Calabi-Yau cones from contact reduction

Conti, Diego; Fino, Anna

doi:10.1007/s10455-010-9202-8

Cited by 8 publications

(28 citation statements)

References 21 publications

(58 reference statements)

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“…Sasakian manifolds are special types of contact manifolds. According to [26,27], an almost contact structure (Φ, η, ξ) on an odd-dimensional Riemannian manifold (M, g M ) is characterized by a nowhere vanishing vector field ξ and a one-form η, satisfying η(ξ) = 1, plus a (1, 1)-tensor Φ such that Φ 2 = −1 + ξ ⊗ η. An almost contact structure is called contact if in addition the one-form satisfies η ∧ (dη) m = 0.…”

Section: Sasakian Manifoldsmentioning

confidence: 99%

Yang–Mills solutions and dyons on cylinders over coset spaces with Sasakian structure

Tormählen

2016

Nuclear Physics B

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We present solutions of the Yang-Mills equation on cylinders R × G/H over coset spaces of odd dimension 2m + 1 with Sasakian structure. The gauge potential is assumed to be SU (m)-equivariant, parametrized by two real, scalar-valued functions. Yang-Mills theory with torsion in this setup reduces to the Newtonian mechanics of a point particle moving in R 2 under the influence of an inverted potential. We analyze the critical points of this potential and present an analytic as well as several numerical finite-action solutions. Apart from the Yang-Mills solutions that constitute SU (m)-equivariant instanton configurations, we construct periodic sphaleron solutions on S 1 × G/H and dyon solutions on iR × G/H.

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Section: Sasakian Manifoldsmentioning

confidence: 99%

Yang–Mills solutions and dyons on cylinders over coset spaces with Sasakian structure

Tormählen

2016

Nuclear Physics B

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“…Since all codimension one subspaces of T are conjugate under SU(n), given a manifold M with a SU(n)-structure P SU(n) , any hypersurface N ⊂ M admits a SU(n − 1)-structure P SU(n−1) such that (N, P SU(n−1) ) is embedded in (M, P SU(n) ) with type W ⊂ T . Thus, Proposition 12 reduces to the known fact that an oriented hypersurface in a manifold with holonomy SU(n) admits a SU(n − 1)-structure which is an integral of I 0W , called a hypo structure (see [13,12]). On the other hand if f is the nearly-Kähler differential operator, the induced differential operator f W is characterized by…”

Section: Hypo and Nearly Hypo Evolution Equationsmentioning

confidence: 99%

“…where t is a coordinate on (a, b) and (α(t), Ω(t), F (t)) is a one-parameter family of SU(n)-structures on N . In this language, finding an embedding of (N, P SU(n−1) ) in (M, P SU(n) ) amounts to finding a solution of certain evolution equations (see [13,12,14]).…”

Section: Hypo and Nearly Hypo Evolution Equationsmentioning

confidence: 99%

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Embedding into manifolds with torsion

Conti

2010

Math. Z.

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Abstract. We introduce a class of special geometries associated to the choice of a differential graded algebra contained in Λ * R n . We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds that can be realized as submanifolds of a Riemannian manifold with special holonomy, to this more general context. In particular, we consider the case of hypersurfaces inside nearly-Kähler and α-Einstein-Sasaki manifolds, proving that the corresponding evolution equations always admit a solution in the real analytic case.There are a number of known results concerning the problem of embedding a generic Riemannian manifold into a manifold with special holonomy. At the very least for this problem to make sense the embedding should be isometric, but often extra conditions are required. A classical example is that of special Lagrangian submanifolds of Calabi-Yau manifolds, introduced in [15] in the context of minimal submanifolds; it is a result of [8] that every compact, oriented real analytic Riemannian 3-manifold can be embedded isometrically as a special Lagrangian submanifold in a 6-manifold with holonomy contained in SU(3) (see also [18] for a generalization). A similar result holds for coassociative submanifolds in 7-manifolds with holonomy contained in G 2 (see [8]); the case of codimension one embeddings in manifolds with special holonomy was considered in [16,13].Whilst the above results have been obtained using characterizations in terms of differential forms, the codimension one embedding problem can be uniformly rephrased using the language of spinors. Indeed, if M is a Riemannian spin manifold with a parallel spinor, then a hypersurface inherits a generalized Killing spinor ψ, namely satisfyingwhere A is a section of the bundle of symmetric endomorphism of T M , corresponding to the Weingarten tensor, and the dot represents Clifford multiplication. One can ask if, given a Riemannian spin manifold (N, g) with a generalized Killing spinor ψ, the spinor can be extended to a parallel spinor on a Riemannian manifold M ⊃ N containing N as a hypersurface; if so, M is Ricci flat and the second fundamental form is determined by the tensor A, which is an intrinsic property of (N, g, ψ). Counterexamples exist in the smooth category [6]; in the real analytic category, the general existence of such an embedding has been established in [3] under the assumption that the tensor A is Codazzi, and in [13] for six-dimensional N .

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“…In Riemannian geometry, dimension eight is of particular interest for various reasons. First of all, an eight-dimensional Riemannian manifold (M, g) can have exceptional holonomy Spin (7) and then (M, g) is Ricci-flat. However, there are even two other irreducible Ricci-flat special holonomy groups from Berger's list [4] in this dimension, namely SU(4), i.e.…”

Section: Introductionmentioning

confidence: 99%

$${\mathrm {SU}}(4)$$ SU ( 4 ) -holonomy via the left-invariant hypo and Hitchin flow

Freibert

2017

Annali di Matematica

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The Hitchin flow constructs eight-dimensional Riemannian manifolds (M, g) with holonomy in Spin( 7) starting with a cocalibrated G 2 -structure on a seven-dimensional manifold. As Sp(2) ⊆ SU(4) ⊆ Spin( 7), one may also obtain Calabi-Yau fourfolds or hyperkähler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics g with Hol(g) ⊆ SU(4) but Hol(g) = Sp(2). We investigate when we actually get Hol(g) = SU(4) and obtain many new explicit examples of Calabi-Yau fourfolds. The results rely on the connection between cocalibrated G 2 -structures and hypo SU(3)-structures and between the Hitchin and the hypo flow and on a systematic study of hypo SU(3)-structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo SU(3)-structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or Sp(2)-holonomy for the Riemannian manifolds obtained by the hypo flow on Lie algebras with initial values lying in certain intrinsic torsion classes and show that for initial values in other torsion classes we always get Riemannian metrics with holonomy equal to SU(4).

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Calabi-Yau cones from contact reduction

Cited by 8 publications

References 21 publications

Yang–Mills solutions and dyons on cylinders over coset spaces with Sasakian structure

Yang–Mills solutions and dyons on cylinders over coset spaces with Sasakian structure

Embedding into manifolds with torsion

$${\mathrm {SU}}(4)$$ SU ( 4 ) -holonomy via the left-invariant hypo and Hitchin flow

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