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Published September 24, 2010 | Submitted
Journal Article Open

Open string theory and planar algebras

Abstract

In this paper, we show that abstract planar algebras are algebras over the topological operad of moduli spaces of stable maps with Lagrangian boundary conditions, which in the case of the projective line are described in terms of real rational functions. These moduli spaces appear naturally in the formulation of open string theory on the projective line. We also show two geometric ways to obtain planar algebras from real algebraic geometry, one based on string topology and one on Gromov–Witten theory. In particular, through the well-known relation between planar algebras and subfactors, these results establish a connection between open string theory, real algebraic geometry and subfactors of von Neumann algebras.

Additional Information

© 2010 IOP Publishing Ltd. Received 19 March 2010, in final form 21 March 2010 Published 16 August 2010. The first author would like to thank Vaughan Jones for useful discussions. The first author is partially supported by NWO. The second author is partially supported by NSF grants DMS-0651925 and DMS-0901221. Part of this work was carried out during the authors' stay at the Max Planck Institute for Mathematics, which we thank for the hospitality and for support. The first author also thanks to Feza Gürsey Institute for the hospitality.

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