Operads and field theory

Abstract

Operády a ich varianty, modulárne a cyklické operády, prirodzene popisujú skladanie objektov rôznych typov. Práca poskytuje prístupný úvod do teórie operád, formalizmu používaného v [1] a modernej aplikácie modulárnych operád vo fyzike [2]. S pomocou príkladov uvedieme Batalin-Vilkovisky formalizmus ako nástroj na kohomologickú integráciu dráhového integrálu v kvantovej teórii pola. Master rovnica, podmienka na akciu, plynie z tohoto formalizmu. Riešenia master rovnice ale taktiež popisujú algebry nad Feynmanovou transformáciou modulárnej operády. Preskúmame master rovnicu takto definovanú na modulárnej operáde a zhrnieme aplikáciu tejto teórie do uzavretej strunovej teórie pola. [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.

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.