Search results for "Fibered knot"
showing 10 items of 12 documents
Construction of Fibred Categories
2019
In Section 5, we introduce methods from classical homological algebra (i.e. using mostly the language of derived categories of abelian categories and their Verdier quotients) to construct the main examples of premotivic categories of interest in this book, while, in Section 6, we study how to check that the localization axiom holds in practice. Section 7 is devoted to the process of obtaining new fibred categories from old ones, by considering homotopy theoretic modules over a ring object.
Fibred Categories and the Six Functors Formalism
2019
In Section 1, we introduce the basic language used in this book, the so-called premotivic categories and their functoriality. This is an extension of the classical notion of fibered categories. They appear with different categorical structures. In Section2, the language of premotivic categories is specialized to that of triangulated categories and to algebraic geometry. We introduce several axioms of such categories which ultimately will lead to the full six functors formalism. An emphasis is given on the study of the main axioms, with a special care about the so-called localization axiom. Then in Section 3, the general theory of descent is formulated in the language of premotivic model cat…
Homology of pseudodifferential operators on manifolds with fibered cusps
2003
The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the 0 0 -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.
Swampland Bounds on the Abelian Gauge Sector
2019
We derive bounds on the number of abelian gauge group factors in six-dimensional gravitational theories with minimal supersymmetry and in their F-theoretic realisations. These bounds follow by requiring consistency of certain BPS strings in the spectrum of the theory, as recently proposed in the literature. Under certain assumptions this approach constrains the number of abelian gauge group factors in six-dimensional supergravity theories with at least one tensor multiplet to be $N \leq 20$ (or $N \leq 22$ in absence of charged matter). For any geometric F-theory realisation with at least one tensor multiplet we establish the bound $N \leq 16$ by demanding unitarity of a heterotic solitonic…
An index formula on manifolds with fibered cusp ends
2002
We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dirac operators associated to a "product fibered cusp metric", the index is given by the integral of t…
AdS$_3$ solutions with exceptional supersymmetry
2018
Among the possible superalgebras that contain the AdS$_3$ isometries, two interesting possibilities are the exceptional $F(4)$ and $G(3)$. Their R-symmetry is respectively SO(7) and $G_2$, and the amount of supersymmetry ${\cal N}=8$ and ${\cal N}=7$. We find that there exist two (locally) unique solutions in type IIA supergravity that realize these superalgebras, and we provide their analytic expressions. In both cases, the internal space is obtained by a round six-sphere fibred over an interval, with an O8-plane at one end. The R-symmetry is the symmetry group of the sphere; in the $G(3)$ case, it is broken to $G_2$ by fluxes. We also find several numerical ${\cal N}=1$ solutions with $G_…
Finitely fibered Rosenthal compacta and trees
2009
We study some topological properties of trees with the interval topology. In particular, we characterize trees which admit a 2-fibered compactification and we present two examples of trees whose one-point compactifications are Rosenthal compact with certain renorming properties of their spaces of continuous functions.
On the Oort conjecture for Shimura varieties of unitary and orthogonal types
2014
In this paper we study the Oort conjecture on Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety $\mathcal{A}_g$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibred surfaces, we show that a Shimura curve $C$ is not contained generically in the Torelli locus if its canonical Higgs bundles contains a unitary Higgs subbundle of rank at least $(4g+2)/5$. From this we prove that a Shimura subvariety of $\mathbf{SU}(n,1)$-type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus $g$, the dimension $n+1$, the degree $2d$ of CM field of the Hermitian space,…
Fibred-categorical obstruction theory
2022
Abstract We set up a fibred categorical theory of obstruction and classification of morphisms that specialises to the one of monoidal functors between categorical groups and also to the Schreier-Mac Lane theory of group extensions. Further applications are provided to crossed extensions and crossed bimodule butterflies, with in particular a classification of non-abelian extensions of unital associative algebras in terms of Hochschild cohomology.
Fibered aspects of Yoneda's regular span
2018
In this paper we start by pointing out that Yoneda's notion of a regular span $S \colon \mathcal{X} \to \mathcal{A} \times \mathcal{B}$ can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category $\mathsf{Fib}(\mathcal{A})$. We study the relationship between these notions and those of internal opfibration and two-sided fibration. This fibrational point of view makes it possible to interpret Yoneda's Classification Theorem given in his 1960 paper as the result of a canonical factorization, and to extend it to a non-symmetric situation, where the fibration given by the product projection $Pr_0 \colon \mathcal{A} \times \mathcal{B} \to \mathcal{A}$ i…