String Field Theory

After more than 50 years from the Veneziano amplitude, the fundamental formulation of string theory (ST) remains elusive. On the one hand, there is the world-sheet formulation that is truly microscopic but which is only valid for infinitesimal string coupling and is highly background dependent. On the other hand, there are other nonperturbative background independent approaches such as supergravity or other quantum theories of (super) Yang–Mills type, which however necessarily miss some of the features of the extended nature of the string (although in some cases they can be holographically equivalent to a ST). In string field theory (SFT), it is possible to keep an exact microscopic world-sheet description together with a complete space–time framework that follows the rules of quantum field theory (QFT) and where nonperturbative contributions can be, at least in principle, coherently accounted for. String field theory is a formulation of ST as a QFT for an infinite number of fields (the various oscillation modes of the string) in spacetime. This formulation allows to better treat some of the shortcomings of the usual on-shell formulation of ST while maintaining at the same time a full microscopic world-sheet approach. The construction of SFTs is such that ST world-sheet amplitudes are reproduced when these are well-defined. But SFT gives a more general construction of amplitudes that is well-defined even when the standard world-sheet approach gives rise to divergences. In this very general framework, all the elementary string interactions are defined so as to provide a solution to the quantum Batalin–Vilkovisky master equation, furnishing a perturbative microscopic definition of the target space path integral of ST. This construction is explicitly realized in terms of (quantum) homotopy algebras for both bosonic strings and superstrings, including Type II, Type I, and heterotic. The construction offered by SFT allows to define the one-particle irreducible (1PI) effective action of ST and thus to give a definition of string perturbation theory where it is possible to discuss quantum effects such as vacuum shifts due to tadpoles and mass renormalization. The explicit knowledge of microscopic ultraviolet SFTs allows to construct the low-energy ST effective action as the Wilsonian action by integrating out the massive string states from the SFT path integral. This top-down construction is safe from infrared divergences and has been very useful for obtaining unambiguous results on nonperturbative contributions, such as D-instanton corrections to perturbative amplitudes and effective superpotentials. String field theories (especially open string field theories [OSFTs]) allow to approach background independence in ST by recasting the plethora of different ST backgrounds in the form of classical solutions to the SFT equation of motion. This program has been fully realized in critical bosonic OSFT, where any D-brane system can be explicitly written as a classical solution of the OSFT on any other D-brane system.