The Geometric Dual of a–Maximisation for Toric Sasaki–Einstein Manifolds

By D. Martelli, J. Sparks, Shing-Tung Yau. 2005

419 citations recorded by the source corpus.

AbstractWe show that the Reeb vector, and hence in particular the volume, of a Sasaki–Einstein metric on the base of a toric Calabi–Yau cone of complex dimension n may be computed by minimising a function Z on $$\mathbb {R}^{n}$$ which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki–Einstein manifold without finding the metric explicitly. For complex dimension n  =  3 the Reeb vector and the volume correspond to the R–symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a–maximisation. We illustrate our results with some examples, including the Yp,q singularities and the complex cone over the second del Pezzo surface.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/05e33c4d209f2ed5b878af322194208fde1e35ac

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Alberto Zaffaroni
2. J. Sparks
3. A. Hanany
4. D. Martelli
5. M. Yamazaki
6. F. Benini
7. Song Sun
8. Shing-Tung Yau

In Nuclear and High Energy Physics

1. Alberto Zaffaroni
2. F. Benini
3. D. Martelli
4. Cyril Closset
5. A. Hanany
6. J. Sparks
7. N. Seiberg
8. D. Gaiotto

Citations in the corpus

• Counting BPS operators in gauge theories: quivers, syzygies and plethystics (2006, background): “[32]), which can be expanded as VR = 3 ∑”
• The large N limit of quiver matrix models and Sasaki-Einstein manifolds (2011, methodology): “In the context of the AdS5/CFT4 correspondence, a-maximization has a geometric counterpart in the volume minimization of SasakiEinstein manifolds(2) [15, 16, 17].”
• Contact terms, unitarity, and F -maximization in three-dimensional superconformal theories (2012, methodology): “As was pointed out in [9,61], the F -maximization principle is closely related to the volume minimization procedure of [62,63].”
• Dirichlet Branes and Mirror Symmetry (2009, background): “For example, a large family of new explicit Sasaki-Einstein metrics has been obtained and studied in this way [350, 351].”
• Examples of asymptotically conical Ricci-flat Kähler manifolds (2008, methodology): “Much of what follows can be found in [43] or [24].”
• Frankel conjecture and Sasaki geometry (2012, background): “In the toric case, it is proved in [44], [25] that the volume functional is proper, so the critical point indeed exists.”
• Integral bounds on curvature and Gromov–Hausdorff limits (2012, background): “ytope given by the Euclidean tangent cone of ∆Z at p. Similarly one can show that C(Y) is also isomorphic to W as polarized affine varieties. It seems possible that one can further adapt the results in [26] to this situation, and determine the Reeb vector field of C(Y) inside the Reeb cone, in terms of”
• Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere (2011, methodology): “The idea of [22] was to allow more general Kähler metrics on X, which by definition induce metrics on Y that are Sasakian, but not necessarily Einstein.”
• F-theorem without supersymmetry (2011, background): “For large N superconformal gauge theories dual to type IIB string theory on AdS5 × Y5, Y5 being a Sasaki-Einstein space, a-maximization is equivalent to the statement [8] that the SasakiEinstein metric on Y5 is a volume minimizer within the set of all Sasakian metrics on this space.”
• Quivers, tilings, branes and rhombi (2005, background): “Recently, it has been shown [ 14 ] that these”
• Obstructions to the Existence of Sasaki–Einstein Metrics (2006, methodology): “We expect that in the particular case that the singularity is toric, neither Lichnerowicz nor Bishop’s bound will obstruct for the critical Reeb vector field b∗ of [13].”
• Brane dimers and quiver gauge theories (2005, methodology): “In recent work [44], a geometrical method was given for performing a-maximization [17] purely in terms of toric data, without needing to know the explicit Calabi-Yau metric on the toric space.”
• From Sasaki-Einstein spaces to quivers via BPS geodesics: Lp,q|r (2005, background): “A similar understanding has been achieved for the volumes in [46].”
• Supersymmetric states of N=4 Yang-Mills from giant gravitons (2006, background): “Over the last few years a number of authors have discovered an infinite number of generalizations of the AdS/CFT conjecture [65, 66, 67, 68, 69, 70, 71, 72, 73].”