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Algebras over operads and properads
Peksová, Lada ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Operads are objects that model operations with several inputs and one output. We define such structures in the context of graphs, namely oriented trees. Then we generalize operads to properads and modular operads by taking general graphs with, or without, orientation. Further we construct the cobar complex of operads and properads and illustrate the construction on the examples of the associative operad Ass and the Frobenius properad Frob. Algebras over the cobar complex of operads correspond to certain homotopy algebras, for our example of Ass it is A1. We find its Maurer-Cartan equation and convert it from coderivations to derivations. Similarly we find the Maurer-Cartan equation for cobar complex of Frobenius properad. Powered by TCPDF (www.tcpdf.org)
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Algebras over operads and properads
Peksová, Lada ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Operads are objects that model operations with several inputs and one output. We define such structures in the context of graphs, namely oriented trees. Then we generalize operads to properads and modular operads by taking general graphs with, or without, orientation. Further we construct the cobar complex of operads and properads and illustrate the construction on the examples of the associative operad Ass and the Frobenius properad Frob. Algebras over the cobar complex of operads correspond to certain homotopy algebras, for our example of Ass it is A1. We find its Maurer-Cartan equation and convert it from coderivations to derivations. Similarly we find the Maurer-Cartan equation for cobar complex of Frobenius properad. Powered by TCPDF (www.tcpdf.org)
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Operads and field theory
Pulmann, Ján ; Jurčo, Branislav (advisor) ; Křižka, Libor (referee)
Operads and their variants, modular and cyclic operads, naturally describe compositions of objects of various types. We provide an accessible introduction to the theory of operads, the formalism for modular operads from [1] and modern application of modular operads to physics, due to Barannikov [2]. Through examples, we introduce Batalin-Vilkovisky formalism as a tool for cohomological integration of path integral in quantum field theories. A master equation, consistency condition for action, follows from this formalism. Solutions to master equation also describe algebras over Feynman transform of a modular operad. We explore the master equation defined in terms of modular operad and review an application to closed string field theory. [1] Martin Doubek, Branislav Jurco, and Korbinian Muenster. Modular operads and the quantum open-closed homotopy algebra. 2013. arXiv: 1308.3223 [math-AT]. [2] Serguei Barannikov. "Modular operads and Batalin-Vilkovisky geometry". In: International Mathematics Research Notices 2007 (2007), rnm075.
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