We consider the extension of classical
–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
–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.
Keywords
moduli space, Klein surfaces, mobius graphs, graph complex,
topological quantum field theories, operads, modular
operads
