On the Semi-Relative Condition for Closed (TOPOLOGICAL) Strings

We provide a simple lagrangian interpretation of the meaning of the $b_0^-$ semi-relative condition in closed string theory. Namely, we show how the semi-relative condition is equivalent to the requirement that physical operators be cohomology classes of the BRS operators acting on the space of local fields {\it covariant} under world-sheet reparametrizations. States trivial in the absolute BRS cohomology but not in the semi-relative one are explicitly seen to correspond to BRS variations of operators which are not globally defined world-sheet tensors. We derive the covariant expressions for the observables of topological gravity. We use them to prove a formula that equates the expectation value of the gravitational descendant of ghost number 4 to the integral over the moduli space of the Weil-Peterson K\"ahler form.Comment: 10 pages, harvmac, CERN-TH-7084/93, GEF-TH-21/199

A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model

We argue that topological matrix models (matrix models of the Kontsevich type) are examples of exact open/closed duality. The duality works at finite N and for generic `t Hooft couplings. We consider in detail the paradigm of the Kontsevich model for two-dimensional topological gravity. We demonstrate that the Kontsevich model arises by topological localization of cubic open string field theory on N stable branes. Our analysis is based on standard worldsheet methods in the context of non-critical bosonic string theory. The stable branes have Neumann (FZZT) boundary conditions in the Liouville direction. Several generalizations are possible.Comment: v2: References added; a new section with generalization to non-zero bulk cosmological constant; expanded discussion on topological localization; added some comment

Introduction to the Basic Concepts of Modern Physics

none2BECCHI C.M.; D'ELIA M.Becchi, CARLO MARIA; D'Elia, Massim