Hypermultiplets, Hyperkahler Cones and Quaternion-Kahler Geometry

Bernard de Wit; Martin Rocek; Stefan Vandoren

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Hypermultiplets, Hyperkahler Cones and Quaternion-Kahler Geometry

Bernard de Wit, Martin Rocek, Stefan Vandoren

High Energy Physics - Theory · Submitted 24 Jan 2001

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The authors' main claim We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets.

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arXiv IDhep-th/0101161 (version 2)

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CitationBernard de Wit, Martin Rocek, Stefan Vandoren. "Hypermultiplets, Hyperkahler Cones and Quaternion-Kahler Geometry." arXiv:hep-th/0101161 (2001). https://pith.science/paper/ABFUOPVP

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We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of perturbative moduli spaces of type-II strings compactified on a Calabi-Yau manifold. As an example of our construction, we study the universal hypermultiplet in detail, and give three inequivalent tensor multiplet descriptions. We also comment on the construction of quaternion-Kahler manifolds that may describe instanton corrections to the moduli space.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study hyperkahler cones and their corresponding quaternion-Kahler spaces.	contribution 0.67	0.86	0.06	asserted 0.83
s1	We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets.	contribution 0.69	0.95	0.70	asserted 1.00
s2	These manifolds characterize the geometry of the hypermultiplet sector of perturbative moduli spaces of type-II strings compactified on a Calabi-Yau manifold.	interpretation 0.93	0.21	0.15	asserted 1.00
s3	As an example of our construction, we study the universal hypermultiplet in detail, and give three inequivalent tensor multiplet descriptions.	contribution 0.96	0.95	0.07	asserted 1.00
s4	We also comment on the construction of quaternion-Kahler manifolds that may describe instanton corrections to the moduli space.	future work 0.78	0.58	0.05	hedged possibility 0.99

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Original abstract

We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of perturbative moduli spaces of type-II strings compactified on a Calabi-Yau manifold. As an example of our construction, we study the universal hypermultiplet in detail, and give three inequivalent tensor multiplet descriptions. We also comment on the construction of quaternion-Kahler manifolds that may describe instanton corrections to the moduli space.

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Cited by 15 Pith papers

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How the literature uses it

50 arXiv papers cite this in 114 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 67 sentences without a confident label

Builds on it 1 paper

In what follows we will prove, generalizing a result of [ref] , that if the function $G(η_I)$ satisfies the $O(2)$ criterion or its extension, then the induced structure is a Swann bundle structure, that is, the hyperkähler space has an $ℍ^×$-action comprising one conformal homothetic and three isometric generators [ref] .

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Uses its method or tool 3 papers

This quotient can be done explicitly following the results of [ref] .

Five-dimensional gauge theory and compactification on a torus · Type IIB viewpoint and embedding into Quaternion-Kähler geometry · 0.96
The situation has considerably improved in recent years, as twistorial techniques [ref] were used to reformulate this problem analytically, in terms of the complex contact structure on the twistor space $$ (a $$-bundle over $_H$), and a suitable set of complex Darboux coordinates on $$.

Quantum hypermultiplet moduli spaces in N=2 string vacua: a review · Introduction · 0.83
The QK metric on $$ can be computed from the holomorphic contact form $$ as indicated below [ref], or by decomposing the metric on the twistor space as in [ref], see [ref] for more details.

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The coefficients are given in [ref] , but it is simple to see what they must be.

Universal correlation functions in rank 1 SCFTs · N=2 supersymmetrization of the Weyl anomaly action · 0.87

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Note that in the quasi-homogeneous case, $w_I$ has anomalous transformations under dilations and $U(1)$ transformations [ref] , compared to the transformations found in [ref] .

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K_=1/2log[(1 + k k̅ - pp̅)(1 + |k-pq|^2 - q q̅)] This reproduces Eq. 7.28 in [ref] upon identifying $k=ζ/2, p=v, q=-u$.

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Note that the metric [ref] now rewrites as s^2_=1/4 ( e^-2K_ ||^2 +ν s_M^2) , consistently with [ref] where [ref] was written in a particular gauge.

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We further deduce that $^̧= g^ ∇_$ is a homothetic conformal Killing vector, ∇_^̧= ^̣_ , and therefore $$ is a hyperkähler cone [ref] .

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Such hyperkähler spaces, known in the mathematics literature as “Swann spaces” and often referred to as “hyperkähler cones” in the physics literature, are the target spaces for rigid $=2$ superconformal sigma models [ref] .

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Let us recall the salient facts about homothetic conformal Killing vector fields (see [ref] for more details).

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Rigid superconformal sigma-models in three dimensions have recently been constructed in component formalism [ref] building on earlier works [ref] .

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