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On the homology of open-closed string theory

In Zwiebach’s study of oriented open-closed string theory [18], he considered a certain moduli space of Riemann Surfaces with boundary having “closed” punctures in the interior and “open” punctures on the boundary coming with parameterizations by the unit disk and upper half disk. While he did not consider it as such, this moduli space forms a 2-colored PROP. The first purpose of this paper is to describe completely the homology of the biggest genus 0 structure inside this PROP. The operad inside of this PROP formed by spheres with no boundary is well known to be homotopy equivalent to the framed little disks operad. It is shown by Getzler that its homology defines a BV-algebra [7]. This extends the result by Cohen [4] showing that the homology of the non-framed little disks operad describes a Gerstenhaber algebra. In [17], Voronov invented the Swiss-cheese operad which is a (non framed) finite dimensional model of the operad inside this PROP formed by Riemann spheres with one or no boundary components. He computed its homology and calls the algebra that it defines a Swiss-cheese algebra. The algebra is defined on a pair of graded vector spaces ( , ) C O V V and consists of a Gestenhaber structure on