Fibration Structure Research Articles

Massive Abelian gauge symmetries and fluxes in F-theory

F-theory compactified on a Calabi-Yau fourfold naturally describes non-Abelian gauge symmetries through the singularity structure of the elliptic fibration. In contrast Abelian symmetries are more difficult to study because of their inherently global nature. We argue that in general F-theory compactifications there are massive Abelian symmetries, such as the uplift of the Abelian part of the U(N) gauge group on D7-branes, that arise from non-Kahler resolutions of the dual M-theory setup. The four-dimensional F-theory vacuum with vanishing expectation values for the gauge fields corresponds to the Calabi-Yau limit. We propose that fluxes that are turned on along these U(1)s are uplifted to non-harmonic four-form fluxes. We derive the effective four-dimensional gauged supergravity resulting from F-theory compactifications in the presence of the Abelian gauge factors including the effects of possible fluxes on the gauging, tadpoles and matter spectrum.

Lefschetz fibration structures on knot surgery 4-manifolds

In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots $K$. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite family of simply connected symplectic 4-manifolds, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.

Conjugacy classes in Möbius groups

Let $${\mathbb H^{n+1}}$$ denote the n + 1-dimensional (real) hyperbolic space. Let $${\mathbb {S}^{n}}$$ denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of $${\mathbb {S}^{n}}$$ is denoted by M(n). Let M o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M o (n) depends on the conjugacy of T and T −1 in M o (n). For an element T in M(n), T and T −1 are conjugate in M(n), but they may not be conjugate in M o (n). In the literature, T is called real if T is conjugate in M o (n) to T −1. In this paper we classify real elements in M o (n). Let T be an element in M o (n). Corresponding to T there is an associated element T o in SO(n + 1). If the complex conjugate eigenvalues of T o are given by $${\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}$$ , then {θ1, . . . , θ k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M o (n) we have parametrized the conjugacy classes of regular elements in M o (n). In the parametrization, when T is not conjugate to T −1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verification of Perelman's collapsing theorem is the last step of Perelman's proof of Thurston's Geometrization Conjecture on the classification of 3-manifolds. Our proof of Perelman's collapsing theorem is almost self-contained, accessible to non-experts and advanced graduate students. Perelman's collapsing theorem for 3-manifolds can be viewed as an extension of implicit function theorem

Seshadri fibrations of algebraic surfaces

AbstractWe refine results of [6] and [10] which relate local invariants – Seshadri constants – of ample line bundles on surfaces to the global geometry – fibration structure. We show that the same picture emerges when looking at Seshadri constants measured at any finite subset of the given surface (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Nonisomorphic Lefschetz fibrations on knot surgery 4-manifolds

In this article we construct an infinite family of simply connected minimal symplectic 4-manifolds, each of which admits at least two nonisomorphic Lefschetz fibration structures with the same generic fiber. We obtain such examples by performing knot surgery on an elliptic surface E(n) using a special type of 2-bridge knots.

Semi-flatland

We study perturbative compactifications of Type II string theory that rely on a fibration structure of the extra dimensions a la SYZ. Non-geometric spaces are obtained by using T-dualities as monodromies. These vacua generically preserve N=1 supersymmetry in four dimensions, and are U-dual to M-theory on G2 manifolds. Several examples are discussed, some of which admit an asymmetric orbifold description. The massless spectrum is matched to that of the dual M-theory compactification on a Joyce manifold when a comparison is possible. We explore the possibility of twisted reductions where left-moving spacetime fermion number Wilson lines are turned on in the fiber. We also give an explanation from this semiflat viewpoint for the Hanany-Witten brane-creation effect and for the equivalence of the Type IIA orientifold on T5/Z2 and Type IIB on S1 x K3.

Fuzzy Fractional Monodromy and the Section–Hyperboloid

A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained. Some ideas for the proof of the theorem are given. A few remarks about the semi-global structure of singular lagrangian fibration are made.

Ample subvarieties and rationally connected fibrations

Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.

Two remarks on sextic double solids

We study the potential density of rational points on double solids ramified along singular reduced sextic surfaces. Also, we investigate elliptic fibration structures on nonsingular sextic double solids defined over a perfect field of characteristic 5.

Cyclic Lagrangian Submanifolds and Lagrangian Fibrations

Let $(M,\omega)$ be a symplectic manifold and $L\subset M$ be a Lagrangian submanifold. In [Oh2], the cyclic condition of $L$ was defined. Y.-G. Oh proved that, in [Oh2], if $(M,\omega)$ is Kahler-Einstein with non-zero scalar curvature and $L$ is minimal, then $L$ is cyclic. In this article, first, we prove that $L$ is cyclic if and only if the ``mean cuvature cohomology class'' of $L$ is rational, when $(M,\omega)$ is Kahler-Einstein with non-zero scalar curvature. Secondly, we see that there are non-cyclic minimal Lagrangian submanifolds when $(M,\omega)$ is a prequantizable Ricci-flat Kahler manifold. Thirdly, if $(M,\omega)$ is Kahler-Einstein with non-zero scalar curvature, there are not minimal Lagrangian fibration structures on $M$ by a result of [Oh2]. Nevertheless we construct Hamiltonian minimal Lagrangian fibration.

Asymptotic form of Gopakumar–Vafa invariants from instanton counting

We study the asymptotic form of the Gopakumar–Vafa invariants at all genera for Calabi–Yau toric threefolds which have the structure of fibration of the A n singularity over P 1. We claim that the asymptotic form is the inverse Laplace transform of the corresponding instanton amplitude in the prepotential of N=2 SU( n+1) gauge theory coupled to external graviphoton fields, which is given by the logarithm of the Nekrasov's partition function.

SUB-SIGNATURE OPERATORS, η-INVARIANTS AND A RIEMANN-ROCH THEOREM FOR FLAT VECTOR BUNDLES

The author presents an extension of the Atiyah-Patodi-Singer invariant for unitary representations [2,3] to the non-unitary case, as well as to the case where the base manifold admits certain finer structures. In particular, when the base manifold has a fibration structure, a Riemann-Roch theorem for these invariants is established by computing the adiabatic limits of the associated η-invariants.

PALP: A Package for Analysing Lattice Polytopes with applications to toric geometry

We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialized to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi–Yau varieties. The package is well tested and optimized in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages. Program summary Program obtainable form: CPC Program Library, Queen's University of Belfast, N. Ireland Title of program: PALP Catalogue identifier: ADSQ Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSQ Computer for which the program is designed: Any computer featuring C Computers on which it has been tested: PCs, SGI Origin 2000, IBM RS/6000, COMPAQ GS140 Operating systems under which the program has been tested: Linux, IRIX, AIX, OSF1 Programming language used: C Memory required to execute with typical data: Negligible for most applications; highly variable for analysis of large polytopes; no minimum but strong effects on calculation time for some tasks Number of bits in a word: arbitrary Number of processors used: 1 Has the code been vectorised or parallelized?: No Number of bytes in distributed program, including test data, etc.: 138 098 Distribution format: tar gzip file Keywords: Lattice polytopes, facet enumeration, reflexive polytopes, toric geometry, Calabi–Yau manifolds, string theory, conformal field theory Nature of problem: Certain lattice polytopes called reflexive polytopes afford a combinatorial description of a very large class of Calabi–Yau manifolds in terms of toric geometry. These manifolds play an essential role for compactifications of string theory. While originally designed to handle and classify reflexive polytopes, with particular emphasis on problems relevant to string theory applications [M. Kreuzer and H. Skarke, Rev. Math. Phys. 14 (2002) 343], the package also handles standard questions (facet enumeration and similar problems) about arbitrary lattice polytopes very efficiently. Method of solution: Much of the code is straightforward programming, but certain key routines are optimized with respect to calculation time and the handling of large sets of data. A double description method (see, e.g., [D. Avis et al., Comput. Geometry 7 (1997) 265]) is used for the facet enumeration problem, lattice basis reduction for extended gcd and a binary database structure for tasks involving large numbers of polytopes, such as classification problems. Restrictions on the complexity of the program: The only hard limitation comes from the fact that fixed integer arithmetic (32 or 64 bit) is used, allowing for input data (polytope coordinates) of roughly up to 10 9. Other parameters (dimension, numbers of points and vertices, etc.) can be set before compilation. Typical running time: Most tasks (typically: analysis of a four dimensional reflexive polytope) can be perfomed interactively within milliseconds. The classification of all reflexive polytopes in four dimensions takes several processor years. The facet enumeration problem for higher (e.g., 12–20) dimensional polytopes varies strongly with the dimension and structure of the polytope; here PALP's performance is similar to that of existing packages [Avis et al., Comput. Geometry 7 (1997) 265]. Unusual features of the program: None

A Fourier-Mukai approach to spectral data for instantons

We study SU(r) instantons on elliptic surfaces with a section and show that they are in one-one correspondence with spectral data consisting of a curve in the dual elliptic surface and a line bundle on that curve. We use relative Fourier-Mukai transforms to analyse their properties and, in the case of the K3 and abelian surface, we show that the moduli space of instantons has a natural Lagrangian fibration structure with respect to the canonical complex symplectic structures.

Global Structure of an Elliptic Fibration

∂ -étale topology is introduced for analytic spaces with boundary as an analog of étale topology for schemes. A locally projective elliptic fibration is bimeromorphically considered as a torsor in the ∂ -étale topology of the associated basic elliptic fibration. The related ∂ -étale cohomology groups have much information on the structure of elliptic fibrations. In particular, an answer to Ueno's extension problem, a relation to Tate–Shafarevich groups and their finiteness properties, characterizations of projective and Kähler elliptic fibrations, and a generalization of logarithmic transformation to arbitrary dimension are obtained. This article is a revised version of [N5].

REFLEXIVE POLYHEDRA, WEIGHTS AND TORIC CALABI-YAU FIBRATIONS

During the last years we have generated a large number of data related to Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive polyhedra. We classified all reflexive polyhedra in three dimensions leading to K3 hypersurfaces and have also completed the four-dimensional case relevant to Calabi-Yau threefolds. In addition, we have analysed for many of the resulting spaces whether they allow fibration structures of the types that are relevant in the context of superstring dualities. In this survey we want to give background information both on how we obtained these data, which can be found at our web site, and on how they may be used. We give a complete exposition of our classification algorithm at a mathematical (rather than algorithmic) level. We also describe how fibration structures manifest themselves in terms of toric diagrams and how we managed to find the respective data. Both for our classification scheme and for simple descriptions of fibration structures the concept of weight systems plays an important role.

Elliptic curves from sextics

Let N be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space N/G is one-dimensional and consists of two components, Ntorus/G and Ngen/G. By quadratic transformations, they are transformed into one-parameter families Cs and Ds of cubic curves respectively. First we study the geometry of Nε/G, &= torus, gen and their structure of elliptic fibration. Then we study the Mordell-Weil torsion groups of cubic curves Cs over Q and Ds over Q(-3) respectively. We show that Cs has the torsion group Z/3Z for a generic s∈Q and it also contains subfamilies which coincide with the universal families given by Kubert [Ku] with the torsion groups Z/6Z,Z/6Z+Z/2Z,Z/9Z, or Z/12Z. The cubic curves Ds has torsion Z/3Z+Z/3Z generically but also Z/3Z+Z/6Z for a subfamily which is parametrized by Q(-3).

Calibrated fibrations

In this paper we investigate the geometry of Calibrated submanifolds and study relations between their moduli-space and geometry of the ambient manifold. In particular for a Calabi-Yau manifold we define Special Lagrangian submanifolds for any Kahler metric on it. We show that for a choice of Kahler metric the Borcea-Voisin threefold has a fibration structure with generic fiber being a Special Lagrangian torus. Moreover we construct a mirror to this fibration. Also for any closed G_2 form on a 7-manifold we study coassociative submanifolds and we show that one example of a G_2 manifold constructed by Joyce in [10] is a fibration with generic fiber being a coassociative 4-torus. Similarly we construct a mirror to this fibration.

FREDHOLM THEORY FOR DEGENERATE PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBERED BOUNDARIES

We consider the calculus Ψ*,* de(X, deΩ½) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm elements in Ψ*,* de(X) by means of the invertibility of an appropriate symbol map, and describe a K-theoretical formula for the Fredholm index extending the Atiyah–Singer formula for closed manifolds. The algebra of operators of order (0, 0) is shown to be a Ψ*-algebra, hence its K-theory coincides with that of its C *-closure, and we give a description of the corresponding cyclic 6-term exact sequence. We define a Wodzicki-type residue trace on an ideal in Ψ*,* de(X, deΩ½), and we show that it coincides with Dixmier's trace for operators of order –dim X in this ideal. This extends a result of Connes for the closed case.