Obstructions to the Existence of Sasaki–Einstein Metrics

By J. Gauntlett, D. Martelli, J. Sparks, Shing-Tung Yau. 2006

185 citations recorded by the source corpus.

We describe two simple obstructions to the existence of Ricci-flat Kähler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kähler–Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R–charge of a gauge invariant chiral primary operator violates the unitarity bound.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d5ff0704300ca212894057b6073811d93f7dbeea

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In Applied Mathematics

1. Chi Li
2. Song Sun
3. G'abor Sz'ekelyhidi
4. Yuchen Liu
5. J. Sparks
6. Robert J. Berman
7. Y. Odaka
8. Simon Donaldson

In Geometry and Topology

1. Chi Li
2. G'abor Sz'ekelyhidi
3. Song Sun
4. J. Sparks
5. A. Hanany
6. D. Martelli
7. Yuchen Liu
8. M. Yamazaki

Citations in the corpus

• Sasaki–Einstein metrics and K–stability (2015, background): “The necessary conditions 2p > q and 2q > p follow from the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau [46], while the existence result was only known previously for (p, q) = (2, 2) and (2, 3), where the latter was shown by Li-Sun [61].”
• Exceptional del Pezzo Hypersurfaces (2008, background): “8 ([10]) Let X be a quasismooth hypersurface in P(a0, .”
• Landscape of superconducting membranes (2009, background): “A necessary condition for existence is [41] miniwi ≥ a > 0, with a given by (30).”
• The large N limit of quiver matrix models and Sasaki-Einstein manifolds (2011, methodology): “More generally, one can compute the volume of a (singular) Sasakian metric on the link Y using the results of [17].”
• Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere (2011, background): “The spaces Yn(~k) probably have a Sasaki-Einstein metric only when n ≤ 2, though, because of the Lichnerowicz obstruction of [27, 40].”
• A class of Sasakian 5-manifolds (2008, background): “These structures have been intensively studied by many authors (see, for instance, [BG2], [BGK], [GMSW], [ GMSY ] and references therein).”
• Weighted Projective Embeddings, Stability of Orbifolds, and Constant Scarla Curvature Kär Metrics (2009, background): “in metrics on odd dimensional manifolds correspond to orbifold Kahler-Einstein metrics on the leaf space of their Reeb vector elds). What rst interested us in this subject was the remarkable work of [GMSY] nding new obstructions to the existence of Ricciat cone metrics on cones over singularities, S”
• Kähler–Einstein metrics and volume minimization (2016, background): “By [28,43,40], there is a Kähler–Einstein edge metric on (V, Δ) if and only if one of the following conditions holds:”
• Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds (2006, background): “The results obtained in this paper has many interesting applications.”
• Counting BPS operators in gauge theories: quivers, syzygies and plethystics (2006, background): “However, for k ≥ 3, [60, 61] recently showed that there is no SasakiEinstein metric, whereby making the AdS/CFT correspondence a little ambiguous here.”
• Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, background): “A holomorphic function f on C(Y ) is called homogeneous with charge μ if Lξf = iμf for some μ > 0 (the name is adopted from [17]).”
• Integral bounds on curvature and Gromov–Hausdorff limits (2012, background): “rem 1.2. There is also an algebraic description of the cone structure on C(Y). A holomorphic function fon C(Y) is called homogeneous with charge µif Lξf= iµf for some µ>0 (the name is adopted from [17]). Let Rd be the space of holomorphic functions on C(Y) with charge d. So we may understand the ”
• Brane tilings and their applications (2008, background): “In [314], it is clarified that some obstructions in Kähler-Einstein metrics can be interpreted in the gauge theory side.”
• Sasaki-Einstein Manifolds (2010, methodology): “9), which was done in [11] for the case of well-formed orbifolds (where all orbifold singularities have complex codimension at least 2), or [53] using the methods developed in [75].”