2,1} is an explicit irregular Sasaki-Einstein metric on the horizon of the complex cone over the first del Pezzo surface. The dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its IR fixed point using the AdS/CFT correspondence. The value we obtain is a quadratic irrational number and, remarkably, agrees with a recent purely field theoretic calculation using a-maximisation.

Paper Structure

This paper contains 16 sections, 107 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. Sasaki--Einstein Metrics on $S^2 \times S^3$
  3. Sasakian--Einstein geometry
  4. The metrics $Y^{p,q}$
  5. Moment maps and convex rational polyhedral cones
  6. Moment maps for torus actions
  7. Toric Calabi--Yau cones
  8. The moment map and its image
  9. Gauged linear sigma models
  10. A brief review
  11. Delzant's construction: from polytopes to gauged linear sigma-models
  12. The topology of the vacuum
  13. Toric Gorenstein canonical singularities
  14. The complex cone over $\mathbb{F}_1$
  15. New non--trivial AdS/CFT predictions
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Toric diagram of $Y^{2,1}$ embedded in the orbifold $\mathbb{C}^3/\mathbb{Z}_3\times \mathbb{Z}_3$.
  • Figure 2: Toric diagrams of $Y^{3,2}$ and $Y^{3,1}$ embedded in the orbifold $\mathbb{C}^3/\mathbb{Z}_4\times \mathbb{Z}_4$.
  • Figure 3: Quiver diagram associated to the complex cone over $dP_1$.
  • Figure 4: Toric diagram of the conifold embedded in the orbifold $\mathbb{C}^3/\mathbb{Z}_2\times\mathbb{Z}_2$.