Gravitational instantons with faster than quadratic curvature decay (II)

Chen, Gao; Chen, Xiuxiong

doi:10.1515/crelle-2017-0026

Cited by 35 publications

(49 citation statements)

References 48 publications

(122 reference statements)

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“…In addition to the ALE spaces, geometers and physicists have studied other types of 'gravitational instantons', namely ALF (asymptotically locally flat), ALG, and ALH spaces [47][48][49][50][51], which are all asymptotic to fibrations over a Euclidean base and contain tori (the fibers) of various dimensions at infinity (and in fact throughout the interior of the space). Relevant to F-theory are the ALG spaces which have a T 2 fibration; 11 in many instances, the deformation theory of the ADE singularities can be reproduced by ALG spaces as well as by ALE spaces.…”

Section: Local Surfacesmentioning

confidence: 99%

“…Most of the analysis of this paper was done in algebraic terms, with the results cast in terms of ALE-fibrations (using Kronheimer's identification of the algebraic and differentio-geometric descriptions of such spaces [31]). However, we noticed in a few places that the algebraic description contained the structure of an elliptic fibration; it would be interesting to know if we could replace that with an ALG-fibration (perhaps using recent results of Hein, Chen, and Chen [48][49][50][51]). If so, we would have even greater confidence in the applicability of these results to F-theory, since an appropriate metric on the space would have been given.…”

Section: Elliptic Fibrations and Alg Fibrationsmentioning

confidence: 99%

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High electric charges in M-theory from quiver varieties

Collinucci

Fazzi

Morrison

et al. 2019

J. High Energ. Phys.

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M-theory on a Calabi-Yau threefold admitting a small resolution gives rise to an Abelian vector multiplet and a charged hypermultiplet. We introduce into this picture a procedure to construct threefolds that naturally host matter with electric charges up to six. These are built as families of Du Val ADE surfaces (or ALE spaces), and the possible charges correspond to the Dynkin labels of the adjoint of the ADE algebra. In the case of charge two, we give a new derivation of the answer originally obtained by Curto and Morrison, and explicitly relate this construction to the Morrison-Park geometry. We also give a procedure for constructing higher-charge cases, which can often be applied to F-theory models. arXiv:1906.02202v1 [hep-th] 5 Jun 20191 Introduction M-theory on certain singular spaces gives rise to non-Abelian gauge symmetries. More specifically, in Calabi-Yau (CY) geometries with non-isolated singularities that admit crepant resolutions (i.e. those that do not alter the CY condition), Dynkin diagrams show up naturally as a pattern traced out by families of small intersecting two-spheres in the resolved geometry (for non-isolated holomorphic two-spheres that come in a complex one-dimensional family). When this occurs, postulating the presence of light M2-branes wrapping such so-called vanishing spheres gives rise to light degrees of freedom that fill out the root system corresponding to the Dynkin diagram. This leads one to expect the effective field theory to contain a non-Abelian gauge multiplet in the limit where areas of the two-spheres approach zero and the geometry becomes singular [1,2]. (For rigid two-spheres, we get light matter multiplets instead, typically charged under the non-Abelian group.) This picture has been exploited for two decades in the field of geometric engineering in string theory and M-theory [3][4][5][6][7][8], as well as in some closely related Ftheory constructions [9][10][11]. In the latter, a direct interpretation of the singular space is not available in all but the simplest cases. However, two strategies connect F-theory to this phenomenon of non-Abelian degrees of freedom: Either one can take a particular limit of M-theory, such that the duality to type IIB compactified on a circle is manifest, and verify the correspondence. Alternatively, one can analyze the F-theory phenomena directly by considering which (p, q)-strings in the type IIB seven-brane background become light. (An important feature of F-theory is in fact the existence of seven-branes of "exotic" types, i.e. beyond the D7-branes and O7-planes of perturbative type IIB.)Much less clear is what to make of possible Abelian gauge symmetries. 1 These can also be related to singularities, but not as directly as in the case of simple Lie algebras. Since the inception of F-theory, it has been known that one way to understand U(1) gauge groups is by seeking sections (or in some cases, multi-sections) of the elliptic fibration of F-theory [11][12][13]. In M-theory, it suffices to think of divisors in the compactification ...

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Section: Local Surfacesmentioning

confidence: 99%

Section: Elliptic Fibrations and Alg Fibrationsmentioning

confidence: 99%

High electric charges in M-theory from quiver varieties

Collinucci

Fazzi

Morrison

et al. 2019

J. High Energ. Phys.

View full text Add to dashboard Cite

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“…for x ∈ X where r(x) denotes the metric distance to a base point o in X [3]. Gravitational instantons can be divided into four categories:…”

Section: Classification Of Noncompact Hyperkähler Manifolds Xmentioning

confidence: 99%

“…Chen-Chen proved that any connected complete gravitational instanton with curvature decay like (2.1) must be asymptotic to some standard model. This is the data of a geometric type [3]. For each geometric type, we have a moduli space of hyperkähler manifolds of that type.…”

Section: Classification Of Noncompact Hyperkähler Manifolds Xmentioning

confidence: 99%

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Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

Fredrickson

2019

SIGMA

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We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.Note: This is the model solution featured in [14,18,27]. The base curve is CP 1 with an irregular singularity at ∞ [15]. Nonabelian Hodge correspondenceThe Hitchin moduli space M is hyperkähler. As a consequence, it has a CP 1 -worth of complex structures, labeled by parameter ζ ∈ CP 1 . Two avatars of the Hitchin moduli space are• the Higgs bundle moduli space (ζ = 0), and• the moduli space of flat GL(n, C)-connections ζ ∈ C × .1 In a local holomorphic coordinate z and a local holomorphic frame for (E, ∂E), if ϕ = Φdz, then ϕ * h = h −1 Φ * hdz.

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Gravitational Instantons and Degenerations of Ricci-flat Metrics on the K3 Surface

Foscolo

2020

Fields Institute Communications

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The study of degenerations of metrics with special holonomy is an important theme unifying the study of convergence of Einstein metrics, the study of complete non-compact manifolds with special holonomy and the construction of spaces with special holonomy by singular perturbation methods. We survey three constructions of degenerating sequences of hyperkähler metrics on the (smooth 4-manifold underlying a complex) K3 surface-the classical Kummer construction, Gross-Wilson's work on collapse along the fibres of an elliptic fibration, and the author's construction of sequences collapsing to a 3-dimensional limit-describing how they fit into the general theory and highlighting the role played in each construction by gravitational instantons, i.e. complete non-compact hyperkähler 4-manifolds with decaying curvature at infinity. Hyperkähler metrics in dimension 4Hyperkähler 4-manifolds are the lowest dimensional non-flat examples of manifolds with special holonomy.Definition 1.1. A Riemannian 4-manifold (M 4 , g) is hyperkähler if the holonomy Hol(g) is contained in SU(2).Despite its integro-di erential definition in terms of parallel transport, the holonomy reduction to SU(2) can be recast in terms of a PDE for a triple of 2-forms satisfying special algebraic properties at each point [18]. Recall that the space of 2-forms on an oriented 4-dimensional vector space carries a natural non-degenerate bilinear form of signature (3, 3). Definition 1.2. Let (M 4 , µ 0 ) be an oriented 4-manifold with volume form µx M at every point x oe M . Given a triple ! of 2-forms on (M, µ 0 ) we consider the matrix Q oe !

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Gravitational instantons with faster than quadratic curvature decay (II)

Cited by 35 publications

References 48 publications

High electric charges in M-theory from quiver varieties

High electric charges in M-theory from quiver varieties

Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

Gravitational Instantons and Degenerations of Ricci-flat Metrics on the K3 Surface

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