{
  "id": "1003.5903",
  "version": "v3",
  "published": "2010-03-30T19:23:28.000Z",
  "updated": "2012-09-05T19:47:33.000Z",
  "title": "Moduli spaces of Klein surfaces and related operads",
  "authors": [
    "Christopher Braun"
  ],
  "comment": "60 pages, reformatted version, with a couple of minor technical corrections",
  "journal": "Algebr. Geom. Topol. 12 (2012) 1831-1899",
  "doi": "10.2140/agt.2012.12.1831",
  "categories": [
    "math.GT",
    "math.AG",
    "math.QA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-09-05T19:47:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "moduli spaces",
      "topological conformal field theories",
      "klein surfaces",
      "topological quantum field theories",
      "open topological conformal field"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.5903B"
    }
  }
}