Sasaki–Einstein Manifolds and Volume Minimisation

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

491 citations recorded by the source corpus.

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat–Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kähler–Einstein metric.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/1ebeb1b31d9ec8644d54c76bba40f62fe7124330

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Citations in the corpus

Sasaki-Einstein Manifolds (2010, methodology): “Proposition 6.2 is proven by writing r′ = r exp(tφ/2) and showing that the derivative of Vol with respect to t at t = 0 is zero, independently of the choice of φ ∈ C∞B (S).”
Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities (2009, methodology): “dmits an Einstein-Kahler metric for 3 ≤ k≤ 8 [37], [33]. Since Σk for k= 1,2 is a toric Fano surface, there is a Einstein-Sasakian metric on the sphere bundle of the canonical line bundle on Σk [13], [27]. Hence we obtain a complete Ricci-ﬂat Kahler metrics in every Kahler class of the canonical lin”
Counting gauge invariants: The Plethystic program (2007, background): “In the case of toric singularities, the situation is similar, the equivariant index of [ 9 ] and the equivalent sum over vertices in (p,q)-webs in [4], reduces the evaluation of the Hilbert series to combinatorics of the toric diagram.”
K-semistability is equivariant volume minimization (2015, methodology): “paper [LL], jointly with Yuchen Liu, we will further conrm this conjecture completely. The key idea to prove our result is to extend the calculations in [Li2] to show a generalization of a result in [MSY] in which they showed that the derivative of a volume functional on the space of Reeb vector eld”
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].”
Asymptotically conical Calabi-Yau manifolds, III (2012, background): “n N Dn0 that diverges to +1at the zero section. In analogy with Section 2.2.3, all we need to know about rhere is an explicit formula for the scaling map t= exp((logt)r@ r). This we can deduce from [11, 20, 39] (see in particular the last example of [11, Section 6]). Indeed, t(Z1;:::;Z8) = (t a 1Z”
Canonical Sasakian Metrics (2006, unknown): “D G ] 1 3 M ar”
THE MASTER SPACE OF N=1 GAUGE THEORIES (2008, background): “A full discussion on the relation of the volume in terms of the Reeb vector is nicely presented in [41].”
Tilings, Chern-Simons Theories and M2 Branes (2008, methodology): “We can still compute volumes of cycles in toric Calabi-Yau singularities [45, 46] and extract from this the dimension of a subset of fields.”
Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II (2015, methodology): “This is an extension of a result of Martelli-Sparks-Yau [27] on the volume minimization property of smooth Sasaki-Einstein metrics.”
Algebraicity of the metric tangent cones and equivariant K-stability (2018, result): “e degree of the orbifold line bundle is the sum of its degrees on irreducible components. Thus we may assume X to be normal. Then [LX18, Proposition 3.10], which generalizes the convexity result from [MSY08], says that ξ →volX,x(ξ) is a strictly convex function for ξ ∈N+ R . Deﬁnition 2.12 (Log Fano”
SQCD: A Geometric Apercu (2008, background): “6Section 5.2 demonstrates that the expression in (3.6) can be written in terms of the plethystic exponential as gNf <Nc(t) = PE [dim[1, 0, . . . , 0; 0, . . . , 1]t2] = PE [N2f t 2] .”
On uniqueness of tangent cones for Einstein manifolds (2012, background): “ere r is the distance to a ﬁxed point. 1 2 TOBIASHOLCKCOLDINGANDWILLIAMP.MINICOZZIII others, and there is a extensive literature of examples; see, e.g., [BGS], [DS], [K1], [K2], [MS1], [MS2], [MSY1], [MSY2], [TY1] and [TY2]. Most examples fall into several diﬀerent classes, including ALE spaces (lik”
Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, background): “Moreover, by [43], A(n) is always an algebraic number.”
Les Houches Lectures on Constructing String Vacua (2008, background): “…natural in weakly coupled type II intersecting D-brane models.5 Moduli stabilization has been more challenging in this setting, due to the absnce of RR fluxes, limited tunability, and technical difficulties in working with the holomorphic vector bundles which are the core of these…”
Minimizing normalized volumes of valuations (2015, methodology): “sition to the current purely algebro-geometric question is however inspired by a corresponding volume minimization phenomenon in Sasaki-Einstein geometry that was discovered by Martelli-Sparks-Yau in [MSY08] (see also [CS12]), which in some sense is a special case of the general procedure discussed ”
Frankel conjecture and Sasaki geometry (2012, methodology): “We first consider transverse Kähler deformation, as discussed in [45, 25].”