Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on toric Sasaki-Einstein seven-manifolds and AdS_4/CFT_3

Dario Martelli, James Sparks

TL;DR

This work constructs and analyzes two infinite families, $Y^{p,k}(\mathbb{C}P^2)$ and $Y^{p,k}(\mathbb{C}P^1\times\mathbb{C}P^1)$, of toric Sasaki–Einstein seven-manifolds whose Calabi–Yau cones are toric. The authors derive detailed toric data and GLSM charges, compute volumes, study homology and supersymmetric submanifolds, and explore M-theory backgrounds with reductions to type IIA, including explicit orbifold limits $S^7/\mathbb{Z}_{3p}$ and $\mathbb{C}^4/\mathbb{Z}_{3p}$. The results interpolate between known homogeneous spaces like $S^7$, $M^{3,2}$, and $Q^{1,1,1}$, providing a geometric foundation for potential ${\cal N}=2$ AdS_4/CFT_3 duals and suggesting avenues for constructing dual Chern–Simons-matter theories. The work highlights algebraicity of volumes, hints at a 3d analogue of $a$-maximisation via $\tau$-minimisation, and points to resolutions and brane-web descriptions as tools to further identify the dual field theories and their operator spectra.

Abstract

We study the geometry and topology of two infinite families Y^{p,k} of Sasaki-Einstein seven-manifolds, that are expected to be AdS_4/CFT_3 dual to families of N=2 superconformal field theories in three dimensions. These manifolds, labelled by two positive integers p and k, are Lens space bundles S^3/Z_p over CP^2 and CP^1 x CP^1, respectively. The corresponding Calabi-Yau cones are toric. We present their toric diagrams and gauged linear sigma model charges in terms of p and k, and find that the Y^{p,k} manifolds interpolate between certain orbifolds of the homogeneous spaces S^7, M^{3,2} and Q^{1,1,1}.

Notes on toric Sasaki-Einstein seven-manifolds and AdS_4/CFT_3

TL;DR

This work constructs and analyzes two infinite families, and , of toric Sasaki–Einstein seven-manifolds whose Calabi–Yau cones are toric. The authors derive detailed toric data and GLSM charges, compute volumes, study homology and supersymmetric submanifolds, and explore M-theory backgrounds with reductions to type IIA, including explicit orbifold limits and . The results interpolate between known homogeneous spaces like , , and , providing a geometric foundation for potential AdS_4/CFT_3 duals and suggesting avenues for constructing dual Chern–Simons-matter theories. The work highlights algebraicity of volumes, hints at a 3d analogue of -maximisation via -minimisation, and points to resolutions and brane-web descriptions as tools to further identify the dual field theories and their operator spectra.

Abstract

We study the geometry and topology of two infinite families Y^{p,k} of Sasaki-Einstein seven-manifolds, that are expected to be AdS_4/CFT_3 dual to families of N=2 superconformal field theories in three dimensions. These manifolds, labelled by two positive integers p and k, are Lens space bundles S^3/Z_p over CP^2 and CP^1 x CP^1, respectively. The corresponding Calabi-Yau cones are toric. We present their toric diagrams and gauged linear sigma model charges in terms of p and k, and find that the Y^{p,k} manifolds interpolate between certain orbifolds of the homogeneous spaces S^7, M^{3,2} and Q^{1,1,1}.

Paper Structure

This paper contains 14 sections, 91 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Metrics and volumes
  3. Review of the metrics of paper3
  4. Volumes
  5. Toric description
  6. $Y^{p,k}(\mathbb{C} P^2)$ family
  7. $Y^{p,k}(\mathbb{C} P^1\times \mathbb{C} P^1)$ family
  8. Homology and supersymmetric submanifolds
  9. Homology
  10. Supersymmetric submanifolds
  11. Supergravity solutions
  12. M-theory and type IIA backgrounds
  13. The orbifolds $S^7/\mathbb{Z}_{3p}$ and $\mathbb{C}^4/\mathbb{Z}_{3p}$
  14. Discussion

Figures (4)

  • Figure 1: Toric diagram for $Y^{p,k}(\mathbb{C} P^2)$. The polytope is bounded by 6 triangular faces.
  • Figure 2: On the left hand side: Toric diagram for the orbifold $Y^{p,3p}(\mathbb{C} P^2)=\mathbb{C}^4/\mathbb{Z}_{3p}$. The polytope is bounded by 4 triangular faces. On the right hand side: Toric diagram for $Y^{2r,3r}(\mathbb{C} P^2)=M^{3,2}/\mathbb{Z}_r$.
  • Figure 3: Toric diagram for $Y^{p,k}(\mathbb{C} P^1\times\mathbb{C} P^1)$. The polytope is bounded by 8 triangular faces.
  • Figure 4: On the left hand side: Toric diagram for $Y^{p,2p}(\mathbb{C} P^1\times\mathbb{C} P^1)$. This is bounded by 4 triangles and a parallelogram, implying that the link of the singularity has worse-than-orbifold singularities. On the right hand side: Toric diagram for $Y^{p,p}(\mathbb{C} P^1\times\mathbb{C} P^1)=Q^{1,1,1}/\mathbb{Z}_p$.