| 1. |
- Backman, Theo, 1986-
(författare)
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Configuration spaces, props and wheel-free deformation quantization
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main theme of this thesis is higher algebraic structures that come from operads and props.The first chapter is an introduction to the mathematical framework needed for the content of this thesis. The chapter does not contain any new results.The second chapter is concerned with the construction of a configuration space model for a particular 2-colored differential graded operad encoding the structure of two A∞ algebras with two A∞ morphisms and a homotopy between the morphisms. The cohomology of this operad is shown to be the well-known 2-colored operad encoding the structure of two associative algebras and of an associative algebra morphism between them.The third chapter is concerned with deformation quantization of (potentially) infinite dimensional (quasi-)Poisson manifolds. Our proof employs a variation on the transcendental methods pioneered by M. Kontsevich for the finite dimensional case. The first proof of the infinite dimensional case is due to B. Shoikhet. A key feature of the first proof is the construction of a universal L∞ structure on formal polyvector fields. Our contribution is a simplification of B. Shoikhet proof by considering a more natural configuration space and a simpler choice of propagator. The result is also put into a natural context of the dg Lie algebras coming from graph complexes; the L∞ structure is proved to come from a Maurer-Cartan element in the oriented graph complex.The fourth chapter also deals with deformation quantization of (quasi-)Poisson structures in the infinite dimensional setting. Unlike the previous chapter, the methods used here are purely algebraic. Our main theorem is the possibility to deformation quantize quasi-Poisson structures by only using perturbative methods; in contrast to the transcendental methods employed in the previous chapter. We give two proofs of the theorem via the theory of dg operads, dg properads and dg props. We show that there is a dg prop morphism from a prop governing star-products to a dg prop(erad) governing (quasi-)Poisson structures. This morphism gives a theorem about the existence of a deformation quantization of (quasi-)Poisson structure. The proof proceeds by giving an explicit deformation quantization of super-involutive Lie bialgebras and then lifting that to the dg properad governing quasi-Poisson structures. The prop governing star-products was first considered by S.A. Merkulov, but the properad governing quasi-Poisson structures is a new construction.The second proof of the theorem employs the Merkulov-Willwacher polydifferential functor to transfer the problem of finding a morphism of dg props to that of finding a morphism of dg operads.We construct an extension of the well known operad of A∞ algebras such that the representations of it in V are equivalent to an A∞ structure on V[[ħ]]. This new operad is also a minimal model of an operad that can be seen as the extension of the operad of associative algebras by a unary operation. We give an explicit map of operads from the extended associative operad to the operad we get when applying the Merkulov-Willwacher polydifferential functor to the properad of super-involutive Lie bialgebras. Lifting this map so as to go between their respective models gives a new proof of the main theorem.
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| 2. |
- Alm, Johan, 1985-
(författare)
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Universal algebraic structures on polyvector fields
- 2014
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The theory of operads is a conceptual framework that has become a kind of universal language, relating branches of topology and algebra. This thesis uses the operadic framework to study the derived algebraic properties of polyvector fields on manifolds.The thesis is divided into eight chapters. The first is an introduction to the thesis and the research field to which it belongs, while the second chapter surveys the basic mathematical results of the field.The third chapter is devoted to a novel construction of differential graded operads, generalizing an earlier construction due to Thomas Willwacher. The construction highlights and explains several categorical properties of differential graded algebras (of some kind) that come equipped with an action by a differential graded Lie algebra. In particular, the construction clarifies the deformation theory of such algebras and explains how such algebras can be twisted by Maurer-Cartan elements.The fourth chapter constructs an explicit strong homotopy deformation of polynomial polyvector fields on affine space, regarded as a two-colored noncommutative Gerstenhaber algebra. It also constructs an explicit strong homotopy quasi-isomorphism from this deformation to the canonical two-colored noncommmutative Gerstenhaber algebra of polydifferential operators on the affine space. This explicit construction generalizes Maxim Kontsevich's formality morphism.The main result of the fifth chapter is that the deformation of polyvector fields constructed in the fourth chapter is (generically) nontrivial and, in a sense, the unique such deformation. The proof is based on some cohomology computations involving Kontsevich's graph complex and related complexes. The chapter ends with an application of the results to properties of a derived version of the Duflo isomorphism.The sixth chapter develops a general mathematical framework for how and when an algebraic structure on the germs at the origin of a sheaf on Cartesian space can be "globalized" to a corresponding algebraic structure on the global sections over an arbitrary smooth manifold. The results are applied to the construction of the fourth chapter, and it is shown that the construction globalizes to polyvector fields and polydifferential operators on an arbitrary smooth manifold.The seventh chapter combines the relations to graph complexes, explained in chapter five, and the globalization theory of chapter six, to uncover a representation of the Grothendieck-Teichmüller group in terms of A-infinity morphisms between Poisson cohomology cochain complexes on a manifold.Chapter eight gives a simplified version of a construction of a family of Drinfel'd associators due to Carlo Rossi and Thomas Willwacher. Our simplified construction makes the connections to multiple zeta values more transparent--in particular, one obtains a fairly explicit family of evaluations on the algebra of formal multiple zeta values, and the chapter proves certain basic properties of this family of evaluations.
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| 3. |
- Granåker, Johan, 1979-
(författare)
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Wheeled Operads in Algebra, Geometry, and Quantization
- 2010
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The theory of generalized operads, the foundational conceptual framework of this thesis, has become a universal language, relating various areas such as algebraic topology, derived categories of algebras, deformation theory, differential geometry and the mathematical theory of quantization.The thesis consists of a preliminary chapter followed by four main chapters. In the first of these, the theory of deformations of morphisms of wheeled properads is treated, extending the non-wheeled case considered by Merkulov and Vallette. The deformation complex of a morphism is defined and shown to be an L-infinity algebra. Several examples of deformation complexes for algebras over wheeled operads are constructed explicitly, and non-trivial extensions of classical complexes computing cohomologies for non-wheeled counterparts are obtained. The Koszulness of the wheeled operads of unimodular Lie and pre-Lie algebras is established.In the second main chapter, Merkulov's definition of BV-manifolds is generalized, extending the prop profile from unimodular Lie bialgebras to unimodular quasi-Lie bialgebras. This allows a larger class of physical models to be treated with operadic methods. An application is given, establishing the equality of two induced structures in an extended BF theory with cosmological term, one by methods of quantum field theory and the other by homotopy transfer of operadic algebras. Also, the non-wheeled properad of quasi-Lie bialgebras is shown to be Koszul.The third main chapter contains, after a brief introduction to de Rham field theories on compactified configuration spaces, computations of some Kontsevich type weights using the logarithmic propagator. These weights turn out to be multiple zeta values, and being coefficients of a morphism of L-infinity algebras they satisfy a relation. This is a new way of proving relations among multiple zeta values, the possible reach of which is an interesting direction for future research.In the fourth main chapter, we extend homotopy algebra to non-wheeled properads. The structure of strong homotopy properad is defined, and it is proven to be homotopy invariant via construction of explicit homotopy transfer formulae.
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| 4. |
- Hellgren, Patrik, 1977-
(författare)
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G-structures and Families of Isotropic Submanifolds in Complex Contact Manifolds
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study a generalized twistor correspondence between irreducible G-structures (with torsion in general) on complex manifolds Z and moduli spaces M of deformations of isotropic homogeneous submanifolds X in complex contact manifolds Y.For any irreducible G-structure on a complex manifold M we present an explicit construction of a contact manifold (a generalized twistor space) Y with contact line bundle L and a family F of isotropic submanifolds X in Y having M as its moduli space. We study those special properties of this family which encode geometric invariants of the original G-structure.Conversely, given a contact manifold (Y,L) and an homogeneous isotropic submanifold X in Y satisfying certain properties, we show that the associated moduli space M of isotropic deformations of X inside Y has an induced G-structure, Gind, and then show how the invariant torsion of Gind can be read off from certain cohomology groups canonically associated with the holomorphic embedding data of X in Y.
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| 5. |
- Strohmayer, Henrik, 1979-
(författare)
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Prop profiles of compatible Poisson and Nijenhuis structures
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- A prop profile of a differential geometric structure is a minimal resolution of an algebraic prop such that representations of this resolution are in one-to-one correspondence with structures of the given type. We begin this thesis with a detailed account of the algebraic tools necessary to construct prop profiles; we treat operads and props, and resolutions of these through Koszul duality. Our main results can be summarized as follows. Firstly, we contribute to the work of S.A. Merkulov on the prop profiles of Poisson and Nijenhuis structures. We prove that the operad of the latter prop profile is Koszul by showing that it has a PBW-basis, and we provide a geometrical interpretation of the former in terms of an L-infinity structure on the structure sheaf of a manifold. Secondly, we construct prop profiles of compatible Poisson and Nijenhuis structures. Representations of minimal resolutions of props correspond to Maurer-Cartan elements of certain Lie algebras associated to the resolved props. Also the differential geometric structures are defined as solutions of Maurer-Cartan equations. We show the correspondence between props and differential geometry by providing explicit isomorphisms between these Lie algebras. Thirdly, in order to construct the prop profiles of compatible Poisson and Nijenhuis structures we study operads of compatible algebraic structures. By studying Cohen-Macaulay properties of posets associated to such operads we prove the Koszulness of a large class of operads of compatible structures.
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| 6. |
- Backman, Theo, 1986-
(författare)
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Compactified Configuration Space of Points on a Line and Homotopies of A_ infty Morphisms
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we construct a configuration space model for a particular 2-colored dg operad encoding the structure of two A_infty algebras with two A_infty morphism and a homotopy between the morphisms. We determine the cohomology of this operad to be the well-known 2-colored operad encoding the structure of a two associative algebras and an associative algebra morphism between them.
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