Torus Symmetry Research Articles

All separable supersymmetric AdS5 black holes

We consider the classification of supersymmetric black hole solutions to five-dimensional STU gauged supergravity that admit torus symmetry. This reduces to a problem in toric Kähler geometry on the base space. We introduce the class of separable toric Kähler surfaces that unify product-toric, Calabi-toric and orthotoric Kähler surfaces, together with an associated class of separable 2-forms. We prove that any supersymmetric toric solution that is timelike, with a separable Kähler base space and Maxwell fields, outside a horizon with a compact (locally) spherical cross-section, must be locally isometric to the known black hole or its near-horizon geometry. An essential part of the proof is a near-horizon analysis which shows that the only possible separable Kähler base space is Calabi-toric. In particular, this also implies that our previous black hole uniqueness theorem for minimal gauged supergravity applies to the larger class of separable Kähler base spaces.

Curvature-induced stabilization and field-driven dynamics of magnetic hopfions in toroidal nanorings

Three dimensional magnetic textures are a cornerstone in magnetism research. In this work, we analyze the stabilization and dynamic response of a magnetic hopfion hosted in a toroidal nanoring with intrinsic Dzyaloshinskii–Moriya interaction simulating FeGe. Our results evidence that unlike their planar counterparts, where perpendicular magnetic anisotropies are necessary to stabilize hopfions, the shape anisotropy originated on the torus symmetry naturally yields the nucleation of these topological textures. We also analyze the magnetization dynamical response by applying a magnetic field pulse to differentiate among several magnetic patterns. Finally, to understand the nature of spin wave modes, we analyze the spatial distributions of the resonant mode amplitudes and phases and describe the differences among bulk and surface modes. Importantly, hopfions lying in toroidal nanorings present a non-circularly symmetric poloidal resonant mode, which is not observed in other systems hosting hopfions.

On the uniqueness of supersymmetric AdS(5) black holes with toric symmetry

We consider the classification of supersymmetric AdS5 black hole solutions to minimal gauged supergravity that admit a torus symmetry. This problem reduces to finding a class of toric Kähler metrics on the base space, which in symplectic coordinates are determined by a symplectic potential. We derive the general form of the symplectic potential near any component of the horizon or axis of symmetry, which determines its singular part for any black hole solution in this class, including possible new solutions such as black lenses and multi-black holes. We find that the most general known black hole solution in this context, found by Chong, Cvetic, Lü and Pope (CCLP), is described by a remarkably simple symplectic potential. We prove that any supersymmetric and toric solution that is timelike outside a smooth horizon, with a Kähler base metric of Calabi type, must be the CCLP black hole solution or its near-horizon geometry.

Apparent counter-rotation in the torus of NGC 1068: influence of an asymmetric wind

ABSTRACT The recent ALMA maps together with observations of H2O maser emission seem to suggest the presence of a counter-rotation in the obscuring torus of NGC 1068. We propose to explain this phenomenon as due to the influence of a wind, considered as radiation pressure, and the effects of torus orientation. In order to test this idea: 1. we make N-body simulation of a clumpy torus taking into account mutual forces between particles (clouds); 2. we apply ray-tracing algorithm with the beams from the central engine to choose the clouds in the torus throat that can be under direct influence of the accretion disk emission; 3. we use semi-analytical model to simulate the influence of the asymmetrical radiation pressure (wind) forced on the clouds in the torus throat. An axis of such a wind is tilted with respect to the torus symmetry axis; 4. we orient the torus relative to an observer and again apply ray-tracing algorithm. In this step the beams go from an observer to the optically thick clouds that allows us to take into account the mutual obscuration of clouds; 5. after projecting on the picture plane, we impose a grid on the resulting cloud distribution and find the mean velocity of clouds in each cells to mimic the ALMA observational maps. By choosing the parameters corresponding to NGC 1068 we obtain the model velocity maps that emulate the effect of an apparent counter-rotation and can explain the discovery made by ALMA.

Existence and uniqueness of asymptotically flat toric gravitational instantons

We prove uniqueness and existence theorems for four-dimensional asymptotically flat, Ricci-flat, gravitational instantons with a torus symmetry. In particular, we prove that such instantons are uniquely characterised by their rod structure, which is data that encodes the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. The proofs involve adapting the methods that are used to establish black hole uniqueness theorems, to a harmonic map formulation of Ricci-flat metrics with torus symmetry, where the target space is directly related to the metric (rather than auxiliary potentials). We also give an elementary proof of the nonexistence of asymptotically flat toric half-flat instantons. Finally, we derive a general set of identities that relate asymptotic invariants such as the mass to the rod structure.

Symmetries of theq-NKdV hierarchy

In this paper, the additional symmetries and the string equation of the [Formula: see text]-NKdV hierarchy are established basing on the Orlov–Shulman’s [Formula: see text] operator, and then some properties are also given. Next, we construct the quantum torus symmetry of the [Formula: see text]-NKdV hierarchy and further derive the quantum torus constraints on the tau function of the [Formula: see text]-NKdV hierarchy. Comparing to the [Formula: see text] infinite-dimensional Lie symmetry, this quantum torus symmetry has a nice algebraic structure with double indices. Finally, we study the ghost symmetry of the multicomponent [Formula: see text]-NKdV hierarchy.

OBSTRUCTIONS TO FREE ACTIONS ON BAZAIKIN SPACES

Apart from spheres and an infinite family of manifolds in dimension seven, Bazaikin spaces are the only known examples of simply connected Riemannian manifolds with positive sectional curvature in odd dimensions. We consider positively curved Riemannian manifolds whose universal covers have the same cohomology as Bazaikin spaces and prove structural results for the fundamental group in the presence of torus symmetry.

Almost non-negatively curved 4-manifolds with torus symmetry

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.

Surgery, polygons and SU(N)‐Floer homology

Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of $SU(N)$-instanton Floer homology with respect to Dehn surgery is studied. In particular, it is shown that there are surgery exact tetragons and pentagons, respectively, for $SU(3)$- and $SU(4)$-instanton Floer homologies. It is also conjectured that $SU(N)$-instanton Floer homology in general admits a surgery exact $(N+1)$-gon. An essential step in the proof is the construction of a family of asymptotically cylindrical metrics on ALE spaces of type $A_n$. This family is parametrized by the $(n-2)$-dimensional associahedron and consists of anti-self-dual metrics with positive scalar curvature. The metrics in the family also admit a torus symmetry.

Quantum Torus Lie Algebra in the q-Deformed Kadomtsev–Petviashvili System

Based on the W∞ symmetry of the q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy, which is a q-deformation of the KP hierarchy, we construct the quantum torus symmetry of the q-KP hierarchy, which further leads to the quantum torus constraint of its tau function. Moreover, we generalize the system to a multi-component q-KP hierarchy that also has the well-known ghost symmetry.

Positive curvature and symmetry in small dimensions

Extending existing work in small dimensions, Dessai computed the Euler characteristic, signature, and elliptic genus for [Formula: see text]-manifolds of positive sectional curvature in the presence of torus symmetry. He also computes the diffeomorphism type by restricting his results to classes of manifolds known to admit non-negative curvature, such as biquotients. The first part of this paper extends Dessai’s calculations to even dimensions up to [Formula: see text]. In particular, we obtain a first characterization of the Cayley plane in such a setting. The second part studies a closely related family of manifolds called positively elliptic manifolds, and we prove a conjecture of Halperin in this context for dimensions up to [Formula: see text] or Euler characteristics up to [Formula: see text].

The shear construction

The twist construction is a method to build new interesting examples of geometric structures with torus symmetry from well-known ones. In fact it can be used to construct arbitrary nilmanifolds from tori. In our previous paper, we presented a generalization of the twist, a shear construction of rank one, which allowed us to build certain solvable Lie algebras from $${\mathbb R}^n$$ via several shears. Here, we define the higher rank version of this shear construction using vector bundles with flat connections instead of group actions. We show that this produces any solvable Lie algebra from $${\mathbb R}^n$$ by a succession of shears. We give examples of the shear and discuss in detail how one can obtain certain geometric structures (calibrated $$ G _2$$ , co-calibrated $$ G _2$$ and almost semi-Kahler) on three-step solvable Lie algebras by shearing almost Abelian Lie algebras. This discussion yields a classification of calibrated $$ G _2$$ -structures on Lie algebras of the form $$(\mathfrak {h}_3\oplus {\mathbb R}^3)\rtimes {\mathbb R}$$ .

Quantum torus symmetries of the CKP and multi-component CKP hierarchies

In this paper, we construct a series of additional flows of the C type Kadomtsev-Petviashvili (CKP) hierarchy and the multi-component CKP hierarchy, and these flows constitute an N-fold direct product of the positive half of the quantum torus symmetry. Comparing to the W∞ infinite dimensional Lie symmetry, this quantum torus symmetry has a nice algebraic structure with double indices.

Wedge Operations and Torus Symmetries II

AbstractA fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K.The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K.We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We call K a seed if K cannot be obtained by wedgings. First, we show that for a fixed positive integer 𝓁, there are at most finitely many seeds of Picard 𝓁 number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.Secondly, we investigate a systematicmethod to find all characteristic maps on K(J) using combinatorial objects called (realizable) puzzles that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy basing on the $W_{\infty}$ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchy. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an $N$-folds direct product of the positive half of the quantum torus symmetries.

Fundamental Groups of Manifolds with Positive Sectional Curvature and Torus Symmetry

In 1965, Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry. These classifications fall into categories according to the strength of their conclusions. We give an overview of these results in the case of torus symmetry and prove new results in each of these categories.

On a generalized conjecture of Hopf with symmetry

A famous conjecture of Hopf states that$\mathbb{S}^{2}\times \mathbb{S}^{2}$does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product$N\times N$can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry.

Wedge operations and torus symmetries

A fundamental result of toric geometry is that there is a bijection between toric varieties and fans. More generally, it is known that some classes of manifolds having well-behaved torus actions, say toric objects, can be classified in terms of combinatorial data containing simplicial complexes. In this paper, we investigate the relationship between the topological toric manifolds over a simplicial complex $K$ and those over the complex obtained by simplicial wedge operations from $K$. Our result provides a systematic way to classify toric objects associated with the class of simplicial complexes obtained from a given $K$ by wedge operations. As applications, we completely classify smooth toric varieties with a few generators and show their projectivity. We also study smooth real toric varieties.

Gauge transformation and symmetries of the commutative multicomponent BKP hierarchy

In this paper, we defined a new multi-component B type Kadomtsev-Petviashvili (BKP) hierarchy that takes values in a commutative subalgebra of After this, we give the gauge transformation of this commutative multicomponent BKP (CMBKP) hierarchy. Meanwhile, we construct a new constrained CMBKP hierarchy that contains some new integrable systems, including coupled KdV equations under a certain reduction. After this, the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi-component BKP hierarchy will be constructed.

Positive curvature and rational ellipticity

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm.