Fibration Structure Research Articles

ON THE STRUCTURE OF HELICAL AXISYMMETRIC SOLUTIONS OF THE NAVIER-STOKES SYSTEM FOR INCOMPRESSIBLE FLUIDS

A class of exact solutions to the Navier-Stokes equations for axisymmetric vortex flows of incompressible fluids is obtained. Invariant manifolds of flows with rotational symmetry relative to a given axis in three-dimensional coordinate space are identified, and the structure of the solutions is described. It is established that typical invariant regions of such flows are rotational figures homeomorphic to a torus, forming a structure of topological fibration, such as in a sphere, cylinder, and more complex configurations composed of such figures. The results are extended to similar solutions of the magnetohydrodynamics (MHD) system and Maxwell’s electrodynamics equations, which possess ℝ3 analogous properties. Examples of axisymmetric vortex vector fields and the topological fibrations they generate on manifolds invariant ℝ3 under the dynamical systems defined by these fields are provided.

Generating Triangulations and Fibrations with Reinforcement Learning

We apply reinforcement learning (RL) to generate fine regular star triangulations of reflexive polytopes, that give rise to smooth Calabi-Yau (CY) hypersurfaces. We demonstrate that, by simple modifications to the data encoding and reward function, one can search for CYs that satisfy a set of desirable string compactification conditions. For instance, we show that our RL algorithm can generate triangulations together with holomorphic vector bundles that satisfy anomaly cancellation and poly-stability conditions in heterotic compactification. Furthermore, we show that our algorithm can be used to search for reflexive subpolytopes together with compatible triangulations that define fibration structures of the CYs.

Asymptotic Weak Gravity Conjecture in M-theory on K3× K3

Abstract The Asymptotic Weak Gravity Conjecture (WGC) has been proposed as a special case of the Tower WGC that probes infinite distances in the moduli space corresponding to weakly coupled gauge regimes. The conjecture has been studied in M-theory on a Calabi–Yau threefold (CY3) with finite volume inducing a 5D effective quantum field theory. In this paper, we extend the scope of the previous study to encompass lower dimensions, particularly we generalize the obtained 5D Asymptotic WGC to the effective field theory (EFT$_{3D}$) coupled to 3D gravity that descends from M-theory compactified on a Calabi–Yau fourfold with an emphasis on $K3\times K3$. We find that the CY4 has three fibration structures labeled as line Type-$\mathbb {T}^{2}$, surface Type-$\mathbb {S}$, and bulk Type-$\mathbb {V}$. The emergent EFT$_{3D}$ is shown to have 2+2 towers of particle states termed as the BPS $\mathcal {T}_{M_{\mathrm{k}}\rightarrow 0}^{\rm{{\small BPS}}}$ and $\mathcal {T}_{M_{\mathrm{k}}\rightarrow \infty }^{\rm{{\small BPS}}}$ as well as the non-BPS $\mathcal {T}_{M_{\mathrm{k}}\rightarrow 0}^{\rm{{\small N-BPS}}}$ and $\mathcal {T}_{M_{\mathrm{k}}\rightarrow \infty }^{\rm{{\small N-BPS}}}$. To ensure the viability of the 3D Asymptotic WGC, we give explicit calculations to thoroughly test the Swampland constraint for both the weakly and strongly gauge coupled regimes. Additional aspects, including the gauge symmetry breaking and duality symmetry, are also investigated.

Back to heterotic strings on ALE spaces. Part II. Geometry of T-dual little strings

This work is the second of a series of papers devoted to revisiting the properties of Heterotic string compactifications on ALE spaces. In this project we study the geometric counterpart in F-theory of the T-dualities between Heterotic ALE instantonic Little String Theories (LSTs) extending and generalising previous results on the subject by Aspinwall and Morrison. Since the T-dualities arise from a circle reduction one can exploit the duality between F-theory and M-theory to explore a larger moduli space, where T-dualities are realised as inequivalent elliptic fibrations of the same geometry. As expected from the Heterotic/F-theory duality the elliptic F-theory Calabi-Yau we consider admit a nested elliptic K3 fibration structure. This is central for our construction: the K3 fibrations determine the flavor groups and their global forms, and are the key to identify various T-dualities. We remark that this method works also more generally for LSTs arising from non-geometric Heterotic backgrounds. We study a first example in detail: a particularly exotic class of LSTs which are built from extremal K3 surfaces that admit flavor groups with maximal rank 18. We find all models are related by a so-called T-hexality (i.e. a 6-fold family of T-dualities) which we predict from the inequivalent elliptic fibrations of the extremal K3.

A QP perspective on topology change in Poisson–Lie T-duality

We describe topological T-duality and Poisson–Lie T-duality in terms of QP (differential graded symplectic) manifolds and their canonical transformations. Duality is mediated by a QP-manifold on doubled non-abelian ‘correspondence’ space, from which we can perform mutually dual symplectic reductions, where certain canonical transformations play a vital role. In the presence of spectator coordinates, we show how the introduction of bibundle structure on correspondence space realises changes in the global fibration structure under Poisson–Lie duality. Our approach can be directly translated to the worldsheet to derive dual string current algebras. Finally, the canonical transformations appearing in our reduction procedure naturally suggest a Fourier–Mukai integral transformation for Poisson–Lie T-duality.

Symmetries of Calabi-Yau prepotentials with isomorphic flops

Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kähler cone made up from an infinite number of individual Kähler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kähler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold.

Optimal symplectic connections and deformations of holomorphic submersions

We give a general construction of extremal Kähler metrics on the total space of certain holomorphic submersions, extending results of Dervan-Sektnan, Fine, and Hong. We consider submersions whose fibres admit a degeneration to Kähler manifolds with constant scalar curvature, in a way that is compatible with the fibration structure. Thus we allow fibres that are K-semistable, rather than K-polystable; this is crucial to moduli theory. On these fibrations we phrase a partial differential equation whose solutions, called optimal symplectic connections, represent a canonical choice of a relatively Kähler metric. We expect this to be the most general construction of a canonical relatively Kähler metric provided all input is smooth. We use the notion of an optimal symplectic connection and the geometry related to it to construct Kähler metrics with constant scalar curvature and extremal metrics on the total space, in adiabatic classes.

Regular points of extremal subsets in Alexandrov spaces

We define regular points of an extremal subset in an Alexandrov space and study their basic properties. We show that a neighborhood of a regular point in an extremal subset is almost isometric to an open subset in Euclidean space and that the set of regular points in an extremal subset has full measure and is dense in it. These results actually hold for strained points in an extremal subset. Applications include the volume convergence of extremal subsets under a noncollapsing convergence of Alexandrov spaces, and the existence of a cone fibration structure of a metric neighborhood of the regular part of an extremal subset. In an appendix, a deformation retraction of a metric neighborhood of a general extremal subset is constructed.

Emergent strings from infinite distance limits

As a refinement of the Swampland Distance Conjecture, we propose that a quantum gravitational theory in an infinite distance limit of its moduli space either decompactifies, or reduces to an asymptotically tensionless, weakly coupled string theory. We support our claim by classifying, as special cases, the behaviour of M-Theory and Type IIA string theory compactifications on Calabi-Yau three-folds at infinite distances in Kähler moduli space.The analysis comprises three parts: we first classify the possible infinite distance limits in the classical Kähler moduli space of a Calabi-Yau three-fold. Each such limit at finite volume is characterized by a universal fibration structure, for which the generic fiber shrinking in the limit is either an elliptic curve, a K3 surface, or an Abelian surface.In the second part we focus on M-Theory and investigate the nature of the towers of asymptotically massless states that arise from branes wrapped on the shrinking fibers. Depending on which of the three classes of fibrations are considered, we obtain decompactification to F-Theory, or a theory with a unique asymptotically tensionless, weakly coupled heterotic or Type II string, respectively. The latter probes a dual D-manifold which is in general non-geometric. In addition to the intrinsic string excitations, towers of states from M2-branes along non-contractible curves become light and correspond to further wrapping and winding modes of the tensionless heterotic or Type II string.In the third part of the analysis, we consider Type IIA string theory on Calabi-Yau three-folds and show that quantum effects obstruct taking finite volume infinite distance limits in the Kähler moduli space. The only possible infinite distance limit which is not a decompactification limit involves K3-fibrations with string scale fiber volume and gives rise to an emergent tensionless heterotic string.

Structure of Fano fibrations of varieties admitting an int-amplified endomorphism

In this paper, we study the structure of Fano fibrations of varieties admitting an int-amplified endomorphism. We prove that if a normal Q-factorial klt projective variety X has an int-amplified endomorphism, then there exists an étale in codimension one finite morphism X˜→X such that X˜ is of Fano type over its Albanese variety. As a corollary, if we further assume that X is smooth and rationally connected, then X is of Fano type.

Extending rationally connected fibrations from ample subvarieties

Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.

Categorical mirror symmetry on cohomology for a complex genus 2 curve

Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs X and Y such that the complex geometry on X mirrors the symplectic geometry on Y. It allows one to deduce symplectic information about Y from known complex properties of X. Strominger-Yau-Zaslow [61] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich [43] conjectured that a complex invariant on X (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of Y (the Fukaya category, see [9], [29], [49], [1]). This is known as homological mirror symmetry. In this project, we first use the construction of “generalized SYZ mirrors” for hypersurfaces in toric varieties following Abouzaid-Auroux-Katzarkov [6], in order to obtain X and Y as manifolds. The complex manifold is the genus 2 curve Σ2 (so of general type c1<0) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model (Y,v0) equipped with a holomorphic function v0:Y→C which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of DbCoh(Σ2) into a cohomological Fukaya-Seidel category of Y as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [55] and in Abouzaid-Seidel [3].

Integral points on varieties defined by matrix factorization into elementary matrices

Let K be a number field and S be a finite set of valuations of K containing the archimedean valuations. Let O be the ring of S-integers. For A∈SL2(O) and k≥1, we define matrix-factorization varieties Vk(A) over O which parametrize factoring A into a product of k elementary matrices beginning with lower triangular; the equations defining Vk(A) are written in terms of Euler's continuant polynomials. We show that the Vk(A) are rational (k−3)-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the O-points of Vk(A). We prove that for k≥4 the O-points on Vk(A) are Zariski dense if Vk(A)(O)≠∅ assuming the group of units O× is infinite. This can then be combined with results on factoring into elementary matrices for SL2(O). One result is that for k≥9 the O-points on Vk(A) are Zariski dense if O× is infinite.

Fibration structure in toric hypersurface Calabi-Yau threefolds

We find through a systematic analysis that all but 29,223 of the 473.8 million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive subpolytope. Such a subpolytope is generally associated with the presence of an elliptic or genus one fibration in the corresponding birational equivalence class of Calabi-Yau threefolds. This extends the growing body of evidence that most Calabi-Yau threefolds have an elliptically fibered phase.

A quick trip through fibration structures

A quick trip through fibration structures

Fibrations of highly singular map germs

We define local fibration structures for real map germs with strictly positive dimensional discriminant: a local fibration structure over the complement of the discriminant, and a complete local fibration structure which includes the stratified discriminant into the picture. We provide new classes of map germs endowed with such local fibration structures.

On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with h1,1 ≥ 140 or h2,1 ≥ 140 that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small h1,1 the fraction of polytopes in the KS database that do not have a genus one or elliptic fibration drops exponentially as h1,1 increases. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.

Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

We show that generalized broken fibrations in arbitrary dimensions admit rank-2 Poisson structures compatible with the fibration structure. After extending the notion of wrinkled fibration to dimension 6 we prove that these wrinkled fibrations also admit compatible rank-2 Poisson structures. In the cases with indefinite singularities we can provide these wrinkled fibrations in dimension 6 with near-symplectic structures.

Enriched duality in double categories: [formula omitted]-categories and [formula omitted]-cocategories

In this work, we explore a double categorical framework for categories of enriched graphs, categories and the newly introduced notion of cocategories. A fundamental goal is to establish an enrichment of V-categories in V-cocategories, which generalizes the so-called Sweedler theory relative to an enrichment of algebras in coalgebras. The language employed is that of V-matrices, and an interplay between the double categorical and bicategorical perspective provides a high-level flexibility for demonstrating essential features of these dual structures. Furthermore, we investigate an enriched fibration structure involving categories of monads and comonads in fibrant double categories a.k.a. proarrow equipments, which leads to natural applications for their materialization as categories and cocategories in the enriched matrices setting.

Compactifications of adjoint orbits and their Hodge diamonds

A recent theorem of [E. Gasparim, L. Grama and L. A. B. San Martin, Lefschetz fibrations on adjoint orbits, Forum Math. 28(5) (2016) 967–980.] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behavior of their fiberwise compactifications. Expressing adjoint orbits and fibers as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenization of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenization, and that extensions of the potential to the compactification must acquire degenerate singularities.