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The operator formalism provides a constructive approach to superstring amplitudes that has two main virtues: it makes it possible to study explicitly all the degeneration limits and it is flexible enough to work in different setups. I will illustrate these two features in a concrete example: the twisted open string partition function at two loops, which describes the interactions between three D-branes in type II theories.

The conventional on-shell techniques of computing string theory amplitudes is not sufficient to compute mass renormalization of massive states and vacuum shift in string theory. The aim of this talk is to define a suitable off-shell continuation of Polyakov approach. Then this off-shell continuation will be used to compute mass-renormalization and perturbative vacuum shift in string theory.

In this talk I will review a recent reformulation of string theory which does not rely on an a priori space-time interpretation or a pre-assumption of locality and include form the onset stringy symmetries such as T-duality. I will explain how this resulting theory, called metastring, leads to formulation where the string is chiral and the target is phase space instead of space-time. I will discuss metastring theory on a flat background and summarize a variety of technical and interpretational ideas. These include a discussion of moduli space of Lorentzian worldsheets, a generalization of the world sheet renormalisation group, a description of the geometry of phase space, a study of the symplectic structure and of closed and open boundary conditions, and the string spectrum and operator algebra. What emerges from these studies is a new quantum notion of space-time that we call modular space-time. This new geometrical concept is fundamental quantum and modular. It is closely linked with T-duality and implements in a precise way a notion of relative locality