Algebraic structures on generalized strings

Abstract

A garland based on a manifold $P$ is a finite set of manifolds homeomorphic to $P$ with some of them glued together at marked points. Fix a manifold $M$ and consider a space $\NN$ of all smooth mappings of garlands based on $P$ into $M$. We construct operations $\bullet$ and $[-,-]$ on the bordism groups $\bor_*(\NN)$ that give $\bor_*(\NN)$ the natural graded commutative assosiative and graded Lie algebra structures. We also construct two auto-homomorphisms $\pr$ and $\li$ of $\bor_*(\NN)$ such that $\pr(\li \alpha_1\bullet \li \alpha_2)= [\alpha_1, \alpha_2]$ for all $\alpha_1, \alpha_2 \in \bor_*(\NN)$. If $P$ is a boundary, then $\pr \circ \li=0$ and thus $\Delta^2=0$ for $\Delta=\li \circ \pr$. We show that under certain conditions the operations $\Delta$ and $\bullet$ give rise to Batalin-Vilkoviski and Gerstenhaber algebra structures on $\bor_*(\NN)$. In a particular case when $P=S^1$, the algebra $\bor_*(\NN)$ is related to the string-homology algebra constructed by Chas and Sullivan.