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2012 Moduli spaces of Klein surfaces and related operads
Christopher Braun
Algebr. Geom. Topol. 12(3): 1831-1899 (2012). DOI: 10.2140/agt.2012.12.1831


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.


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Christopher Braun. "Moduli spaces of Klein surfaces and related operads." Algebr. Geom. Topol. 12 (3) 1831 - 1899, 2012.


Received: 30 March 2010; Revised: 25 August 2011; Accepted: 8 May 2012; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1254.30071
MathSciNet: MR2980000
Digital Object Identifier: 10.2140/agt.2012.12.1831

Primary: 30F50, 32G15
Secondary: 18D50, 57R56, 81T40

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.12 • No. 3 • 2012
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