Abstract. Using the relation between diffeomorphisms in the bulk and Weyl transformations on the boundary, we study the Weyl transformation properties of the bulk metric on-shell and of the boundary action. We obtain a universal formula for one of the classes of trace anomalies in any even dimension in terms of the parameters of the gravity action.

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We derive a Kontsevich-type matrix model for the c = 1 string directly from the W ∞ solution of the theory. The model that we obtain is different from previous proposals, which are proven to be incorrect. Our matrix model contains the Penner and Kontsevich cases, and we study its quantum effective action. The simplicity of our model leads to an encouraging interpretation in the context of background-independent non-critical string field theory.

Supersymmetric field theories can be studied exactly on off-shell "localizing" supergravity backgrounds. We show that these supergravity configurations can be identified with BRST invariant configurations of background topological gravity coupled to background topological gauge multiplets. We apply this topological point of view to two-dimensional N = (2, 2) supersymmetric matter theories to obtain, in a simple and straightforward way, a complete classification of localizing supersymmetric backgrounds in two dimensions. We recover all known localizing backgrounds and (infinitely) many more that have not been explored so far. The newly found localizing backgrounds are characterized by quantized fluxes for both graviphotons of the N = (2, 2) supergravity multiplet. The BRST invariant topological backgrounds are parametrized by both Killing vectors and S 1 -equivariant cohomology of the two-dimensional spacetime. We completely reconstruct the supergravity backgrounds from the topological data: some of the supergravity fields are twisted versions of the topological backgrounds, but others are composite, in that they are nonlinear functionals of topological fields. Moreover, we show that the supersymmetric Ω-deformation is nothing but the background value of the ghost-for-ghost of topological gravity, a result which holds for higher dimensions too.

We illustrate a method for computing the number of physical states of open string theory at the stable tachyonic vacuum in level truncation approximation. The method is based on the analysis of the gauge-fixed open string field theory quadratic action that includes Fadeev-Popov ghost string fields. Computations up to level 9 in the scalar sector are consistent with Sen's conjecture about the absence of physical open string states at the tachyonic vacuum. We also derive a long exact cohomology sequence that relates relative and absolute cohomologies of the BRS operator at the non-perturbative vacuum. We use this exact result in conjunction with our numerical findings to conclude that the higher ghost number non-perturbative BRS cohomologies are non-empty.

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