Circle Bundle Research Articles

Hyperkähler sigma model and field theory on Gibbons-Hawking spaces

We describe a novel deformation of the 3-dimensional sigma model with hyperk\"ahler target, which arises naturally from the compactification of a 4-dimensional $\mathcal{N}=2$ theory on a hyperk\"ahler circle bundle (Gibbons-Hawking space). We derive the condition for which the deformed sigma model preserves 4 out of the 8 supercharges. We also study the contribution from a NUT center to the sigma model path integral, and find that supersymmetry implies it is a holomorphic section of a certain holomorphic line bundle over the hyperk\"ahler target. We study explicitly the case where the original 4-dimensional theory is pure $U(1)$ super Yang-Mills, and show that the contribution from a NUT center in this case is simply the Jacobi theta function.

Line bundles and the Thom construction in noncommutative geometry

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct \mathbb{Z} - and \mathbb{N} -graded algebras, the \mathbb{Z} -graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the \mathbb{N} -graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.

Topological T-duality for general circle bundles

We extend topological T-duality to the case of general circle bundles. In this setting we prove existence and uniqueness of T-duals. We then show that T-dual spaces have isomorphic twisted cohomology, twisted $K$-theory and Courant algebroids. A novel feature is that we must consider two kinds of twists in de Rham cohomology and $K$-theory, namely by degree 3 integral classes and a less familiar kind of twist using real line bundles. We give some examples of T-dual non-oriented circle bundles and calculate their twisted $K$-theory.

Twistor Space for Rolling Bodies

On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g is not antiselfdual, the distribution D is (2,3,5) in T(M). We show that if M is a Cartesian product of two Riemann surfaces (S1,g1) and (S2,g2), and if g=g1--g2, then the circle bundle T(S1 x S2) is just the configuration space for the physical system of two solid bodies B1 and B2, bounded by the surfaces S1 and S2 and rolling on each other. The condition for the two bodies to roll on each other `without slipping or twisting' identifies the restricted velocity space for such a system with the tautological distribution D on T(S1 x S2). We call T(S1 x S2) the twistor space, and D the twistor distribution for the rolling bodies. Among others we address the following question: "For which pairs of bodies does the restricted velocity distribution (which we identify with the twistor distribution D) have the simple Lie group G2 as its group of symmetries?" Apart from the well known situation when the boundaries S1 and S2 of the two bodies have constant curvatures whose ratio is 1:9, we unexpectedly find three different types of surfaces, which when bounding a body that rolls `without slipping or twisting' on a plane, have D with the symmetry group G2. Although we have found the differential equations for the curvatures of S1 and S2 that gives D with G2 symmetry, we are unable to solve them in full generality so far.

A Floer–Gysin exact sequence for Lagrangian submanifolds

We establish a Floer-theoretical analog of the classical Gysin long exact sequence from algebraic topology for circle bundles. We study algebraic and functorial properties of this sequence and derive applications to computations of Lagrangian Floer homologies as well as to questions on the topology of Lagrangian submanifolds.

Trivializations of differential cocycles

Associated to a differential character is an integral cohomology class, referred to as the characteristic class, and a closed differential form, referred to as the curvature. The characteristic class and curvature are equal in de Rham cohomology, and this is encoded in a commutative square. In the Hopkins–Singer model, where differential characters are equivalence classes of differential cocycles, there is a natural notion of trivializing a differential cocycle. In this paper, we extend the notion of characteristic class, curvature, and de Rham class to trivializations of differential cocycles. These structures fit into a commutative square, and this square is a torsor for the commutative square associated to characters with degree one less. Under the correspondence between degree two differential cocycles and principal circle bundles with connection, we recover familiar structures associated to global sections.

Weighted Circle Actions on the Heegaard Quantum Sphere

Weighted circle actions on the quantum Heeqaard 3-sphere are considered. The fixed point algebras, termed quantum weighted Heegaard spheres, and their representations are classified and described on algebraic and topological levels. On the algebraic side, coordinate algebras of quantum weighted Heegaard spheres are interpreted as generalised Weyl algebras, quantum principal circle bundles and Fredholm modules over them are constructed, and the associated line bundles are shown to be non-trivial by an explicit calculation of their Chern numbers. On the topological side, the C*-algebras of continuous functions on quantum weighted Heegaard spheres are described and their K-groups are calculated.

A COUNTEREXAMPLE TO GUILLEMIN'S ZOLLFREI CONJECTURE

We construct Zollfrei Lorentzian metrics on every non-trivial orientable circle bundle over an orientable closed surface. Further we prove a weaker version of Guillemin's conjecture assuming global hyperbolicity of the universal cover.

Equivariant Willmore surfaces in conformal homogeneous three spaces

The complete classification of homogeneous three spaces is well known for some time. Of special interest are those with rigidity four which appear as Riemannian submersions with geodesic fibres over surfaces with constant curvature. Consequently their geometries are completely encoded in two values, the constant curvature, c, of the base space and the so called bundle curvature, r. In this paper, we obtain the complete classification of equivariant Willmore surfaces in homogeneous three spaces with rigidity four. All these surfaces appear by lifting elastic curves of the base space. Once more, the qualitative behaviour of these surfaces is encoded in the above mentioned parameters (c,r). The case where the fibres are compact is obtained as a special case of a more general result that works, via the principle of symmetric criticality, for bundle-like conformal structures in circle bundles. However, if the fibres are not compact, a different approach is necessary. We compute the differential equation satisfied by the equivariant Willmore surfaces in conformal homogeneous spaces with rigidity of order four and then we reduce directly the symmetry to obtain the Euler Lagrange equation of 4r2-elasticae in surfaces with constant curvature, c. We also work out the solving natural equations and the closed curve problem for elasticae in surfaces with constant curvature. It allows us to give explicit parametrizations of Willmore surfaces and Willmore tori in those conformal homogeneous 3-spaces.

CR immersions and Lorentzian geometry

Using tools from Lorentzian geometry (arising from the presence of the Fefferman metric) we prove a Takahashi type theorem (for a class of pseudohermitian immersions covered by connection-preserving equivariant immersions among the total spaces of the canonical circle bundles) thus relating the geometry of a pseudohermitian immersion from a strictly pseudoconvex CR manifold $$M$$ into an odd dimensional sphere, to the spectrum of the sublaplacian on $$M$$ .

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kahler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szego kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Englis (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.

Skyrmions from gravitational instantons

We propose a construction of Skyrme fields from holonomy of the spin connection of gravitational instantons. The procedure is implemented for Atiyah–Hitchin and Taub–NUT instantons. The skyrmion resulting from the Taub–NUT is given explicitly on the space of orbits of a left translation inside the whole isometry group. The domain of the Taub–NUT skyrmion is a trivial circle bundle over the Poincaré disc. The position of the skyrmion depends on the Taub–NUT mass parameter, and its topological charge is equal to two.

Conformal symmetries of self-dual hyperbolic monopole metrics

We determine the group of conformal automorphisms of the self-dual metrics on $n \# \mathbb{CP}^{2}$ due to LeBrun for $n \geq 3$, and Poon for $n = 2$. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space $\mathcal{H}^{3}$ minus a finite number of points, called monopole points. We show that for $n \geq 3$, any conformal automorphism is a lift of an isometry of $\mathcal{H}^{3}$ which preserves the set of monopole points. Furthermore, we prove that for $n = 2$, such lifts form a subgroup of index $2$ in the full automorphism group, which we show to be a semi-direct product $(\mathrm{U}(1) \times \mathrm{U}(1)) \rtimes \mathrm{D}_{4}$, where $\mathrm{D}_{4}$ is the dihedral group of order $8$.

Partial hyperbolicity on 3-dimensional nilmanifolds

Every partially hyperbolic diffeomorphism on a 3-dimensional nilmanifold is leaf conjugate to a nilmanifold automorphism. Moreover, if the nilmanifold is not the 3-torus, the center foliation is an invariant circle bundle.

Complex structures on product of circle bundles over complex manifolds

Soient L ¯ i →X i des fibrés en droites holomorphes sur des variétés complexes compactes, pour i=1,2. Soit S i le fibré en cercles associé par rapport à un produit scalaire hermitienne sur L ¯ i . On construit des structures complexes sur S=S 1 ×S 2 dites de type scalaire, diagonal, ou linéaire. Bien que des structures de type scalaire existent toujours, on construit des structures plus générales de type diagonal mais non-scalaire dans le cas où les L ¯ i sont des (ℂ * ) n i fibrés équivariants qui vérifient certaines hypothèses supplémentaires. Les structures complexes de type linéaire sont des variétés des drapeaux (généralisées) et les L i sont des fibrés en droites amples négatifs. Lorsque H 1 (X 1 ;ℝ)=0 et c 1 (L ¯ 1 ) est non-nulle la variété compacte S n’admet pas de structure symplectique et donc elle est non-Kählerienne par rapport à toute structure complexe.

CONFORMAL COURANT ALGEBROIDS AND ORIENTIFOLD T-DUALITY

We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs (L, H) with L a flat line bundle and H ∈ H3(M, L) a degree 3 class with coefficients in L. As a special case gerbes for the crossed module (U(1) → ℤ2) can be used to twist TM ⊕ T*M into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if L2= 1. The structure of Conformal Courant algebroids on circle bundles leads us to construct a T-duality for orientifolds with free involution. This incarnation of T-duality yields an isomorphism of 4-periodic twisted cohomology. We conjecture that the isomorphism extends to an isomorphism in twisted KR-theory and give some calculations to support this claim.

M5/D4 brane partition function on a circle bundle

We consider Abelian M5 brane on a six-manifold which we take as a circle bundle over a five-manifold M . We compute the zero mode part of the M5 brane partition function using Chern-Simons theory and Hamiltonian formulation respectively and find an agreement. We also show that the D4 brane on M shares exactly the same zero mode partition function again using the Hamiltonian formulation. For the oscillator modes we find that KK modes associated with the circle compactification are missing from the D4 brane. By making an infinitesimal noncommutative deformation we have instanton threshold bound states. We explicitly compute the instanton partition function up to instanton charge three, and show a perfect match with a corresponding contribution inside the M5 brane partition function, thus providing a very strong supporting evidence that D4 brane is identical with M5 brane which extends beyond the BPS sector. We comment on the modular properties of the M5 brane partition function when compactified on T 2 times a four-manifold. We briefly discuss a case of a singular fibration.

Quantization via Deformation of Prequantization

We introduce the notion of a "Souriau bracket" on a prequantum circle bundle $Y$ over a phase space $X$ and explain how a deformation of $Y$ in the direction of this bracket provides a genuine quantization of $X$.

Quasi-Einstein metrics on hypersurface families

We construct quasi-Einstein metrics on some hypersurface families. The hypersurfaces are circle bundles over the product of Fano, Kähler–Einstein manifolds. The quasi-Einstein metrics are related to various gradient Kähler–Ricci solitons constructed by Dancer and Wang and some Hermitian, non-Kähler, Einstein metrics constructed by Wang and Wang on the same manifolds.

Non-Abelian localization for supersymmetric Yang-Mills-Chern-Simons theories on a Seifert manifold

We derive non-Abelian localization formulae for supersymmetric Yang-Mills-Chern-Simons theory with matters on a Seifert manifold M, which is the three-dimensional space of a circle bundle over a two-dimensional Riemann surface \Sigma, by using the cohomological approach introduced by Kallen. We find that the partition function and the vev of the supersymmetric Wilson loop reduces to a finite dimensional integral and summation over classical flux configurations labeled by discrete integers. We also find the partition function reduces further to just a discrete sum over integers in some cases, and evaluate the supersymmetric index (Witten index) exactly on S^1x\Sigma. The index completely agrees with the previous prediction from field theory and branes. We discuss a vacuum structure of the ABJM theory deduced from the localization.