## Algebraic & Geometric Topology

### Moduli spaces of Klein surfaces and related operads

Christopher Braun

#### 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
Revised: 25 August 2011
Accepted: 8 May 2012
First available in Project Euclid: 19 December 2017

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

#### 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

#### References

• A Alexeevski, S Natanzon, Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves, Selecta Math. 12 (2006) 307–377
• N L Alling, N Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics 219, Springer, Berlin (1971)
• M Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. (1988) 175–186
• P Buser, M Seppälä, Symmetric pants decompositions of Riemann surfaces, Duke Math. J. 67 (1992) 39–55
• J Chuang, A Lazarev, Dual Feynman transform for modular operads, Commun. Number Theory Phys. 1 (2007) 605–649
• J Conant, K Vogtmann, On a theorem of Kontsevich, Algebr. Geom. Topol. 3 (2003) 1167–1224
• K Costello, The $A_\infty$ operad and the moduli space of curves
• K Costello, A dual version of the ribbon graph decomposition of moduli space, Geom. Topol. 11 (2007) 1637–1652
• E Getzler, M M Kapranov, Modular operads, Compositio Math. 110 (1998) 65–126
• V Ginzburg, M Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994) 203–272
• J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157–176
• J Kock, Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts 59, Cambridge Univ. Press (2004)
• M Kontsevich, Feynman diagrams and low-dimensional topology, from: “First European Congress of Mathematics, Vol. II (Paris, 1992)”, Progr. Math. 120, Birkhäuser, Basel (1994) 97–121
• A D Lauda, H Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, Topology Appl. 155 (2008) 623–666
• C I Lazaroiu, On the structure of open-closed topological field theory in two dimensions, Nuclear Phys. B 603 (2001) 497–530
• C C M Liu, Moduli of $J$–holomorphic curves with Lagrangian boundary conditions and open Gromov–Witten invariants for an $S^1$–equivariant pair
• G Moore, D–branes, RR–fields and $K$–theory, lecture (2001) Available at \setbox0\makeatletter\@url http://online.itp.ucsb.edu/online/mp01/moore1/ {\unhbox0
• G Moore, Some comments on branes, $G$–flux, and $K$–theory, from: “Strings 2000. Proceedings of the International Superstrings Conference (Ann Arbor, MI)”, volume 16 (2001) 936–944
• S M Natanzon, Klein surfaces, Uspekhi Mat. Nauk 45 (1990) 47–90, 189
• R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299–339
• G Segal, Topological structures in string theory, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001) 1389–1398 Topological methods in the physical sciences (London, 2000)
• M Seppälä, Moduli spaces of stable real algebraic curves, Ann. Sci. École Norm. Sup. 24 (1991) 519–544
• V Turaev, P Turner, Unoriented topological quantum field theory and link homology, Algebr. Geom. Topol. 6 (2006) 1069–1093