Sasaki-Einstein manifolds and volume minimisation
type: Item
instance of: scholarly article
title: Sasaki-Einstein manifolds and volume minimisation
zbMATH Open document ID: 1161.53029
author: Dario Martelli
author: James Sparks
author: Shing-Tung Yau
published in: Communications in Mathematical Physics
publication date: 2008-06-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/0603021
review text: The authors study the following objects: Given a \((2n-1)\)-dimensional Riemannian manifold \((L, g_{L})\), we say that it is Sasakian if the metric cone \(X=\mathbb{R}^{+}\times L\) is a Kähler manifold, where the cone metric \(g_{X}\) is defined to be \(g_{X}=dr^{2}+ r^{2}g_{L}\), \(r>0\) is the coordinate on \(\mathbb{R}^{+}\). Hence \(X\) admits a parallel complex structure \(J\), and then, the Reeb vector vector field \(\xi\) can be defined as \(\xi = J(r\frac{\partial}{\partial r})\), where \( r\frac{\partial}{\partial r}\) is the vector field on \(X\) which is the infinitesimal generator of the one-parameter family of dilation diffeomorphisms \((r, p) \longmapsto (e^{t}r, p)\). Restricted to a vector field on the link \(L\) of the cone \(X\), the orbits of \(\xi\) define a foliation of \(L\). So, if the Kähler metric \(g_{X}\) on \(X\) is actually Ricci-flat, then the Riemannian manifold \((L, g_{L})\) is called Sasakian-Einstein. The authors show that the Einstein-Hilbert action \(\mathcal{S}(g_{L})=\int_{L}[s(g_{L}) + 2(n-1)(3-2n)]d\mu\), restricted to the space of Sasakian metrics on \(L\), where \(s(g_{L})\) is the scalar curvature of \(g_{L}\), and \(d\mu\) is the associated Riemannian measure, is just the volume functional for the link: \(S(g_{L})= 4(n-1) vol(L)\). In the remainder of the paper, by reducing their problem to studying the volume of the link, they prove that the functional \(S (g_{L}) \) depends only on the Reeb vector field. Using the Duistermaat-Heckman formula, the authors derive a very explicit formula for the volume of \(L\) at a critical Reeb vector field \(\xi\) and they prove later that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. Since physical study of the so-called AdS/CFT correspondence has placed the importance on the understanding of the geometry of Sasaki-Einstein manifolds, the authors also show that their results provide, in complex dimension \(n=3\) and via…
reviewed by: Fortuné Massamba
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 5294705
zbMATH Keywords: Sasaki-Einstein manifolds
zbMATH Keywords: metric cone
zbMATH Keywords: volume functional
zbMATH Keywords: link
zbMATH Keywords: Reeb vector field
zbMATH Keywords: surperconformal field theories
MaRDI profile type: Publication
OpenAlex ID: W3104222275
arXiv ID: hep-th/0603021
cites work: The large-\(N\) limit of superconformal field theories and supergravity
cites work: Superconformal field theory on threebranes at a Calabi-Yau singularity
cites work: Branes at conical singularities and holography
cites work: Non-spherical horizons. I
cites work: On Kähler-Einstein metrics on certain Kähler manifolds with \(C_ 1(M)>0\)
cites work: Kähler-Einstein metrics on complex surfaces with \(C_ 1>0\)
cites work: Q5488820
cites work: Supersymmetric AdS 5 solutions of M-theory
cites work: A new infinite class of Sasaki-Einstein manifolds
cites work: Toric Sasaki-Einstein metrics on \(S^2\times S^3\)
cites work: Q5311073
cites work: A note on Einstein–Sasaki metrics in D ⩾ 7
cites work: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay
cites work: The exact superconformal \(R\)-symmetry maximizes \(a\)
cites work: The holographic Weyl anomaly
cites work: The geometric dual of \(a\)-maximisation for toric Sasaki-Einstein manifolds
cites work: Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals
cites work: New results for AdS/CFT and beyond
cites work: D-brane gauge theories from toric singularities and toric duality
cites work: From toric geometry to quiver gauge theory: The equivalence of a‐maximization and Z‐minimization
cites work: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
cites work: Five-dimensional supergravity dual of \(a\)-maximization
cites work: Current correlators and AdS/CFT geometry
cites work: Comments on anomalies and charges of toric-quiver duals
cites work: On \(\eta\)-Einstein Sasakian geometry
cites work: The exact superconformal \(R\)-symmetry minimizes \(\tau _{RR}\)
cites work: On the variation in the cohomology of the symplectic form of the reduced phase space
cites work: Addendum to ``On the variation in the cohomology of the symplectic form of the reduced phase space''
cites work: Q5513583
cites work: Q4039830
cites work: Counting chiral primaries in \(\mathcal N=1\), \(d=4\) superconformal field theories
cites work: An index for 4 dimensional super conformal theories
cites work: Moment maps on symplectic cones
cites work: Certain conditions for a Riemannian manifold to be isometric with a sphere
cites work: A note on toric contact geometry
cites work: Q4720334
cites work: Q3995493
cites work: Real Killing spinors and holonomy
cites work: Sur la Structure du Groupe d’Homéomorphismes Analytiques d’une Certaine Variété Kaehlérinne
cites work: Q3694298
cites work: Q3976715
cites work: The index of elliptic operators. III
cites work: Q3961493
cites work: Equivariant index formulas for orbifolds
cites work: Quantum deformations from toric geometry
cites work: Integration and homogeneous functions
cites work: ABCD of instantons
cites work: Partition functions of pure spinors
cites work: 3-Sasakian manifolds
cites work: Q3473832
DOI: 10.1007/S00220-008-0479-4
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