Implicit metric on a deformation of the Atiyah-Hitchin manifold

Nonlinearity, 1996

We consider 3-monopoles symmetric under inversion symmetry. We show that the moduli space of these monopoles is an Atiyah-Hitchin submanifold of the 3monopole moduli space. This allows what is known about 2-monopole dynamics to be translated into results about the dynamics of 3-monopoles. Using a numerical ADHMN construction we compute the monopole energy density at various points on two interesting geodesics. The first is a geodesic over the two-dimensional rounded cone submanifold corresponding to right angle scattering and the second is a closed geodesic for three orbiting monopoles.

1996

The moduli space describing the low-energy dynamics of BPS multi-monopoles for several charge configurations is presented. We first prove the conjectured form of the moduli space of n − 1 distinct monopoles in a spontaneously broken SU (n) gauge theory. We further propose the solution where one of the charge components has two units, hence asymptotically corresponds to embeddings of two monopoles of one charge type and the rest different. The latter hyperkähler metrics possess features of the two-monopole Atiyah-Hitchin metric. We also conjecture classes of solutions to multi-monopole moduli spaces with arbitrary charge and no more than two units in each component, which models the gluing together of Atiyah-Hitchin metrics. Our approach here uses the generalized Legendre transform technique to find the new hyperkähler manifolds and rederive previously conjectured ones.

Modern Physics Letters A, 2000

Using a geometric realization of the SU (2) R symmetry and a procedure of factorisation of the gauge and SU (2) R charges, we study the small instanton singularities of the Higgs branch of supersymmetric U (1) r gauge theories with eight supercharges. We derive new solutions for the moduli space of vacua preserving manifestly the eight supercharges. In particular, we obtain an extension of the ordinary ADE singularities for hyperKahler manifolds and show that the classical moduli space of vacua is in general given by cotangent bundles of compact weighted projective spaces.

Lecture Notes in Physics, 1995

Communications in Mathematical Physics, 2018

We advocate that the generalized Kronheimer construction of the Kähler quotient crepant resolution M ζ -→ C 3 /Γ of an orbifold singularity where Γ ⊂ SU(3) is a finite subgroup naturally defines the field content and the interaction structure of a superconformal Chern-Simons Gauge Theory. This latter is supposedly the dual of an M2-brane solution of D = 11 supergravity with C × M ζ as transverse space. We illustrate and discuss many aspects of this type of constructions emphasizing that the equation p p p∧ p p p = 0 which provides the Kähler analogue of the holomorphic sector in the hyperKähler moment map equations canonically defines the structure of a universal superpotential in the CS theory. Furthermore the kernel D Γ of the above equation can be described as the orbit with respect to a quiver Lie group G Γ of a special locus that has also a universal definition. We provide an extensive discussion of the relation between the coset manifold G Γ /F Γ , the gauge group F Γ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the so named tautological vector bundles that are in one-to-one correspondence with the nontrivial irreps of Γ. These first Chern classes are represented by (1,1)-forms on M ζ and provide a basis for the cohomology group H 2 (M ζ ). We also discuss the relation with conjugacy classes of Γ and we provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of C 2 /Γ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.

Physics Letters B, 1988

Using recently established results on the superconformal moduli space and their relation to the geometry of N= 2 SUGRA, we give an explicit formula for the K~ihler potential for the chiral multiplets associated to deformation of the K~hler class [ ( 1, 1 ) forms]. The formula holds for all compactifications on (2, 2) systems and it requires only topological informations about the internal superconformal theory (i.e. the intersection numbers). The metric turns out to be the unique K~hler metric which is conformal to the one proposed by Strominger some time ago. From this fact we infer that the cone in H2(K, Dq) consisting of K~ihler classes coincides with a class of convex cones whose remarkable geometrical properties were already noticed in the contexts of N= 2 supergravity and Jordan algebras. Our formula gives a closed expression for this K~ihler cone.

Journal Fur Die Reine Und Angewandte Mathematik, 2004

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymmetric sigma models with Wess-Zumino term.

Journal of High Energy Physics, 2015

We use the twistorial construction of D-instantons in Calabi-Yau compactifications of type II string theory to compute an explicit expression for the metric on the hypermultiplet moduli space affected by these non-perturbative corrections. In this way we obtain an exact quaternion-Kähler metric which is a non-trivial deformation of the local c-map. In the four-dimensional case corresponding to the universal hypermultiplet, our metric fits the Tod ansatz and provides an exact solution of the continuous Toda equation. We also analyze the fate of the curvature singularity of the perturbative metric by deriving an S-duality invariant equation which determines the singularity hypersurface after inclusion of the D(-1)-instanton effects.

Nuclear Physics B, 2001

We obtain a simple explicit expression for the hyper-Kähler Calabi metric on the cotangent bundle of CP n+1 , for all n, in which it is constructed as a metric of cohomogeneity one with SU (n + 2)/U (n) principal orbits. These results enable us to obtain explicit expressions for an L 2-normalisable harmonic 4-form in D = 8, and an L 2-normalisable harmonic 6-form in D = 12. We use the former in order to obtain an explicit resolved M2-brane solution, and we show that this solution is invariant under all three of the supersymmetries associated with the covariantly-constant spinors in the 8-dimensional Calabi metric. We give some discussion of the corresponding dual N = 3 three-dimensional field theory. Various other topics are also addressed, including superpotentials for the Calabi metrics and the metrics of exceptional G 2 and Spin(7) holonomy in D = 7 and D = 8. We also present complex and quaternionic conifold constructions, associated with the cone metrics whose resolutions are provided by the Stenzel T * S n+1 and Calabi T * CP n+1 metrics. In the latter case we relate the construction to the hyper-Kähler quotient. We then use the hyper-Kähler quotient to give a quaternionic rederivation of the Calabi metrics.

Communications in Mathematical Physics, 1987

We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a symplectic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.