Wheeled Operads in Algebra, Geometry, and Quantization

Sammanfattning: 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.