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We define a theory of descendent integration on the moduli spaces of stablepointed disks. The descendent integrals are proved to be coefficients of the$\tau$-function of an open KdV heirarchy. A relation between the integrals anda representation of half the Virasoro algebra is also proved. The constructionof the theory requires an in depth study of homotopy classes of multivaluedboundary conditions. Geometric recursions based on the combined structure ofthe boundary conditions and the moduli space are used to compute the integrals.We also provide a detailed analysis of orientations.  Our open KdV and Virasoro constraints uniquely specify a theory of highergenus open descendent integrals. As a result, we obtain an open analog(governing all genera) of Witten's conjectures concerning descendent integralson the Deligne-Mumford space of stable curves.

Intersection theory on moduli of disks, open KdV and Virasoro