# Open string theory and planar algebras

## TL;DR

This paper establishes a deep connection between open string theory, planar algebras, and real algebraic geometry by demonstrating that planar algebras are algebras over certain topological operads related to moduli spaces.

## Contribution

It introduces a novel interpretation of planar algebras as algebras over the moduli space operad and links open string theory with real algebraic geometry and subfactor theory.

## Key findings

- Planar algebras are algebras over the operad of moduli spaces of stable maps with Lagrangian boundary conditions.
- Two geometric methods to derive planar algebras from real algebraic geometry are proposed.
- The work connects open string theory, real algebraic geometry, and subfactors of von Neumann algebras.

## Abstract

In this note we show that abstract planar algebras are algebras over the topological operad of moduli spaces of stable maps with Lagrangian boundary conditions, which in the case of the projective line are described in terms of real rational functions. These moduli spaces appear naturally in the formulation of open string theory on the projective line. We also show two geometric ways to obtain planar algebras from real algebraic geometry, one based on string topology and one on Gromov-Witten theory. In particular, through the well known relation between planar algebras and subfactors, these results establish a connection between open string theory, real algebraic geometry, and subfactors of von Neumann algebras.