Nocommutative structures inquantum field theory

by Lada Peksova
Doctoral thesis
Date of Examination:2020-10-22
Date of issue:2021-10-19
Advisor:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Ivo Sachs
Referee:Dr. Roman Golovko
Persistent Address: http://dx.doi.org/10.53846/goediss-8886

 

 

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Abstract

English

In this thesis, structures defined via modular operads and properads are generalized to their non-commutative analogs. We define the connected sum for modular operads. This way we are able to construct the graded commutative product on the algebra over Feynman transform of the modular operad. This forms a Batalin-Vilkovisky algebra with symmetry given by the modular operad. We transfer this structure to the cohomology via the Homological perturbation lemma. In particular, we consider these constructions for Quantum closed and Quantum open modular operad. As a parallel project, we introduce an associative analog of Frobenius properad, called Open Frobenius properad. We construct the cobar complex over it and in the spirit of Barannikov, we interpret algebras over the cobar complex as homological differential operators. Furthermore, we present the IBA∞-algebras as an analog of well-known IBL∞-algebras.
Keywords: modular operads, properads, Batalin-Vilkovisky algebras, Quantum homotopy algebras, IBL∞ and IBA∞ algebras
 
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