Moduli spaces of Klein surfaces and related operads

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-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.

60 pages, reformatted version, with a couple of minor technical corrections

References in corpus (6)

• Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
• Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair
• Unoriented topological quantum field theory and link homology
• The A-infinity operad and the moduli space of curves
• A dual point of view on the ribbon graph decomposition of moduli space
• Dual Feynman transform for modular operads

Cited by in corpus (8)

• Crossed simplicial groups and structured surfaces
• Involutive A-infinity algebras and dihedral cohomology
• M-theory interpretation of the real topological string
• A Klein TQFT: the local Real Gromov-Witten theory of curves
• Structured Topological Field Theories via Crossed Simplicial Groups
• Cohomology and deformations of oriented dialgebras
• Localisations of cobordism categories and invertible TFTs in dimension two
• Open-closed modular operads, Cardy condition and string field theory