Moduli spaces of Klein surfaces and related operads

Braun, Christopher (2012) Moduli spaces of Klein surfaces and related operads. Algebraic and Geometric Topology, 12 (3). pp. 1831-1899. ISSN 1472-2747

Other (pdf)
1003.5903v3 - Accepted Version

Download (504kB)


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.

Item Type:
Journal Article
Journal or Publication Title:
Algebraic and Geometric Topology
Additional Information:
Date of Acceptance: 08/05/2012
Uncontrolled Keywords:
ID Code:
Deposited By:
Deposited On:
11 Sep 2015 12:18
Last Modified:
27 Nov 2020 03:11