2 edition of **Construction of hyperkähler metrics for complex adjoint orbits** found in the catalog.

Construction of hyperkähler metrics for complex adjoint orbits

SeМЃrgio d"Amorim Santa Cruz

- 277 Want to read
- 36 Currently reading

Published
**1995**
by typescript in [s.l.]
.

Written in English

**Edition Notes**

Thesis (Ph.D.) - University of Warwick, 1995.

Statement | Sérgio d"Amorim Santa Cruz. |

ID Numbers | |
---|---|

Open Library | OL19085625M |

We introduce, for complex semisimple adjoint orbits, the associated spectral curve and construct the twistor space as a union of certain regular adjoint orbits; we also exhibit the family of. Abstract. Kronheimer’s work in this period was mainly concerned with the subject of “hyperkahler geometry”. Recall that a Kahler manifold is a Riemannian manifold with a complex structure I on its tangent spaces which is preserved by the parallel transport of the Levi-Civita connection: equivalently, it is a manifold whose holonomy group reduces to the unitary group.

Santa-Cruz, Twistor geometry for hyperkahler metrics on complex adjoint orbits, Ann. Global Anal. Geom. 15 () – Crossref, ISI, Google Scholar 8. A. The Orbits of a Compact Lie Group for the Adjoint Representation.. B. The Canonical Complex Structure. C. The G-Invariant Ricci Form D. The Symplectic Structure of Kirillov-Kostant-Souriau E. The Invariant Kahler Metrics on the Orbits F. Compact Homogeneous Kahler Manifolds G. The Space of Orbits H. Examples

modulispace of metrics near gdoesnot split into the productof complex and k ahler moduli as above, e.g. for a given hyperk ahler metric there is a whole sphere S2 of complex structures compatible with g. Hence mirror symmetry as formulated above for Calabi-Yau manifolds needs to be reformulated for. Lino Anderson da Silva Grama CV Lattes in studying the symplectic geometry of Landau-Ginzburg models defined in generalized toric varieties and in noncompact adjoint orbits of complex and semisimple Lie groups, andresearching applications to homological mirror symmetry. Calabi-Yau metrics on canonical bundles of complex flag manifolds.

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In Gaiotto, Moore and Neitzke presented a new construction of hyperkähler metrics on the total spaces of certain complex integrable systems, represented as a torus fibration M over a base space B, except for a divisor D in B, in which the torus fiber degenerates into a nodal torus.

Other examples include coadjoint orbits of complex semisimple Lie algebras [3, 8, 10], resolutions of surface singularities [9], cotangent bundles of complex reductive groups [11], cotangent bundles of holomorphic symmetric spaces [3].

More generally, cotangent bundles of complex manifolds have one on a neighbourhood of the zero section [5, 7]. I briefly review a new construction of hyperkahler metrics on total spaces of complex integrable systems, which we described in joint work with Davide Gaiotto and Greg Moore.

The key ingredient in the construction is a collection of integers which govern “quantum corrections” to the metric, and which obey the wall-crossing formula of Cited by: Section 4specializes the previous sections to examine the adjoint orbits of a complex semisimple group.

Adjoint orbits are defined inand then introduces regularadjoint orbits. Next, presents some results relating Jordan decompositions to the study of adjoint by: 5. known explicit hyperk¨ahler metrics have been constructed. The manifold M is a total space of a complex integrable system and it can Construction of hyperkähler metrics for complex adjoint orbits book expressed as follows.

There exists a complex manifold B, a divisor D⊂ B and a subset M′ ⊂ M such that M′ is a torus ﬁbration over B′:= B\D. On the divisor D, the torus ﬁbers of M degenerate, as Cited by: 1. by (φ,Σ). Dualizing the complex linear ﬁelds Σ to chiral ﬁelds φ˜ the ﬁnal result is a super-symmetric N = (2,2) sigma model in terms of (φ,φ˜) which is guaranteed by construction to have N = (4,4) supersymmetry, and thus to deﬁne a hyperka¨hler metric.

In equations, these steps are. Both the metrics gsf and gdepend on a real parameter R>0; in the limit as R!1, the torus bers of M0collapse, in either corrections g gsf are exponentially suppressed in Rwhen we are away from D: so as R!1, glooks very close to gsf except in a small neighborhood of the singular bers.

We show that on an arbitrary nilpotent orbit in where is a direct sum of classical simple Lie algebras, there is a G-invariant hyperKähler structure obtainable as a hyperKäher quotient of the flat hyperKähler manifold ℝ 4N ≅ℍ ïncidences between various low-dimensional simple Lie groups lead to some nilpotent orbits being described as hyperKähler quotients (in some cases in fact.

Then there exists a bijection between the set of orbits in Δ, with each of the sets of adjoint and coadjoint orbits of G. Proof. Any coadjoint orbit of G uniquely determines an orbit Y through, say (l, p) in Π. The space Y is a bundle over the orbit through p ∈ V ⁎ whose fibre over p is the coadjoint orbit through l ∈ h p ⁎.

In this paper we develop two co-adjoint orbit constructions for the phase spaces of the generalised Sl(2) and Sl(3) KdV hierarchies. This involves the construction of two group actions in terms of Yang-Baxter operators, and a hamiltonian reduction of the co-adjoint orbits.

The Poisson brackets are reproduced by the Kirillov construction. Remark Bielawski [7] has informed us of an alternative proof under the additional assumption that the hyperkahler metrics are complete. Complex structures From now on we shall assume that M and G are as in the statement of Theorem There will be an open dense set M in M consisting of the union of the principal orbits.

geometry of complex manifolds, holomorphic symplectic geometry, geomet-ric representation theory, Hodge theory and many others. The most recent addition to the list is the link between hyperkahler geometry and theoretical physics: it turns out that hyperk¨ahler manifolds play a critical part in the.

In an earlier paper, the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite-dimensional hyperKähler quotient of a flat vector space. This paper uses that quotient construction to compute hyperKühler potentials explicitly for orbits.

Abstract: We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the Chern-Ricci form, the Hermitian scalar curvature and the Nijenhuis tensor in terms of root data.

We also discuss when the Chern-Ricci. We study the hyperkähler metrics associated to minimal singularities in the nilpotent variety of a semisimple Lie group. We show that Kronheimer's 4-dimensional ALE spaces are naturally realized within the context of coadjoint orbits and can be thought of as certain moduli spaces ofSU(2) invariants instantons on ℝ4∖{O} with appropriate boundary conditions.

Different types of gradient flows that arise from different metrics including the so-called normal metric on adjoint orbits of a Lie group and the Kähler metric are compared.

It is discussed how a Kähler metric can arise from a complex structure induced by the Hilbert transform. It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKhler metrics with a single function that is a global potential for each of the Khler structures (a hyperKhler potential).

It is by now well known that adjoint orbits of complex semisimple Lie groups carry hyperkähler metrics (cf.). For regular semisimple orbits, the most general construction is due to Alekseevsky and Graev and Santa-Cruz, who associate a U (k)-invariant pseudo-hyperkähler structure to any reduced spectral curve S ∈ | O (2 k) | (provided S satisfies a reality condition).

Index Theory in Physics and the Local Index Theorem Chapter PhysicalMotivationandOverview ricandSelf–AdjointOperators,FormallySelf-AdjointandEssentiallySelf-Adjoint.

Spectral Theory. Metrics on the Space of Closed Operators. Trace Class and Hilbert-SchmidtOperators. Uniﬁcations on Almost-Complex Manifolds. Holonomy. complex group and Kis its maximal compact subgroup.

Quotient construction of the hyperk ahler structure. The following is due to Hitchin [Hi] for Xbeing a curve and Fujiki [Fu] in higher dimensions. First, x a C1principal G-bundle P!Xand a reduction P K!Xof the structure group to K. Acknowledgements I would like to thank my supervisor, Professor Nigel Hitchin, for many things: for having kindly accepted me as his student in unusual circumstances; for introduc.The generalised Legendre transform construction of hyperkähler metrics is studied further, showing that many known hyperkähler metrics (including the ones on coadjoint orbits) arise in this way.An alternative way to introduce complex manifolds is through almost com-plex structures.

De nition An almost complex structure on an smooth manifold Mis an endomorphism J: TM!TMof the tangent bundle such that J2 = Id, where Id is the identity map.

In other words an almost complex structure equips the tangent space.