#### Volume 12, issue 3 (2012)

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Moduli spaces of Klein surfaces and related operads

### Christopher Braun

Algebraic & Geometric Topology 12 (2012) 1831–1899
##### 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 ${A}_{\infty }$–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.

##### Keywords
moduli space, Klein surfaces, mobius graphs, graph complex, topological quantum field theories, operads, modular operads
##### Mathematical Subject Classification 2000
Primary: 32G15, 30F50
Secondary: 57R56, 18D50, 81T40