Hyperkahler Research Articles

Hyperkähler manifolds with torsion obtained from hyperholomorphic bundles

Hyperkähler manifolds with torsion obtained from hyperholomorphic bundles

Cotangent bundles of 4-dimensional hypercomplex Lie groups

We study geometrical structures on the cotangent bundle T * G of a Lie group G which are left-invariant with respect to the Lie group structure on T * G determined by a left-invariant affine structure ∇ on G. In particular, we investigate the existence of conformally hyper-Kahler metrics and hyper-Kahler with torsion (HKT) structures on the cotangent bundle of hypercomplex 4-dimensional Lie groups. By applying Inonu-Wigner contractions to compact semisimple Lie algebras we obtain non semisimple Lie algebras endowed with invariant HKT structures.

RG flows from Spin(7), CY 4-fold and HK manifolds to AdS, Penrose limits and pp waves

We obtain explicit realizations of holographic renormalization group (RG) flows from M-theory, from E^{2,1} \times Spin(7) at UV to AdS_4 \times \tilde{S^7} (squashed S^7) at IR, from E^{2,1} \times CY4 at UV to AdS_4 \times Q^{1,1,1} at IR, and from E^{2,1} \times HK (hyperKahler) at UV to AdS_4 \times N^{0,1,0} at IR. The dual type IIA string theory configurations correspond to D2-D6 brane systems where D6 branes wrap supersymmetric four-cycles. We also study the Penrose limits and obtain the pp-wave backgrounds for the above configurations. Besides, we study some examples of non-supersymmetric and supersymmetric flows in five-dimensional gauge theories.

Lagrangian Submanifolds in Hyperkähler Manifolds, Legendre Transformation

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection. We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperkahler manifolds. We give a Plucker type formula for Lagrangian intersections under this transformation.

Hyper-Hamiltonian dynamics

We introduce an extension of hamiltonian dynamics, defined on hyperkahler manifolds, which we call ``hyperhamiltonian dynamics''. We show that this has many of the attractive features of standard hamiltonian dynamics. We also discuss the prototypical integrable hyperhamiltonian systems, i.e. quaternionic oscillators.

M-fivebranes wrapped on supersymmetric cycles. II

We construct D=11 supergravity solutions dual to the twisted field theories arising when M-theory fivebranes wrap supersymmetric cycles. The cases considered are M-fivebranes wrapped on (i) a complex Lagrangian four-cycle in a D=8 hyper-Kaehler manifold corresponding to a D=2 field theory with (2,1) supersymmetry (ii) a product of two holomorphic two-cycles in a product of two Calabi-Yau two-folds corresponding to a D=2 field theory with (2,2) supersymmetry and (iii) a product of a holomorphic two-cycle and a SLAG three-cycle in a product of a Calabi-Yau two-fold and a Calabi-Yau three-fold corresponding to a quantum mechanics with two supercharges. In each case we construct BPS equations and find IR superconformal fixed points corresponding to new examples of AdS/CFT duality arising from the twisted field theories.

3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite familiesof new 3-Sasakian manifolds M (p1,p2, p3) andM (p1,p2, p3, p4) in dimension 11 and 15 respectively. The metric cone on (p1,p2, p3) is a generalization ofthe Kronheimer hyperkahler metric on the regular maximalnilpotent orbit of sl (3, C)whereas the cone on M (p1,p2, p3, p4)generalizes the hyperkahler metric onthe 16-dimensional orbit of so(6, C).These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p1,p2, p3).These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O(Θ) of compact self-dual positivescalar curvature Einstein metrics with orbifoldsingularities and with only one Killing vector field.

Curvature and characteristic numbers of hyper-Kähler manifolds

Characteristic numbers of compact hyper-Kahler manifolds are expressed in graph-theoretical form, considering them as a special case of the curvature invariants introduced by L. Rozansky and E. Witten. The appropriate graphs are generated by wheels, and the recently proved Wheeling theorem is used to give a formula for the $\mathscr{L}$2-norm of the curvature of an irreducible hyper-Kahler manifold in terms of the volume and Pontryagin numbers. The formula involves the multiplicative sequence that is the square root of the Â-polynomial.

HyperKähler potentials in cohomogeneity two

HyperKähler potentials in cohomogeneity two

On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

The study of higher dimensional hyperkahler manifolds has attracted much attention: we have [Wk], [Bg1,2,3,4], [Fj1,2], [Bv1], [Vb1,2], [Sl1,2], [HS], [Huy], [Gu3,4,5] etc. It is evident that there are only a few known examples of these manifolds and the obvious question is: can we classify them as in the case of complex dimension 4? The Riemann-Roch formula plays an important role in the surface case, which yields K-3 surfaces as the only irreducible examples. However, for the higher dimensional case, the Riemann-Roch formula is not enough to give a picture of both the Hodge diamond and the existence of holomorphic sections of line bundles. In [Gu5], we combined the results of the Riemann-Roch formula in [Sl1,2] (see also [LW]) and the representations generated by the Kahler classes (see [Vb2], [LL], [Bg4]) to give a picture of the Hodge diamonds of irreducible compact hyperkahler manifolds of complex dimension 4. Theorem 1 (reproduced here) gives an upper bound b2 ≤ 23 for the second Betti number and was obtained independently by Beauville [Bv2] (unpublished). He kindly let me publish alone. The bound is obtained by applying Verbitsky’s work. In [HS] there is also an upper bound for the Euler characteristic but there seems as yet no lower bound, nor any bound for the Betti numbers. However, the method in [HS] actually gives us a way to calculate what we call generalized Chern numbers (which are only defined on hyperkahler manifolds) by Rozansky-Witten invariants, some of which in turn can be calculated as Chern numbers. Combining this approach with the method in [Bg4] we obtain an inequality in the opposite direction to the one in [HS] and apply it to our situation. Surprisingly, once we already have the bound on b2 this gives a more natural and much stronger inequality than the one we manipulated from the Riemann-Roch formula in [Gu5]. Therefore, we obtain our: Main Theorem. If M is an irreducible compact hyperkahler manifold of complex dimension 4, then 3 ≤ b2 ≤ 23. Moreover,

M(atrix)-theory scattering in the noncommutative (2,0) theory

We study the Quantum-Mechanics on the hyper-Kahler manifold that is the blow-up of an $A_1$-singularity. This system is relevant for M(atrix)-theory as it was conjectured to describe scattering in the "noncommutative" deformation of a free 5+1D tensor multiplet in the sector with two units of longitudinal light-like momentum.

ON HYPER-KAHLER SINGULARITIES

Using a geometric realization of the SU (2)R symmetry and a procedure of factorization 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 hyper-Kahler manifolds and show that the classical moduli space of vacua is in general given by cotangent bundles of compact weighted projective spaces.

A Geometric Interpretation of the χ y Genus¶on Hyper-Kähler Manifolds

The group SL(2) acts on the space of cohomology groups of any hyper-Kahler manifold X. The \chi_{y} genus of a hyper-Kahler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the \chi_{y} genus.

Quaternionic metrics from harmonic superspace: Lagrangian approach and quotient construction

Starting from the most general harmonic superspace action of self-interacting Q + hypermultiplets in the background of N=2 conformal supergravity, we derive the general action for the bosonic sigma model with a generic 4 n-dimensional quaternionic-Kähler (QK) manifold as the target space. The action is determined by the analytic harmonic QK potential and supplies an efficient systematic procedure of the explicit construction of QK metrics by the given QK potential. We find out this action to have two flat limits. One gives the hyper-Kähler (HK) sigma model with a 4 n-dimensional target manifold, while another yields a conformally-invariant sigma model with 4( n+1)-dimensional HK target. We work out the harmonic superspace version of the QK quotient construction and use it to give a new derivation of QK extensions of four-dimensional Taub–NUT and Eguchi–Hanson metrics. We analyze in detail the geometrical and symmetry structure of the second metric. The QK sigma model approach allows us to reveal the enhancement of its SU(2)⊗ U(1) isometry to SU(3) or SU(1,2) at the special relations between its free parameters: the Sp(1) curvature (“Einstein constant”) and the “mass”.

Noncommutative Instantons and Twistor Transform

Recently N.Nekrasov and A.Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of the 4-dimensional real affine space. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative projective plane, certain complexes of sheaves on a noncommutative 3-dimensional projective space, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative projective plane has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative projective plane. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative R^4 than the one considered by Nekrasov and Schwarz (a q-deformed R^4).

HyperKähler Potentials via Finite-Dimensional Quotients

It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKahler metrics with a single function that is a global potential for each of the Kahler structures (a hyperKahler potential). In an earlier paper, the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite-dimensional hyperKahler quotient of a flat vector space. This paper uses that quotient construction to compute hyperKuhler potentials explicitly for orbits of elements with small Jordan blocks. It is seen that the Kahler potentials of Biquard and Gauduchon for SL(n, C)-orbits of elements with X2 = 0, are in fact hyperKahler potentials.

Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds

We prove that for $n>1$ one cannot immerse $S^{2n}$ as a minimal Lagrangian manifold into a hyperKahler manifold. More generally we show that any minimal Lagrangian immersion of an orientable closed manifold $L^{2n}$ into a hyperKahler manifold $H^{4n}$ must have nonvanishing second Betti number $\beta_2$ and that if $\beta_2=1$ , $L^{2n}$ is a Kahler manifold and more precisely a Kahler submanifold in $H^{4n}$ w.r.t. one of the complex structures on $H^{4n}$ . In addition we derive a result for the other Betti numbers.

THE MODULI SPACE OF COMPLEX LAGRANGIAN SUBMANIFOLDS IN THE HYPER-KAEHLER MANIFOLD

The deformation of a compact complex Lagrangian submanifold in a hyper-Kaehler manifold and the moduli space are studied. It is proved that the moduli space Mcl is a special Kaehler manifold, where special means that there is a real flat torsionfree symplectic connection ∇ satisfying ( is a complex structure of Mcl). Thus, following [4], one knows that T * Mcl is a hyper-Kaehler manifold and then that Mcl is a complex Lagrangian submanifold in T * Mcl.

Special K�hler Manifolds

We give an intrinsic definition of the special geometry which arises in global N= 2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kähler manifold is so related to an integrable system. The cotangent bundle of a special Kähler manifold carries a hyperkähler metric. We also define special geometry in supergravity in terms of the special geometry in global supersymmetry.

Branes for Higgs phases and exact conformal field theories

We consider multicenter supergravity solutions corresponding to Higgs phases of supersymmetric Yang-Mills theories with Z_N symmetric vacua. In certain energy regimes, we find a description in terms of a generalized wormhole solution that corresponds to the SL(2,R)/U(1) \times SU(2)/U(1) exact conformal field theory. We show that U-dualities map these backgrounds to purely gravitational ones and comment on the relation to the black holes arising from intersecting D1 and D5 branes. We also discuss supersymmetric properties of the various solutions and the relation to 2-dim solitons, on flat space, of the reduced axion-dilaton-gravity equations. Finally, we address the problem of understanding other supergravity solutions from the multicenter ones. As prototype examples we use rotating D3 branes and NS5 and D5 branes associated to non-Abelian duals of 4-dim hyper-Kahler metrics with SO(3) isometry.