Moduli spaces of Klein surfaces and related operads

TL;DR

This paper extends the theory of topological quantum field theories to include unorientable surfaces using operads, introducing Mobius graphs to study moduli spaces of Klein surfaces and their homology.

Contribution

It introduces a new operad for associative algebras with involution, identifies its Koszul dual, and generalizes graph decompositions to Klein surfaces.

Findings

Mobius graph decomposition of Klein surface moduli spaces

Homology computation via Mobius graph complex

Generalization of ribbon graph decomposition to unorientable surfaces

Abstract

We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul and we identify the dual dg operad governing A-infinity algebras with involution in terms of Mobius graphs which are a generalisation of ribbon graphs. We then generalise open topological conformal field theories to open Klein topological conformal field theories and give a generators and relations description of the open KTCFT operad. We deduce an analogue of the ribbon graph decomposition of the moduli spaces of Riemann surfaces: a Mobius graph decomposition of the moduli spaces of Klein surfaces (real algebraic curves). The Mobius graph complex then computes the homology of these moduli spaces. We also obtain a different graph complex computing the homology of the moduli spaces of admissible stable symmetric Riemann surfaces which are partial compactifications of the moduli spaces of Klein surfaces.