Hyperkahler Research Articles

On gauged linear sigma models with torsion

We study a broad class of two dimensional gauged linear sigma models (GLSMs) with o-shell N = (2; 2) supersymmetry that flow to nonlinear sigma models (NLSMs) on noncompact geometries with torsion. These models arise from coupling chiral, twisted chiral, and semichiral multiplets to known as well as to a new N = (2; 2) vector multiplet, the constrained semichiral vector multiplet (CSVM). We discuss three kinds of models, corresponding to torsionful deformations of standard GLSMs realizing Kahler, hyperkahler, and Calabi-Yau manifolds. The (2; 2) supersymmetry guarantees that these spaces are generalized Kahler. Our analysis of the geometric structure is performed at the classical level, but we also discuss quantum aspects such as R-symmetry anomalies. We provide an explicit example of a generalized Kahler structure on the conifold.

Teichmüller space for hyperkähler and symplectic structures

Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and $Diff_0$ the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifold M are cohomology classes which can be represented by a minimal rational curve on a deformation of M. We determine the Teichmuller space of hyperkahler structures on a hyperkahler manifold, identifying any of its connected components with an open subset of the Grassmannian $SO(b_2-3,3)/SO(3)\times SO(b_2-3)$ consisting of all Beauville-Bogomolov positive 3-planes in $H^2(M, R)$ which are not orthogonal to any of the MBM classes. This is used to determine the Teichmuller space of symplectic structures of Kahler type on a hyperkahler manifold of maximal holonomy. We show that any connected component of this space is naturally identified with the space of cohomology classes $v\in H^2(M,\R)$ with $q(v,v)>0$, where $q$ is the Bogomolov-Beauville-Fujiki form on $H^2(M,\R)$.

Poisson structures on twistor spaces of hyperkähler and HKT manifolds

We characterize HKT structure in terms of nondegenrate complex Poisson bivector on hypercomplex manifold. We extend the characterization to the twistor space. After considering the flat case in detail, we show that the twistor space of hyperkaehler manifold admits a holomorphic Poisson structure. We briefly mention the relation to quaternionic and hypercomplex deformations on tori and K3 surfaces

Mathematical Theory and Physical Mechanics for Planetary Ionospheric Physics

In a dynamic system, e.g., Geometrodynamics geophysical isomorphisms from plasmasphere^i to ionosphere^ii, e.g., Upper-atmospheric lightning (UAL, sferics), Middle-atmospheric lightning and Lower-atmospheric lightning (MAL, LAL, sferics) and Terrestrial and Subterranean Perturbation Regimes (TSTPR, terics) real Physical space is represented as (M,g) R^ 5→(M,g) R^4 brane. F-theory propagates QED continuous polyphasic flux to (Mg) R^4 brane is postulated utilizing Universal constants (K), c.f. Newton's Laws of Motion; c; Phi; Boltzmann's Constant loge S = k W ; Gaussian distributions; Maxwell's Equations; Planck time and Planck Space constants; a; Psi. Constants are propagated from hypothesized compaction and perturbation of topological gauged-energy string landscape (Mg) R^4 d-brane applied to electromagnetic and gravitational Geophysical sweep-out phenomena, e.g. Birkeland currents, ring currents, sferics, terics and given tensorized fields of ionized plasma events^iii and energy phenomena of the near Astrophysical medium. These can be computed from Calabi-Yau manifolds as CP^4 in density matrices of Hilbert space, Hyper-Kahler or 4-Kahler manifolds across weighted projective space. e.g., in Gaussian Unitary Ensembles (GUE) where as a joint probability for eigenvalues and-vectors 3 2 4 1 1 k i j j i k e Z η η λ β βη λ λ − < = Π Π − (1) from dispersion k^2=w^2 p_0 from Boltzmann's constant H [1] and Trubnikov's 0, 1, 2, 3 tensors [2,3].

K-symplectic structures and absolutely trianalytic subvarieties in hyperkähler manifolds

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures $(M,I,J',K')$ containing $I$. For a generic complex structure $I$ on $M$, all complex subvarieties of $(M,I)$ are absolutely trianalytic. It is known that a normalization $Z'$ of a trianalytic subvariety is smooth; we prove that $b_2(Z')$ is no smaller than $b_2(M)$ when $M$ has maximal holonomy (that is, $M$ is IHS). To study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of the hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold $X$ is a k-dimensional space $R$ of closed 2-forms on $X$ which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in $R$ is a smooth, non-degenerate quadric hypersurface in $R$. We consider absolutely trianalytic tori in a hyperkahler manifold $M$ of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where $k=b_2(M)$. We show that the tangent bundle $TX$ of a k-symplectic manifold is a Clifford module over a Clifford algebra $Cl(k-1)$. Then an absolutely trianalytic torus in a hyperkahler manifold $M$ with $b_2(M)\geq 2r+1$ is at least $2^{r-1}$-dimensional.

Symmetry and quaternionic integrable systems

Given a hyperkahler manifold M, the hyperkahler structure defines a triple of symplectic structures on M; with these, a triple of Hamiltonians defines a so called hyperhamiltonian dynamical system on M. These systems are integrable when can be mapped to a system of quaternionic oscillators. We discuss the symmetry of integrable hyperhamiltonian systems, i.e. quaternionic oscillators; and conversely how these symmetries characterize, at least in the Euclidean case, integrable hyperhamiltonian systems.

Pfaffian bundles on cubic surfaces and configurations of planes

We construct a canonical birational map between the moduli space of Pfaffian vector bundles on a cubic surface and the space of complete pentahedra inscribed in the cubic surface. The universal situation is also considered and we obtain a rationality result. As a by-product, we provide an explicit normal form for five general lines in \(\mathbb {P}_5\). Applications to the geometry of Palatini threefolds and Debarre–Voisin Hyper-Kahler manifolds are also discussed.

Manifolds with holonomy $$U^*(2m)$$ U ∗ ( 2 m )

We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup \(U^*(2m)\), a real form of \(GL(2m,\mathbf {C})\), otherwise denoted by \(SL(m,\mathbf {H}) \cdot U(1)\). We show in particular how examples may be generated from quaternionic Kahler or hyperkahler manifolds with a circle action.

Some model theory of fibrations and algebraic reductions

Let \(p= \mathrm{tp }(a/A)\) be a stationary finite rank type in an arbitrary stable theory and \({\mathbb {P}}\) an \(A\)-invariant family of partial types. The following property is introduced and characterised: whenever \(c\) is definable over \((A,a)\) and \(a\) is not algebraic over \((A,c)\), then \(\mathrm{tp }(c/A)\) is almost internal to \({\mathbb {P}}\). The characterisation involves among other things an apparently new notion of “descent” for stationary types. Motivation comes partly from results in Sect. 2 of (Campana et al. in J Differ Geom 85(3):397–424, 2010) where structural properties of generalised hyperkahler manifolds are given. The model-theoretic results obtained here are applied back to the complex-analytic setting to prove that the algebraic reduction of a nonalgebraic (generalised) hyperkahler manifold does not descend. The results are also applied to the theory of differentially closed fields, where examples coming from differential-algebraic groups are given.

On involutions with middle-dimensional fixed-point locus and holomorphic-symplectic manifolds

Let $$g$$ be an involution of a compact closed manifold $$X$$ such that the fixed-point set $$X^{g}$$ is middle dimensional. Under the assumption that the normal bundle of the fixed-point set is either the tangent or co-tangent bundle conditions on the equivariant invariants of $$X$$ arise. In particular if $$X$$ is a holomorphic-symplectic manifold and $$g$$ an anti holomorphic-symplectic involution one arrives at a generalisation of Beauville’s result that for $$X$$ a hyper-Kahler manifold the $$\hat{A}$$ genus of $$X^{g}$$ is one.

Self-dual supersymmetric nonlinear sigma models

In four-dimensional N=1 Minkowski superspace, general nonlinear sigma models with four-dimensional target spaces may be realised in term of CCL (chiral and complex linear) dynamical variables which consist of a chiral scalar, a complex linear scalar and their conjugate superfields. Here we introduce CCL sigma models that are invariant under U(1) "duality rotations" exchanging the dynamical variables and their equations of motion. The Lagrangians of such sigma models prove to obey a partial differential equation that is analogous to the self-duality equation obeyed by U(1) duality invariant models for nonlinear electrodynamics. These sigma models are self-dual under a Legendre transformation that simultaneously dualises (i) the chiral multiplet into a complex linear one; and (ii) the complex linear multiplet into a chiral one. Any CCL sigma model possesses a dual formulation given in terms of two chiral multiplets. The U(1) duality invariance of the CCL sigma model proves to be equivalent, in the dual chiral formulation, to a manifest U(1) invariance rotating the two chiral scalars. Since the target space has a holomorphic Killing vector, the sigma model possesses a third formulation realised in terms of a chiral multiplet and a tensor multiplet. The family of U(1) duality invariant CCL sigma models includes a subset of N=2 supersymmetric theories. Their target spaces are hyper Kahler manifolds with a non-zero Killing vector field. In the case that the Killing vector field is triholomorphic, the sigma model admits a dual formulation in terms of a self-interacting off-shell N=2 tensor multiplet. We also identify a subset of CCL sigma models which are in a one-to-one correspondence with the U(1) duality invariant models for nonlinear electrodynamics. The target space isometry group for these sigma models contains a subgroup U(1) x U(1).

Nonlinear sigma models with AdS supersymmetry in three dimensions

In three-dimensional anti-de Sitter (AdS) space, there exist several realizations of N-extended supersymmetry, which are traditionally labelled by two non-negative integers p>=q such that p+q=N. Different choices of p and q, with N fixed, prove to lead to different restrictions on the target space geometry of supersymmetric nonlinear sigma-models. We classify all possible types of hyperkahler target spaces for the cases N=3 and N=4 by making use of two different realizations for the most general (p,q) supersymmetric sigma-models: (i) off-shell formulations in terms of N=3 and N=4 projective supermultiplets; and (ii) on-shell formulations in terms of covariantly chiral scalar superfields in (2,0) AdS superspace. Depending on the type of N=3,4 AdS supersymmetry, nonlinear sigma-models can support one of the following target space geometries: (i) hyperkahler cones; (ii) non-compact hyperkahler manifolds with a U(1) isometry group which acts non-trivially on the two-sphere of complex structures; (iii) arbitrary hyperkahler manifolds including compact ones. The option (iii) is realized only in the case of critical (4,0) AdS supersymmetry. As an application of the (4,0) AdS techniques developed, we also construct the most general nonlinear sigma-model in Minkowski space with a non-centrally extended N=4 Poincare supersymmetry. Its target space is a hyperkahler cone (which is characteristic of N=4 superconformal sigma-models), but the sigma-model is massive. The Lagrangian includes a positive potential constructed in terms of the homothetic conformal Killing vector the target space is endowed with. This mechanism of mass generation differs from the standard one which corresponds to a sigma-model with the ordinary N=4 Poincare supersymmetry and which makes use of a tri-holomorphic Killing vector.

Zariski $F$-decomposition and Lagrangian fibration on hyperkähler manifolds

For a compact HyperKahler manifold X , we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a Lagrangian fibration.

The semi-chiral quotient, hyperkähler manifolds and T-duality

We study the construction of generalized Kahler manifolds, described purely in terms of N=(2,2) semichiral superfields, by a quotient using the semichiral vector multiplet. Despite the presence of a b-field in these models, we show that the quotient of a hyperkahler manifold is hyperkahler, as in the usual hyperkahler quotient. Thus, quotient manifolds with torsion cannot be constructed by this method. Nonetheless, this method does give a new description of hyperkahler manifolds in terms of two-dimensional N=(2,2) gauged non-linear sigma models involving semichiral superfields and the semichiral vector multiplet. We give two examples: Eguchi-Hanson and Taub-NUT. By T-duality, this gives new gauged linear sigma models describing the T-dual of Eguchi-Hanson and NS5-branes. We also clarify some aspects of T-duality relating these models to N=(4,4) models for chiral/twisted-chiral fields and comment briefly on more general quotients that can give rise to torsion and give an example.

ON CHERN–SIMONS QUIVERS AND TORIC GEOMETRY

We discuss a class of three-dimensional [Formula: see text] Chern–Simons (CS) quiver gauge models obtained from M-theory compactifications on singular complex four-dimensional hyper-Kähler (HK) manifolds, which are realized explicitly as a cotangent bundle over two-Fano toric varieties V2. The corresponding CS gauge models are encoded in quivers similar to toric diagrams of V2. Using toric geometry, it is shown that the constraints on CS levels can be related to toric equations determining V2.

A Model for the HyperKahler Metrics Building Program

Its by now well known that the harmonic superspace provides astrong framework for constructing general hyperKahler metrics. Origi-nal results in this issue are given by the Taub-Nut and Egushi-Hansonmetrics exhibiting both U(1) Pauli-Gursey isomertie as shown by Gib-bon et all in [1]. Its also shown that there exist series of potentialdepending explicitly on the harmonics variables u on the sphere S2.Based on these knowledge and on previous contributions to the pro-gram of metrics building [2, 3], we contribute by presenting a specialhyperKahler potential H+4 leading to an -parametrized Liouville equa-tion on the harmonic superspace.

Three-dimensional (p, q) AdS superspaces and matter couplings

We introduce N-extended (p,q) AdS superspaces in three space-time dimensions, with p+q=N and p>=q, and analyse their geometry. We show that all (p,q) AdS superspaces with X^{IJKL}=0 are conformally flat. Nonlinear sigma-models with (p,q) AdS supersymmetry exist for p+q<=4 (for N>4 the target space geometries are highly restricted). Here we concentrate on studying off-shell N=3 supersymmetric sigma-models in AdS_3. For each of the cases (3,0) and (2,1), we give three different realisations of the supersymmetric action. We show that (3,0) AdS supersymmetry requires the sigma-model to be superconformal, and hence the corresponding target space is a hyperkahler cone. In the case of (2,1) AdS supersymmetry, the sigma-model target space must be a non-compact hyperkahler manifold endowed with a Killing vector field which generates an SO(2) group of rotations of the two-sphere of complex structures.

Extended supersymmetric sigma models in AdS4 from projective superspace

There exist two superspace approaches to describe N=2 supersymmetric nonlinear sigma models in four-dimensional anti-de Sitter (AdS_4) space: (i) in terms of N=1 AdS chiral superfields, as developed in arXiv:1105.3111 and arXiv:1108.5290; and (ii) in terms of N=2 polar supermultiplets using the AdS projective-superspace techniques developed in arXiv:0807.3368. The virtue of the approach (i) is that it makes manifest the geometric properties of the N=2 supersymmetric sigma-models in AdS_4. The target space must be a non-compact hyperkahler manifold endowed with a Killing vector field which generates an SO(2) group of rotations on the two-sphere of complex structures. The power of the approach (ii) is that it allows us, in principle, to generate hyperkahler metrics as well as to address the problem of deformations of such metrics. Here we show how to relate the formulation (ii) to (i) by integrating out an infinite number of N=1 AdS auxiliary superfields and performing a superfield duality transformation. We also develop a novel description of the most general N=2 supersymmetric nonlinear sigma-model in AdS_4 in terms of chiral superfields on three-dimensional N=2 flat superspace without central charge. This superspace naturally originates from a conformally flat realization for the four-dimensional N=2 AdS superspace that makes use of Poincare coordinates for AdS_4. This novel formulation allows us to uncover several interesting geometric results.

Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence

When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kahler manifolds with a quaternionic isometry and, on the other hand, hyperkahler manifolds with a rotational isometry, further equipped with a hyperholomorphic circle bundle with a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wall-crossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wall-crossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wall-crossing and the theory of cluster algebras.

AdS5 supersymmetry in N = 1 superspace

We use N=1 superspace to construct the supersymmetric matter couplings of vector and hyper multiplets in a five-dimensional anti-de Sitter spacetime background. For hypermultiplets, we find that AdS_5 supersymmetry requires the scalar fields to lie on a hyper-Kahler manifold endowed with a certain type of holomorphic Killing vector.