Homological algebra related to surfaces with boundary

TL;DR

This paper introduces IBL$_ Infty$-algebras, a homotopy algebraic structure connecting string topology, symplectic field theory, and Lagrangian Floer theory for surfaces with boundary.

Contribution

It defines and explores IBL$_ Infty$-algebras, unifying three areas of mathematical physics through a common algebraic framework.

Findings

IBL$_\infty$-algebras generalize involutive bi-Lie algebras

The framework applies to string topology, symplectic field theory, and Floer theory

Provides new tools for studying surfaces with boundary

Abstract

In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, which we call IBL$_\infty$-algebras.