课程题目：Operads, Homotopy Algebras and Strings

主讲人：Branislav Jurco 教授
      Charles University, Prague 

授课时间：7月2日、9日、14日、15日、16日、17日      上午9:00--11:00

授课地点：数学学院管理科研楼1518教室

课程简介：

The aim of the lectures is to elucidate the algebraic structures of the string field theory from the operadic point of view.

It has been known already for some time that the homotopy algebras considered by Zwiebach in in the context of closed string field theory can be understood as algebras over the Feynman transform of the modular envelope of the cyclic operad Com. This packs the otherwise complicated axioms into standard constructions over Com, an easily understood algebraic object and opens up the way for applications of operad homotopy theory to study of these algebras.

The operadic point of view will be extended to the analogous algebras in open and open-closed string theory. 

For open strings, the role of the modular envelope of Com is played by the operad QO (Quantum Open), the modular envelope of the cyclic operad Ass, which will be described explicitly. The Feynman transform FQO of QO, an analogue of bar construction will be discussed in the realm of modular operads, and the resulting axioms of algebras over FQO will also be described explicitly.

According to Barannikov's theory, algebras over the Feynman transform are equivalently described by solutions of a "quantum master equation" in certain generalized (in general noncommutative) BV algebra. The closed case leads to the ordinary (commutative) BV structures.

Also, a 2-coloured modular operad QOC (Quantum Open Closed), describing the algebraic structures, in the open-closed case will be introduced. The generalized BV algebra will be made explicit, thus making contact with the work by Kajiura and Stasheff , which is deeply based on the open-closed string field theory description by Zwiebach.

In the course of lectures the necessary background material will be explained too.

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