Recall that perturbative string theory is a higher dimensional version of the Feynman perturbation series in quantum field theory. This Feynman perturbation series may be understood as computing the path integral over the Lagrangian of the given quantum field theory. String field theory is the attempt to identifiy this Lagrangian description corresponding to the string perturbation series.
So string field theory is the attempt to complete the following analogy:
Feynman perturbation series : QFT Lagrangian :: String pertrubation theory : String field theory .
The original hope was that string field theory would be a way to embed the string perturbation series presription into a more coherent non-perturbative framework.
The most detailed insight that has come out of the study of string field theory is the full understanding of the role of the “tachyon” field in bosonic perturbative string theory. In the bosonic version of the theory one of the excitations of the string is a quantum that appears to have imaginary mass. Such “tachyonic” quanta appear in ordinary field theory when the perturbation series is developed around an extremum of the QFT action functional that is not a local minimum, but a local maximum: it indicates that the classical configuratoin around which the perturbation series computes the quantum corrections is dynamically unstable and time evolution will tend to evolve it to the next local minimum. In the perturbative quantum description the movement to the next local minimum manifests itself in the condensation (as in Bose-Einstein condensation?) of the tachyon field. This is called tachyon condensation.
Shortly after its conception it was suspected that the tachyon that appears in the perturbation theory of the bosonic string is similarly an indication that the bosonic string’s perturbation series has to be understood as being a perturbation about a local maximum of some acvtion functional. String field theory aimed to provide that notion of action functional. And indeed, in bosonic string field theory one has a kind of higher action functional and may compute the “tachyon potential” that it implies. It indeed has a local maximum at the point about which the ordinary bosonic string perturbation series is a perturbative expansion, while a local minimum is foun nearby.
Sen conjectured the statement, now known as Sen’s conjecture that the “depth” of this tachyon potential, i.e. the energy density difference between this local maximum and this local minimum corresponds precisely to the energy density of the space-fillind D25-brane that is seen in perturbative string theory. This would mean that the condensation of the bosonic string’s tachyon corresponds to the decay of the unstable space-filling D25 brane.
The detailed quantititative confrmation of Sen’s conjecture has been one of the main successes of string field theory. In the course of this a detailed algebraic description of the “true bosonic string vacuum”, i.e. of the theory at that local tachyon potential minimum has been found. However, the algebraic expressions involved tend to be hard to handle in their complexity.
The shortcoming of the current development of string field theory can be summarized as follows:
it has beeen studied as a theory of a classical action functional. Little is known about the true quantum effects of the string field theory action functional.
the best understanding exists for bosonic open string field theory, while closed and supersymmetric string field theory has remained much less accessible.
See A Survey of Cohomological Physics and n-categorical physics.
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