Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 036, 33 pages      arXiv:0910.5929
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

Open/Closed String Topology and Moduli Space Actions via Open/Closed Hochschild Actions

Ralph M. Kaufmann
Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067, USA

Received October 30, 2009, in final form April 10, 2010; Published online April 30, 2010

In this paper we extend our correlation functions to the open/closed case. This gives rise to actions of an open/closed version of the Sullivan PROP as well as an action of the relevant moduli space. There are several unexpected structures and conditions that arise in this extension which are forced upon us by considering the open sector. For string topology type operations, one cannot just consider graphs, but has to take punctures into account and one has to restrict the underlying Frobenius algebras. In the moduli space, one first has to pass to a smaller moduli space which is closed under open/closed duality and then consider covers in order to account for the punctures.

Key words: string topology; Hochschild complex; double sided bar complex; foliations; open/closed field theory; moduli spaces; clusters of points.

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