Tese (doutorado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2017. / Made available in DSpace on 2017-10-24T03:21:31Z (GMT). No. of bitstreams: 1 347946.pdf: 810479 bytes, checksum: 1c2f1fb7f8c41b3cc057f71e30a878d5 (MD5) Previous issue date: 2017 / Sejam G um grupo finito que age em uma categoria de fusão C, H um subgrupo de G e M uma categoria módulo sobre C. Se M é um C-módulo H-equivariante, existe a equivariantização MH. O presente trabalho caracteriza os objetos simples em MH, em que M é uma categoria módulo indecomponível semissimples, bem como traz um estudo detalhado das ferramentas usadas para tal.<br> / Abstract : Let G be a group acting on a fusion category C, H a subgroup of G and M a module category over C. If M is a H-equivariant C-module then there exists the equivariantization MH. This work characterizes simple objects in MH, where M is a semisimple indecomposable module category, as well as presents a detailed study of the tools used for it.

There is a well-known correspondence between two-dimensional topological quantum field theories (2-D TQFTs) and commutative Frobenius algebras. Every 2-D TQFT also gives rise to a diffeomorphism invariant of closed, orientable two-manifolds, which may be investigated via the associated commutative Frobenius algebras. We investigate which such diffeomorphism invariants may arise from TQFTs, and in the process uncover a distinction between two fundamentally different types of commutative Frobenius algebras ("weak" Frobenius algebras and "strong" Frobenius algebras). These diffeomorphism invariants form the starting point for our investigation into marked cobordism categories, which generalize the local cobordism relations developed by Dror Bar-Natan during his investigation of Khovanov's link homology. We subsequently examine the particular class of 2-D TQFTs known as "universal sl(n) TQFTs". These TQFTs are at the algebraic core of the link invariants known as sl(n) link homology theories, as they provide the algebraic structure underlying the boundary maps in those homology theories. We also examine the 3-manifold diffeomorphism invariants known as skein modules, which were first introduced by Marta Asaeda and Charles Frohman. These 3-manifold invariants adapt Bar-Natan's marked cobordism category (as induced by a specific 2-D TQFT) to embedded surfaces, and measure which such surfaces may be embedded within in 3-manifold (modulo Bar-Natan's local cobordism relations). Our final results help to characterize the structure of such skein modules induced by universal sl(n) TQFTs.

The Landau-Ginzburg (LG) B-Model is a significant feature of singularity theory and mirror symmetry. Krawitz in 2010, guided by work of Kaufmann, provided an explicit construction for the LG B-model when using diagonal symmetries of a quasihomogeneous, nondegenerate polynomial. In this thesis we discuss aspects of how to generalize the LG B-model construction to allow for nondiagonal symmetries of a polynomial, and hence nonabelian symmetry groups. The construction is generalized to the level of graded vector space and the multiplication developed up to an unknown factor. We present complete examples of nonabelian LG B-models for the polynomials x^2y + y^3, x^3y + y^4, and x^3 + y^3 + z^3 + w^2.

We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category Rep(S_t), to the additive Karoubi envelope of the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions. We then generalize the above results to any group G, the case where G is the trivial group corresponding to the case mentioned above. Thus, to every group G we associate a linear monoidal category Par(G) that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of Par(G) into the group Heisenberg category associated to G. This embedding intertwines the natural actions of both categories on modules for wreath products of G. Finally, we prove that the additive Karoubi envelope of Par(G) is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.

Landau-Ginzburg mirror symmetry takes place in the context of affine singularities in CN. Given such a singularity defined by a quasihomogeneous polynomial W and an appropriate group of symmetries G, one can construct the FJRW theory (see [3]). This construction fills the role of the A-model in a mirror symmetry proposal of Berglund and H ubsch [1]. The conjecture is that the A-model of W and G should match the B-model of a dual singularity and dual group (which we denote by WT and GT). The B-model construction is based on the Milnor ring, or local algebra, of the singularity. We verify this conjecture for a wide class of singularities on the level of Frobenius algebras, generalizing work of Krawitz [10]. We also review the relevant parts of the constructions.

There are several reasons to be interested in conformal field theories in two dimensions. Apart from arising in various physical applications, ranging from statistical mechanics to string theory, conformal field theory is a class of quantum field theories that is interesting on its own. First of all there is a large amount of symmetries. In addition, many of the interesting theories satisfy a finiteness condition, that together with the symmetries allows for a fully non-perturbative treatment, and even for a complete solution in a mathematically rigorous manner. One of the crucial tools which make such a treatment possible is provided by category theory. This thesis contains results relevant for two different classes of conformal field theory. We partly treat rational conformal field theory, but also derive results that aim at a better understanding of logarithmic conformal field theory. For rational conformal field theory, we generalize the proof that the construction of correlators, via three-dimensional topological field theory, satisfies the consistency conditions to oriented world sheets with defect lines. We also derive a classifying algebra for defects. This is a semisimple commutative associative algebra over the complex numbers whose one-dimensional representations are in bijection with the topological defect lines of the theory. Then we relax the semisimplicity condition of rational conformal field theory and consider a larger class of categories, containing non-semisimple ones, that is relevant for logarithmic conformal field theory. We obtain, for any finite-dimensional factorizable ribbon Hopf algebra H, a family of symmetric commutative Frobenius algebras in the category of bimodules over H. For any such Frobenius algebra, which can be constructed as a coend, we associate to any Riemann surface a morphism in the bimodule category. We prove that this morphism is invariant under a projective action of the mapping class group ofthe Riemann surface. This suggests to regard these morphisms as candidates for correlators of bulk fields of a full conformal field theories whose chiral data are described by the category of left-modules over H.

Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions, the A and B models, which are created based on a choice of an affine singularity with a group of symmetries. Both models are vector spaces equipped with multiplication and a pairing (making them Frobenius algebras), and they are also Frobenius manifolds. We give a result relating stabilization of singularities in classical singularity to its counterpart in the Landau-Ginzburg setting. The A model comes from so-called FJRW theory and can be de fined up to a full cohomological field theory. The structure of this model is determined by a generating function which requires the calculation of certain numbers, which we call correlators. In some cases the their values can be computed using known techniques. Often, there is no known method for finding their values. We give new computational methods for computing concave correlators, including a formula for concave genus-zero, four-point correlators and show how to extend these results to find other correlator values. In many cases these new methods give enough information to compute the A model structure up to the level of Frobenius manifold. We give the FJRW Frobenius manifold structure for various choices of singularities and groups.

In this thesis we compute the Frobenius manifold of the Landau-Ginzburg A-model (FJRW theory) for certain polynomials. Specifically, our computations apply to polynomials that are sums of An and Dn singularities, paired with the corresponding maximal symmetry group. In particular this computation applies to several K3 surfaces. We compute the necessary correlators using reconstruction, the concavity axiom, and new techniques. We also compute the Frobenius manifold of the D3 singularity.

Le thème général de cette thèse est l’exploitation de l’interaction entre la sémantique dénotationnelle et la syntaxe. Des sémantiques satisfaisantes ont été découvertes pour les preuves en logique intuitionniste et linéaire, mais dans le cas de la logique classique, la solution du problème est connue pour être particulièrement difficile. Ce travail commence par l’étude d’une interprétation concrète des preuves classiques dans la catégorie des ensembles ordonnés et bimodules, qui mène à l’extraction d’invariants significatifs. Suit une généralisation de cette sémantique concrète, soit l’interprétation des preuves classiques dans une catégorie compacte fermée où chaque objet est doté d’une structure d’algèbre de Frobenius. Ceci nous mène à une définition de réseaux de démonstrations pour la logique classique. Le concept de correction, l’élimination des coupures et le problème de la “full completeness” sont abordés au moyen d’un enrichissement naturel dans les ordres sur la catégorie de Frobenius, produisant une catégorie pour l'élimination des coupures et un concept de ressources pour la logique classique. Revenant sur notre première sémantique concrète, nous montrons que nous avons une représentation fidèle de la catégorie de Frobenius dans la catégorie des ensembles ordonnés et bimodules. / The general theme of this thesis is the exploitation of the fruitful interaction between denotational semantics and syntax. Satisfying semantics have been discovered for proofs in intuitionistic and certain linear logics, but for the classical case, solving the problem is notoriously difficult.This work begins with investigations of concrete interpretations of classical proofs in the category of posets and bimodules, resulting in the definition of meaningful invariants of proofs. Then, generalizing this concrete semantics, classical proofs are interpreted in a free symmetric compact closed category where each object is endowed with the structure of a Frobenius algebra. The generalization paves a way for a theory of proof nets for classical proofs. Correctness, cut elimination and the issue of full completeness are addressed through natural order enrichments defined on the Frobenius category, yielding a category with cut elimination and a concept of resources in classical logic. Revisiting our initial concrete semantics, we show we have a faithful representation of the Frobenius category in the category of posets and bimodules.

Dans cette thèse, on présente la théorie algébrique IH par le biais de générateurs et d’équations.Le modèle libre de IH est la catégorie des sous-espaces linéaires sur un corps k. Les termes de IH sont des diagrammes de cordes, qui, selon le choix de k, peuvent exprimer différents types de réseaux et de formalismes graphiques, que l’on retrouve dans des domaines scientifiques divers, tels que les circuits quantiques, les circuits électriques et les réseaux de Petri. Les équations de IH sont obtenues via des lois distributives entre algèbres de Hopf – d’où le nom “Interacting Hopf algebras” (algèbres de Hopf interagissantes). La caractérisation via les sous-espaces permet de voir IH comme une syntaxe fondée sur les diagrammes de cordes pour l’algèbre linéaire: les applications linéaires, les espaces et leurs transformations ont chacun leur représentation fidèle dans le langage graphique. Cela aboutit à un point de vue alternatif, souvent fructueux, sur le domaine.On illustre cela en particulier en utilisant IH pour axiomatiser la sémantique formelle de circuits de calculs de signaux, pour lesquels on s’intéresse aux questions de la complète adéquation et de la réalisabilité. Notre analyse suggère un certain nombre d’enseignements au sujet du rôle de la causalité dans la sémantique des systèmes de calcul. / We present by generators and equations the algebraic theory IH whose free model is the category oflinear subspaces over a field k. Terms of IH are string diagrams which, for different choices of k, expressdifferent kinds of networks and graphical formalisms used by scientists in various fields, such as quantumcircuits, electrical circuits and Petri nets. The equations of IH arise by distributive laws between Hopfalgebras - from which the name interacting Hopf algebras. The characterisation in terms of subspacesallows to think of IH as a string diagrammatic syntax for linear algebra: linear maps, spaces and theirtransformations are all faithfully represented in the graphical language, resulting in an alternative, ofteninsightful perspective on the subject matter. As main application, we use IH to axiomatise a formalsemantics of signal processing circuits, for which we study full abstraction and realisability. Our analysissuggests a reflection about the role of causality in the semantics of computing devices.