We consider the extension of classical –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 –algebras with involution in terms of Möbius 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 Möbius graph decomposition of the moduli spaces of Klein surfaces (real algebraic curves). The Möbius 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.
"Moduli spaces of Klein surfaces and related operads." Algebr. Geom. Topol. 12 (3) 1831 - 1899, 2012. https://doi.org/10.2140/agt.2012.12.1831