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Moduli spaces of Klein surfaces and related operads

Christopher Braun

Algebraic & Geometric Topology 12 (2012) 1831–1899

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–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 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
Received: 30 March 2010
Revised: 25 August 2011
Accepted: 8 May 2012
Published: 7 September 2012
Christopher Braun
Centre for Mathematical Science
City University London
Northampton Square
United Kingdom