On operad structures of moduli spaces and string theory

TL;DR

This paper explores how the topology of moduli spaces of punctured Riemann spheres underpins algebraic structures in string theory, revealing their origin from operad theory and conformal field theory.

Contribution

It demonstrates that algebraic structures in string theory arise naturally from the operad of punctured Riemann surfaces and their relation to conformal field theory.

Findings

Algebraic structures like homotopy Lie and Batalin-Vilkovisky algebras are derived from moduli space topology.

Operads of punctured Riemann spheres encode the algebraic operations in string theory.

Conformal field theories are shown to be algebras over these operads.

Abstract

Recent algebraic structures of string theory, including homotopy Lie algebras, gravity algebras and Batalin-Vilkovisky algebras, are deduced from the topology of the moduli spaces of punctured Riemann spheres. The principal reason for these structures to appear is as simple as the following. A conformal field theory is an algebra over the operad of punctured Riemann surfaces, this operad gives rise to certain standard operads governing the three kinds of algebras, and that yields the structures of such algebras on the (physical) state space naturally.