String field theory (SFT) is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. This is accomplished at the level of perturbation theory by finding a collection of vertices for joining and splitting strings, as well as string propagators, that give a Feynman diagram-like expansion for string scattering amplitudes. In most string field theories, this expansion is encoded by a classical action found by second-quantizing the free string and adding interaction terms. As is usually the case in second quantization, a classical field configuration of the second-quantized theory is given by a wave function in the original theory. In the case of string field theory, this implies that a classical configuration, usually called the string field, is given by an element of the free string Fock space. The principal advantages of the formalism are that it allows the computation of off-shell amplitudes and, when a classical action is available, gives non-perturbative information that cannot be seen directly from the standard genus expansion of string scattering. In particular, following the work of Ashoke Sen, it has been useful in the study of tachyon condensation on unstable D-branes. It has also had applications to topological string theory, non-commutative geometry, and strings in low dimensions. String field theories come in a number of varieties depending on which type of string is second quantized: Open string field theories describe the scattering of open strings, closed string field theories describe closed strings, while open-closed string field theories include both open and closed strings. In addition, depending on the method used to fix the worldsheet diffeomorphisms and conformal transformations in the original free string theory, the resulting string field theories can be very different. Using light cone gauge, yields light-cone string field theories whereas using BRST quantization, one finds covariant string field theories. There are also hybrid string field theories, known as covariantized light-cone string field theories which use elements of both light-cone and BRST gauge-fixed string field theories. A final form of string field theory, known as background independent open string field theory, takes a very different form; instead of second quantizing the worldsheet string theory, it second quantizes the space of two-dimensional quantum field theories.
Light-cone string field theory Light-cone string field theories were introduced by Stanley Mandelstam and developed by Mandelstam, Michael Green, John Schwarz and Lars Brink. An explicit description of the second-quantization of the light-cone string was given by Michio Kaku and Keiji Kikkawa. Light-cone string field theories were the first string field theories to be constructed and are based on the simplicity of string scattering in light-cone gauge. For example, in the bosonic closed string case, the worldsheet scattering diagrams naturally take a Feynman diagram-like form, being built from two ingredients, a propagator,
and two vertices for splitting and joining strings, which can be used to glue three propagators together,
These vertices and propagators produce a single cover of the moduli space of n {\displaystyle n} -point closed string scattering amplitudes so no higher order vertices are required. Similar vertices exist for the open string. When one considers light-cone quantized superstrings, the discussion is more subtle as divergences can arise when the light-cone vertices collide. To produce a consistent theory, it is necessary to introduce higher order vertices, called contact terms, to cancel the divergences. Light-cone string field theories have the disadvantage that they break manifest Lorentz invariance. However, in backgrounds with light-like Killing vectors, they can considerably simplify the quantization of the string action. Moreover, until the advent of the Berkovits string it was the only known method for quantizing strings in the presence of Ramond–Ramond fields. In recent research, light-cone string field theory played an important role in understanding strings in pp-wave backgrounds.
Free covariant string field theory An important step in the construction of covariant string field theories (preserving manifest Lorentz invariance) was the construction of a covariant kinetic term. This kinetic term can be considered a string field theory in its own right: the string field theory of free strings. Since the work of Warren Siegel, it has been standard to first BRST-quantize the free string theory and then second quantize so that the classical fields of the string field theory include ghosts as well as matter fields. For example, in the case of the bosonic open string theory in 26-dimensional flat spacetime, a general element of the Fock-space of the BRST quantized string takes the form (in radial quantization in the upper half plane),
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