Fibration Structure Research Articles

Five-brane webs and highest weight representations

We consider BPS-counting functions mathcal{Z} N,M of N parallel M5-branes probing a transverse ℤM orbifold geometry. These brane web configurations can be dualised into a class of toric non-compact Calabi-Yau threefolds which have the structure of an elliptic fibration over (affine) AN −1. We make this symmetry of mathcal{Z} N,M manifest in particular regions of the parameter space of the setup: we argue that for specific choices of the deformation parameters, the supercharges of the system acquire particular holonomy charges which lead to infinitely many cancellations among states contributing to the partition function. The resulting (simplified) mathcal{Z} N,M can be written as a sum over weights forming a single irreducible representation of the Lie algebra mathfrak{a} N −1 (or its affine counterpart). We show this behaviour explicitly for an extensive list of examples for specific values of (N, M ) and generalise the arising pattern for generic configurations. Finally, for a particular compact M5-brane setup we use this form of the partition function to make the duality N ↔ M manifest.

The affine invariant of proper semitoric integrable systems

This paper initiates the study of semitoric integrable systems with two degrees of freedom and with proper momentum-energy map, but with possibly nonproper S1-momentum map. This class of systems includes many standard examples, such as the spherical pendulum. To each such system we associate a subset of , invariant under a natural notion of isomorphism and encoding the integral affine structure of the singular Lagrangian fibration, in the spirit of Delzant polygons for toric systems.

Fibrations in CICY threefolds

In this work we systematically enumerate genus one fibrations in the class of 7, 890 Calabi-Yau manifolds defined as complete intersections in products of projective spaces, the so-called CICY threefolds. This survey is independent of the description of the manifolds and improves upon past approaches that probed only a particular algebraic form of the threefolds (i.e. searches for “obvious” genus one fibrations as in [1, 2]). We also study K3-fibrations and nested fibration structures. That is, K3 fibrations with potentially many distinct elliptic fibrations. To accomplish this survey a number of new geometric tools are developed including a determination of the full topology of all CICY threefolds, including triple intersection numbers. In 2, 946 cases this involves finding a new “favorable” description of the manifold in which all divisors descend from a simple ambient space. Our results consist of a survey of obvious fibrations for all CICY threefolds and a complete classification of all genus one fibrations for 4, 957 “Kähler favorable” CICYs whose Kähler cones descend from a simple ambient space. Within the CICY dataset, we find 139, 597 obvious genus one fibrations, 30, 974 obvious K3 fibrations and 208, 987 nested combinations. For the Kähler favorable geometries we find a complete classification of 377, 559 genus one fibrations. For one manifold with Hodge numbers (19, 19) we find an explicit description of an infinite number of distinct genus-one fibrations extending previous results for this particular geometry that have appeared in the literature. The data associated to this scan is available here [3].

Lefschetz fibrations on knot surgery 4-manifolds via Stallings twist

In this article we construct a family of knot surgery 4-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface E(2) using connected sums of fibered knots obtained by Stallings twist from a slice knot 31♯3∗1 . By comparing their monodromy groups induced from the corresponding monodromy factorizations, we show that they admit mutually nonisomorphic Lefschetz fibration structures.

Dual little strings from F-theory and flop transitions

A particular two-parameter class of little string theories can be described by M parallel M5-branes probing a transverse affine AN − 1 singularity. We previously discussed the duality between the theories labelled by (N, M) and (M, N). In this work, we propose that these two are in fact only part of a larger web of dual theories. We provide evidence that the theories labelled by (N, M) and left(frac{NM}{k},kright) are dual to each other, where k = gcd(N,M). To argue for this duality, we use a geometric realization of these little string theories in terms of F-theory compactifications on toric, non-compact Calabi-Yau threefolds XN,M which have a double elliptic fibration structure. We show explicitly for a number of examples that XNM/k,k is part of the extended moduli space of XN,M, i.e. the two are related through symmetry transformations and flop transitions. By working out the full duality map, we provide a simple check at the level of the free energy of little string theories.

Some Landau–Ginzburg models viewed as rational maps

Gasparim, Grama and San Martin (2016) showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.

Multiple fibrations in Calabi-Yau geometry and string dualities

In this work we explore the physics associated to Calabi-Yau (CY) n-folds that can be described as a fibration in more than one way. Beginning with F-theory vacua in various dimensions, we consider limits/dualities with M-theory, type IIA, and heterotic string theories. Our results include many M-/F-theory correspondences in which distinct F-theory vacua - associated to different elliptic fibrations of the same CY n-fold - give rise to the same M-theory limit in one dimension lower. Examples include 5-dimensional correspondences between 6-dimensional theories with Abelian, non-Abelian and superconformal structure, as well as examples of higher rank Mordell-Weil geometries. In addition, in the context of heterotic/F-theory duality, we investigate the role played by multiple K3- and elliptic fibrations in known and novel string dualities in 8-, 6- and 4-dimensional theories. Here we systematically summarize nested fibration structures and comment on the roles they play in T-duality, mirror symmetry, and 4-dimensional compactifications of F-theory with G-flux. This investigation of duality structures is made possible by geometric tools developed in a companion paper [1].

Local–global principle for 0-cycles on fibrations over rationally connected bases

We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation for 0-cycles on algebraic varieties that possess a fibration structure. The issue is to establish the exactness of a local-to-global sequence (E) of Chow groups of 0-cycles. Recently, Harpaz and Wittenberg proved the exactness of (E) for rationally connected fibrations whose smooth fibres verify the exactness of (E) and whose base is either the projective space or a curve verifying the exactness of (E). In the present paper, we prove the exactness of (E) for fibrations whose bases are Châtelet surfaces or smooth proper models of homogeneous spaces of connected linear algebraic groups with connected stabilizers. We require that either the fibration is valuatively split in codimension 1 and most closed fibres satisfy weak approximation for 0-cycles, or the generic fibre has a 0-cycle of degree 1 and (E) is exact for most fibres.

F-theory and 2d (0, 2) theories

F-theory compactified on singular, elliptically fibered Calabi-Yau five-folds gives rise to two-dimensional gauge theories preserving N=(0,2) supersymmetry. In this paper we initiate the study of such compactifications and determine the dictionary between the geometric data of the elliptic fibration and the 2d gauge theory such as the matter content in terms of (0,2) superfields and their supersymmetric couplings. We study this setup both from a gauge-theoretic point of view, in terms of the partially twisted 7-brane theory, and provide a global geometric description based on the structure of the elliptic fibration and its singularities. Global consistency conditions are determined and checked against the dual M-theory compactification to one dimension. This includes a discussion of gauge anomalies, the structure of the Green-Schwarz terms and the Chern-Simons couplings in the dual M-theory supersymmetric quantum mechanics. Furthermore, by interpreting the resulting 2d (0,2) theories as heterotic worldsheet theories, we propose a correspondence between the geometric data of elliptically fibered Calabi-Yau five-folds and the target space of a heterotic gauged linear sigma-model (GLSM). In particular the correspondence between the Landau-Ginsburg and sigma-model phase of a 2d (0,2) GLSM is realized via different T-branes or gluing data in F-theory.

Poisson structures on wrinkled fibrations

We provide local formulae for Poisson bivectors and symplectic forms on the leaves of Poisson structures associated with wrinkled fibrations on smooth 4-manifolds. When such a fibration structure does not have 2-spheres in its fibres, the associated Poisson structure is integrable.

Open book structures on semi-algebraic manifolds

Given a $C^2$ semi-algebraic mapping $F: \mathbb{R}^N \rightarrow \mathbb{R}^p,$ we consider its restriction to $W\hookrightarrow \mathbb{R^{N}}$ an embedded closed semi-algebraic manifold of dimension $n-1\geq p\geq 2$ and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection $\frac{F}{\Vert F \Vert}:W\setminus F^{-1}(0)\to S^{p-1}$. Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering $W$ as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of $F$ with the canonical projection $\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}$ and prove that the fibers of $\frac{F}{\Vert F \Vert}$ and $\frac{\pi\circ F}{\Vert \pi\circ F \Vert}$ are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection $\frac{F}{\Vert F \Vert}$ and $W\cap F^{-1}(0).$ Similar formulae are proved for mappings obtained after composition of $F$ with canonical projections.

Fibered stable varieties

We show that if a stable variety (in the sense of Kollár and Shepherd-Barron) admits a fibration with stable fibers and base, then this fibration structure deforms (uniquely) for all small deformations. During our proof we obtain a Bogomolov-Sommese type vanishing for vector bundles and reflexive differential n − 1 n-1 -forms as well.

Topological invariants and fibration structure of complete intersection Calabi-Yau four-folds

We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compute the full Hodge diamond as well as additional topological invariants such as Chern classes and intersection numbers. Using this data, we conclude that there are at least 36,779 topologically distinct manifolds in our list. We also study the fibration structure of these manifolds and find that 99.95 percent can be described as elliptic fibrations. In total, we find 50,114,908 elliptic fibrations, demonstrating the multitude of ways in which many manifolds are fibered. A sub-class of 26,088,498 fibrations satisfy necessary conditions for admitting sections. The complete data set can be downloaded at http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/Cicy4folds/index.html .

N = 2 heterotic-type II duality and bundle moduli

Heterotic string compactifications on a $K3$ surface $\mathfrak{S}$ depend on a choice of hyperk\"ahler metric, anti-self-dual gauge connection and Kalb-Ramond flux, parametrized by hypermultiplet scalars. The metric on hypermultiplet moduli space is in principle computable within the $(0,2)$ superconformal field theory on the heterotic string worldsheet, although little is known about it in practice. Using duality with type II strings compactified on a Calabi-Yau threefold, we predict the form of the quaternion-K\"ahler metric on hypermultiplet moduli space when $\mathfrak{S}$ is elliptically fibered, in the limit of a large fiber and even larger base. The result is in general agreement with expectations from Kaluza-Klein reduction, in particular the metric has a two-stage fibration structure, where the $B$-field moduli are fibered over bundle and metric moduli, while bundle moduli are themselves fibered over metric moduli. A more precise match must await a detailed analysis of $R^2$-corrected ten-dimensional supergravity.

On the reconstruction problem in mirror symmetry

Let π:M→B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M0∨ over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B0⊂B of π. Transition functions of M0∨ are obtained via A∞ homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.

Effect of formaldehyde on the photochromic properties of ordered molybdenum oxide thin films produced by hydrothermal process

Molybdenum oxide thin films are prepared with different volume ratios of formaldehyde solution and sodium molybdate solution ranging from 0 to 0.5:10 via hydrothermal method. The results characterized from XRD and TEM demonstrate that all of the as-prepared films are MoO3 hexagonal structures. The SEM images reveal that the surface of the film formed without formaldehyde consists of hexagonal prisms, and the surface morphology of the film to be composed of fibration structures when the volume ratio of formaldehyde solution and sodium molybdate solution is 0.1:10. When increasing the volume ratio of formaldehyde solution and sodium molybdate solution, the surface morphologies of the film transformation to flower-like structures are observed. It is found that the higher the content of formaldehyde, the more complete the flower-like structures. The most ordered flower-like molybdenum oxide film which has the best photochromic performance exhibits a maximum color difference value of 8.921 when the volume ratio of formaldehyde solution and sodium molybdate solution is 0.3:10.

Toric elliptic fibrations and F-theory compactifications

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discriminant components, this description is far more accurate than previous studies. The discriminant locus and Kodaria fibers groups of all 102581 elliptic fibrations are computed. The maximal gauge group is SU(27), which would naively be in contradiction with 6-dimensional anomaly cancellation.

Twistors and electromagnetic knots

Since its discovery in 1931, the Hopf fibration has played an important role in physics in seemingly unrelated situations ranging from qubits to Taub-NUT spaces in general relativity [1]. Recently the structure of this fibration has been considered in relation to solutions of the source-free Maxwell equations and led to linked and knotted forms of electromagnetic fields [2, 3]. In twistor theory, it was the structure of the Hopf fibration that gave a twistor its name [4]. Here we present a deeper correspondence between a (non-null) twistor and the knotted electromagnetic fields, in which the Poynting vector plays a central role.

Moduli stabilisation for chiral global models

Abstract We combine moduli stabilisation and (chiral) model building in a fully con-sistent global set-up in Type IIB/F-theory. We consider compactifications on Calabi-Yau orientifolds which admit an explicit description in terms of toric geometry. We build globally consistent compactifications with tadpole and Freed-Witten anomaly cancellation by choosing appropriate brane set-ups and world-volume fluxes which also give rise to SU(5) or MSSM-like chiral models. We fix all the Kähler moduli within the Kähler cone and the regime of validity of the 4D effective field theory. This is achieved in a way compatible with the local presence of chirality. The hidden sector generating the non-perturbative effects is placed on a del Pezzo divisor that does not have any chiral intersection with any other brane. In general, the vanishing D-term condition implies the shrinking of the rigid divisor supporting the visible sector. However, we avoid this problem by generating r < n D-term conditions on a set of n intersecting divisors. The remaining (n − r) flat directions are fixed by perturbative corrections to the Kähler potential. We illustrate our general claims in an explicit example. We consider a K3-fibred Calabi-Yau with four Kähler moduli, that is a hypersurface in a toric ambient space and admits a ‘simple’ F-theory up-lift. We present explicit choices of brane set-ups and fluxes which lead to three different phenomenological scenarios: the first with GUT-scale strings and TeV-scale SUSY by fine-tuning the background fluxes; the second with an exponentially large value of the volume and TeV-scale SUSY without fine-tuning the background fluxes; and the third with a very anisotropic configuration that leads to TeV-scale strings and two micron-sized extra dimensions. The K3 fibration structure of the Calabi-Yau three-fold is also particularly suitable for cosmological purposes.

The structure of a Laurent polynomial fibration in n variables

Bass, Connell and Wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain R is isomorphic to the symmetric algebra of a finitely generated projective R-module of rank n. In this paper we prove a corresponding structure theorem for a ring A which is a locally Laurent polynomial algebra in n variables over an integral domain R, viz., we show that A is isomorphic to an R-algebra of the form ( Sym R ( Q ) ) [ I − 1 ] , where Q is a direct sum of n finitely generated projective R-modules of rank one and I is a suitable invertible ideal of the symmetric algebra Sym R ( Q ) . Further, we show that any faithfully flat algebra over a Noetherian normal domain R, whose generic and codimension-one fibres are Laurent polynomial algebras in n variables, is a locally Laurent polynomial algebra in n variables over R.