Algebraic & Geometric Topology

Moduli spaces of Klein surfaces and related operads

Christopher Braun

Full-text: Open access

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–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.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 3 (2012), 1831-1899.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715420

Digital Object Identifier
doi:10.2140/agt.2012.12.1831

Mathematical Reviews number (MathSciNet)
MR2980000

Zentralblatt MATH identifier
1254.30071

Subjects
Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 30F50: Klein surfaces
Secondary: 57R56: Topological quantum field theories 18D50: Operads [See also 55P48] 81T40: Two-dimensional field theories, conformal field theories, etc.

Keywords
moduli space Klein surfaces mobius graphs graph complex topological quantum field theories operads modular operads

Citation

Braun, Christopher. Moduli spaces of Klein surfaces and related operads. Algebr. Geom. Topol. 12 (2012), no. 3, 1831--1899. doi:10.2140/agt.2012.12.1831. https://projecteuclid.org/euclid.agt/1513715420


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