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,[1] it has been useful in the study of tachyon condensation on unstable D-branes.
[9][10][11][12][13] An explicit description of the second-quantization of the light-cone string was given by Michio Kaku and Keiji Kikkawa.
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
-point closed string scattering amplitudes so no higher order vertices are required.
[17] 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.
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), where
In the worldsheet string theory, the unphysical elements of the Fock space are removed by imposing the condition
In the case of the open bosonic string a gauge-unfixed action with the appropriate symmetries and equations of motion was originally obtained by André Neveu, Hermann Nicolai and Peter C.
[22] For the bosonic closed string, construction of a BRST-invariant kinetic term requires additionally that one impose
The kinetic term is then Additional considerations are required for the superstrings to deal with the superghost zero-modes.
The best studied and simplest of covariant interacting string field theories was constructed by Edward Witten.
The cubic vertex, is a trilinear map which takes three string fields of total ghostnumber three and yields a number.
[25][26] Second, following the work of Martin Schnabl [27] one can seek analytic solutions by carefully picking an ansatz which has simple behavior under star multiplication and action by the BRST operator.
The resulting geometry is completely flat except for a single curvature singularity where the midpoints of the three propagators meet.
It follows that, for on-shell amplitudes, the n-point open string amplitudes computed using Witten's open string field theory are identical to those computed using standard worldsheet methods.
[31][32] There are two main constructions of supersymmetric extensions of Witten's cubic open string field theory.
The first is very similar in form to its bosonic cousin and is known as modified cubic superstring field theory.
The first consistent extension of Witten's bosonic open string field theory to the RNS string was constructed by Christian Preitschopf, Charles Thorn and Scott Yost and independently by Irina Aref'eva, P. B. Medvedev and A. P.
This action has been shown to reproduce tree-level amplitudes and has a tachyon vacuum solution with the correct energy.
[35] The one subtlety in the action is the insertion of picture changing operators at the midpoint, which imply that the linearized equations of motion take the form Because
[36] However, such solutions would have operator insertions near the midpoint and would be potentially singular, and importance of this problem remains unclear.
It has been shown to reproduce correctly tree level amplitudes[39] and has been found, numerically, to have a tachyon vacuum with appropriate energy.
[40][41] The known analytic solutions to the classical equations of motion include the tachyon vacuum[42] and marginal deformations.
A formulation of superstring field theory using the non-minimal pure-spinor variables was introduced by Berkovits.
In an attempt to resolve the allegedly problematic midpoint insertion of the modified cubic theory, Berkovits and Siegel proposed a superstring field theory based on a non-minimal extension of the RNS string,[44] which uses a midpoint insertion with no kernel.
In general, a manifestly BV invariant, quantizable action takes the form[48] where
[50][51] A formulation of the NS sector of the heterotic string was given by Berkovits, Okawa and Zwiebach.