Notes on string topology

TL;DR

This paper introduces the field of String Topology, surveying its foundational concepts, recent developments, and future research directions, including connections to quantum field theories and Gromov-Witten invariants.

Contribution

It provides a comprehensive overview of String Topology, highlighting key structures, approaches, and open problems in this emerging area of mathematics.

Findings

Introduction of BV-algebra structure on loop space homology

Survey of homotopy theoretic approaches and operad models

Discussion of connections to quantum field theories and Gromov-Witten invariants

Abstract

This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topology by introducing a BV-algebra structure on the homology of a loop space of a manifold, then discuss the homotopy theoretic approach to String Topology, using the Thom-Pontrjagin construction, the cacti operad, and fat graphs. We review quantum field theories and indicate how string topology fits into the general picture. Other topics include an open-closed version of string topology, a Morse theoretic interpretation, relation to Gromov-Witten invariants, and "brane'' topology, which deals with sphere spaces. The paper is a joint account of the lecture series given by each of us at the 2003 Summer School on String Topology and Hochschild Homology in Almeria, Spain.