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C N Pope - One of the best experts on this subject based on the ideXlab platform.
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time dependent multi centre solutions from new metrics with Holonomy sim n 2
Classical and Quantum Gravity, 2008Co-Authors: G W Gibbons, C N PopeAbstract:The classifications of Holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, Sim(n − 2). Ricci-flat metrics with Holonomy were constructed by Kerr and Goldberg, and a single four-dimensional example with a nonzero cosmological constant was exhibited by Ghanam and Thompson. Here we reduce the problem of finding the general n-dimensional Einstein metric of Sim(n − 2) Holonomy, with and without a cosmological constant, to solving a set linear generalized Laplace and Poisson equations on an (n − 2)-dimensional Einstein base manifold. Explicit examples may be constructed in terms of generalized harmonic functions. A dimensional reduction of these multi-centre solutions gives new time-dependent Kaluza–Klein black holes and monopoles, including time-dependent black holes in a cosmological background whose spatial sections have non-vanishing curvature.
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cohomogeneity one manifolds of spin 7 and g 2 Holonomy
Physical Review D, 2002Co-Authors: Mirjam Cvetic, G W Gibbons, C N PopeAbstract:In this paper, we look for metrics of cohomogeneity one in D = 8 and D = 7 dimensions with Spin(7) and G2 Holonomy respectively. In D = 8, we first consider the case of principal orbits that are S 7 , viewed as an S 3 bundle over S 4 with triaxial squashing of the S 3 fibres. This gives a more general system of first-order equations for Spin(7) Holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over CP 3 , with a non-trivial parameter that characterises the homogeneous squashing of CP 3 . We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k, l) = SU(3)/U(1), where the integers k and l characterise the embedding of U(1). We find new ALC and AC metrics of Spin(7) Holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N(k, l), and numerical and perturbative results for ALC families with AC limits. We then study D = 7 metrics of G2 Holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU(3)/(U(1) × U(1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S 3 × S 3 . Additional topics include a detailed and explicit discussion of the Einstein metrics
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supersymmetric domain walls from metrics of special Holonomy
arXiv: High Energy Physics - Theory, 2001Co-Authors: G W Gibbons, C N Pope, K S StelleAbstract:Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These metrics may be obtained from known cohomogeneity one metrics of special Holonomy by taking a "Heisenberg limit", based on an In\"on\"u-Wigner contraction of the isometry group. Associated with each such metric is an Einstein metric with negative cosmological constant on a solvable group manifold. We discuss the relevance of our metrics to the resolution of singularities in domain-wall spacetimes and some applications to holography. The extremely simple forms of the explicit metrics suggest that they will be useful for many other applications. We also give new but incomplete inhomogeneous metrics of Holonomy SU(3), $G_2$ and Spin(7), which are $T_1$, $T_2$ and $T_3$ bundles respectively over hyper-K\"ahler four-manifolds.
Zoltán Muzsnay - One of the best experts on this subject based on the ideXlab platform.
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metrizability of Holonomy invariant projective deformation of sprays
Canadian Mathematical Bulletin, 2020Co-Authors: S G Elgendi, Zoltán MuzsnayAbstract:In this paper, we consider projective deformation of the geodesic system of Finsler spaces by Holonomy invariant functions: Starting by a Finsler spray $S$ and a Holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\lambda P C$. We prove that for any Holonomy invariant nontrivial function $P$ and for almost every value $\lambda\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the Holonomy invariant function is necessarily one of the principal curvatures of the geodesic structure.
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Tangent Lie Algebra of a Diffeomorphism Group and Application to Holonomy Theory
The Journal of Geometric Analysis, 2020Co-Authors: Balázs Hubicska, Zoltán MuzsnayAbstract:In this paper we introduce the notion of tangent space $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G of a (not necessary smooth) subgroup $${\mathcal {G}}$$ G of the diffeomorphism group $${\mathcal {D}}i\!f\!f^{\infty }(M)$$ D i f f ∞ ( M ) of a compact manifold M . We prove that $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G is a Lie subalgebra of the Lie algebra of smooth vector fields on M . The construction can be generalized to subgroups of any (finite- or infinite-dimensional) Lie groups. The tangent Lie algebra $${\mathcal {T}}_{o} {\mathcal {G}}$$ T o G introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example we discuss in detail the tangent structure of the Holonomy group and fibered Holonomy group of Finsler manifolds.
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Finsler 2-manifolds with maximal Holonomy group of infinite dimension
Differential Geometry and Its Applications, 2015Co-Authors: Zoltán Muzsnay, Péter T. NagyAbstract:Abstract In this paper we are investigating the Holonomy structure of Finsler 2-manifolds. We show that the topological closure of the Holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant–Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian Holonomy structures.
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tangent lie algebras to the Holonomy group of a finsler manifold
arXiv: Differential Geometry, 2012Co-Authors: Zoltán Muzsnay, Péter T. NagyAbstract:Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the Holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal Holonomy algebra by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. At the end we introduce the notion of the Holonomy algebra of a Finsler manifold by all conjugates of infinitesimal Holonomy algebras by parallel translations with respect to the Berwald connection. We prove that this Holonomy algebra is tangent to the Holonomy group.
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Finsler 2-manifolds whose Holonomy group is the diffeomorphism group of the circle
arXiv: Differential Geometry, 2012Co-Authors: Zoltán Muzsnay, Péter T. NagyAbstract:In this paper we show that the topological closure of the Holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian Holonomy structures.
Sergei Gukov - One of the best experts on this subject based on the ideXlab platform.
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m theory and singularities of exceptional Holonomy manifolds
arXiv: High Energy Physics - Theory, 2004Co-Authors: Bobby Samir Acharya, Sergei GukovAbstract:M theory compactifications on G_2 Holonomy manifolds, whilst supersymmetric, require singularities in order to obtain non-Abelian gauge groups, chiral fermions and other properties necessary for a realistic model of particle physics. We review recent progress in understanding the physics of such singularities. Our main aim is to describe the techniques which have been used to develop our understanding of M theory physics near these singularities. In parallel, we also describe similar sorts of singularities in Spin(7) Holonomy manifolds which correspond to the properties of three dimensional field theories. As an application, we review how various aspects of strongly coupled gauge theories, such as confinement, mass gap and non-perturbative phase transitions may be given a simple explanation in M theory.
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gauge theory at large n and new g2 Holonomy metrics
Nuclear Physics, 2001Co-Authors: Jaume Gomis, Andreas Brandhuber, Steven S Gubser, Sergei GukovAbstract:Abstract We find a one-parameter family of new G 2 Holonomy metrics and demonstrate that it can be extended to a two-parameter family. These metrics play an important role as the supergravity dual of the large N limit of four-dimensional supersymmetric Yang–Mills. We show that these G 2 Holonomy metrics describe the M theory lift of the supergravity solution describing a collection of D6-branes wrapping the supersymmetric three-cycle of the deformed conifold geometry for any value of the string coupling constant.
R F Picken - One of the best experts on this subject based on the ideXlab platform.
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on two dimensional Holonomy
Transactions of the American Mathematical Society, 2010Co-Authors: Joao Martins, R F PickenAbstract:We define the thin fundamental categorical group P2(M,�) of a based smooth manifold (M,�) as the categorical group whose objects are rank-1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂: E ! G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M,�) ! C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1-form on P, a principal G bundle over M, and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.
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Holonomy and parallel transport for abelian gerbes
Advances in Mathematics, 2002Co-Authors: Marco Mackaay, R F PickenAbstract:In this paper, we establish a one-to-one correspondence between U(1)-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The Holonomy of a gerbe with group U(1) on a simply connected manifold M is a group morphism from the thin second homotopy group to U(1), satisfying a smoothness condition, where a homotopy between maps from [0,1]2 to M is thin when its derivative is of rank ⩽2. For the non-simply connected case, Holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its Holonomy is carried out in detail for the simply connected case.
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Holonomy and parallel transport for abelian gerbes
arXiv: Differential Geometry, 2000Co-Authors: Marco Mackaay, R F PickenAbstract:In this paper we establish a one-to-one correspondence between $S^1$-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The Holonomy of a gerbe with group $S^1$ on a simply connected manifold $M$ is a group morphism from the thin second homotopy group to $S^1$, satisfying a smoothness condition, where a homotopy between maps from $[0,1]^2$ to $M$ is thin when its derivative is of rank $\leq 2$. For the non-simply connected case, Holonomy is replaced by a parallel transport functor between two monoidal Lie groupoids. The reconstruction of the gerbe and connection from its Holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids.
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an axiomatic definition of Holonomy
International Journal of Mathematics, 1994Co-Authors: A Caetano, R F PickenAbstract:A group of loops is associated to every smooth pointed manifold M using a strong homotopy relation. It is shown that the Holonomy of a connection on a principal G-bundle may be presented as a group morphism and that every such morphism satisfying a natural smoothness condition is the Holonomy of some unique connection up to isomorphism.
Dominic Joyce - One of the best experts on this subject based on the ideXlab platform.
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riemannian Holonomy groups and calibrated geometry
2007Co-Authors: Dominic JoyceAbstract:The Holonomy group Hol(g) of a Riemannian n-manifold (M, g) is a global invariant which measures the constant tensors on the manifold. It is a Lie subgroup of SO(n), and for generic metrics Hol(g) = SO(n). If Hol(g) is a proper subgroup of SO(n) then we say g has special Holonomy. Metrics with special Holonomy are interesting for a number of different reasons. They include Kahler metrics with Holonomy U(m), which are the most natural class of metrics on complex manifolds, Calabi-Yau manifolds with Holonomy SU(ra), and hyperkahler manifolds with Holonomy Sp(m).
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compact manifolds with special Holonomy
2000Co-Authors: Dominic JoyceAbstract:The book starts with a thorough introduction to connections and Holonomy groups, and to Riemannian, complex and Kahler geometry. Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with Holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperkahler manifolds). These are constructed and studied using complex algebraic geometry. The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional Holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated. The first known examples of these manifolds were discovered by the author in 1993-5. This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.
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compact 8 manifolds with Holonomy spin 7
Inventiones Mathematicae, 1996Co-Authors: Dominic JoyceAbstract:In Berger’s classification [4] of the possible Holonomy groups of a nonsymmetric, irreducible riemannian manifold, there are two special cases, the exceptional Holonomy groups G2 in 7 dimensions and Spin(7) in 8 dimensions. Bryant [6] proved that such metrics exist locally, using the theory of exterior differential systems, and gave some explicit examples. Later, Bryant and Salamon [7] constructed explicit, complete metrics with Holonomy G2 and Spin(7). In two previous papers [12], [13], the author constructed many examples of compact riemannian 7-manifolds with Holonomy G2. This paper will construct examples of compact riemannian 8-manifolds with Holonomy Spin(7), using similar methods. We believe that these are the first examples known. Since metrics with Holonomy Spin(7) are ricci-flat, these are also new examples of compact, ricci-flat riemannian 8-manifolds. Let M be an 8-manifold. A Spin(7)structure on M can be encoded in a 4-form Ω on M , a special 4-form satisfying the condition that the stabilizer of Ω at each point should be isomorphic to Spin(7). By an abuse of notation, we usually identify the Spin(7)structure with its associated 4-form Ω. Since Spin(7) ⊂ SO(8), the Spin(7)structure also induces a riemannian metric g and an orientation on M . It turns out that the Holonomy group Hol(g) of g is a subgroup of Spin(7), with Spin(7)structure Ω, if and only if dΩ = 0. The quantity dΩ is called the torsion of the Spin(7)structure Ω, and Ω is called torsion-free if dΩ = 0. Very briefly, the plan of the paper splits into four steps, as follows. Firstly, a compact 8-manifold M is given. Secondly, a Spin(7)structure Ω on M is found, with small torsion, i.e. dΩ is small. Thirdly, Ω is deformed to a nearby Spin(7)structure Ω with dΩ = 0. Thus Ω is torsion-free, and if g is the associated metric then Hol(g) ⊂ Spin(7). Fourthly, it is shown that Hol(g) is Spin(7), and not some proper subgroup. The structure of the paper is designed around this division of the construction into four steps. There are six chapters. This first chapter is of introductory material. The second chapter has the full statements of the main results. In
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compact riemannian 7 manifolds with Holonomy g sb 2 ii
Journal of Differential Geometry, 1996Co-Authors: Dominic JoyceAbstract:This is the second of two papers about metrics of Holonomy G2 on compact 7manifolds. In our first paper [15] we established the existence of a family of metrics of Holonomy G2 on a single, compact, simply-connected 7-manifold M , using three general results, Theorems A, B and C. Our purpose in this paper is to explore the theory of compact Riemannian 7-manifolds with Holonomy G2 in greater detail. By relying on Theorems A-C we will be able to avoid the emphasis on analysis that characterized [15], so that this sequel will have a more topological flavour. The paper has four chapters. The first chapter consists of introductory material. Section 1.1 gives some elementary geometric and topological material on compact 7-manifolds with torsion-free G2structures. Then §1.2 describes the Holonomy groups SU(2) and SU(3), and §1.3 explains the concept of asymptotically locally Euclidean Riemannian manifolds (shortened to ALE spaces) with special Holonomy. Recall that in [15], a compact 7-manifold M was defined by desingularizing a quotient T /Γ of the 7-torus by a finite group of isometries Γ ∼= Z2. The subject of Chapters 2 and 3 is a generalization of this idea. Chapter 2 defines a general construction for compact 7-manifolds with torsion-free G2structures, which works by desingularizing quotients T /Γ for finite groups Γ. The ALE spaces with Holonomy SU(2) and SU(3) discussed in §1.3 are an essential ingredient in performing this desingularization. The central result of Chapter 2 is Theorem 2.2.3, which states that given a suitable finite group Γ and certain other data, one may construct a compact 7manifold M from T /Γ that admits torsion-free G2structures. This result is proved using Theorems A-C of [15]. Chapter 3 is devoted entirely to examples of this construction. We give many examples of compact 7-manifolds with Holonomy G2, and determine their basic topological invariants — the Betti numbers and fundamental group. Finally, in Chapter 4 we discuss some areas of interest, and give a number of open problems. This paper is not written to be read independently of [15]. The language and results of [15] will be used freely, in particular the introductory material in [15, §1.1]. For reference we reproduce here the model 3and 4forms φ, ∗φ defining the flat G2structure on R, as given in [15, §1.1]: