Moduli spaces of punctured Poincar\'e disks

TL;DR

This paper explores the moduli space of marked particles on the Poincaré disk, extending classical associativity concepts and connecting to various areas in mathematical physics and geometry.

Contribution

It introduces a geometric and combinatorial construction of the moduli space of marked particles on the Poincaré disk, generalizing Tamari's associativity.

Findings

Connections to Kontsevich's deformation quantization

Relation to Voronov's swiss-cheese operad

Relevance to open-closed string theory

Abstract

The Tamari lattice and the associahedron provide methods of measuring associativity on a line. The real moduli space of marked curves captures the space of such associativity. We consider a natural generalization by considering the moduli space of marked particles on the Poincar\'{e} disk, extending Tamari's notion of associativity based on nesting. A geometric and combinatorial construction of this space is provided, which appears in Kontsevich's deformation quantization, Voronov's swiss-cheese operad, and Kajiura and Stasheff's open-closed string theory.