20
1 Introduction
Another reason to use SFT is gauge invariance: it is always easier to describe a
system when its gauge invariance is manifest. We have explained that string theory
contains Yang–Mills and graviton fields with the corresponding (spacetime) gauge
invariances (non-Abelian gauge symmetry and diffeomorphisms). In fact, these
symmetries are enhanced to an enormous gauge invariance when taking into account
the higher-spin fields. This invariance is hidden in the standard formulation and
cannot be exploited fully. On the other hand, the full gauge symmetry is manifest in
string field theory.
Finally, the worldvolume description of p-brane is difficult because there is no
analogue of the Polyakov action. If one could find a first-principle description of
SFT which does not rely on CFT and first-quantization, then one may hope to
generalize it to build a brane field theory.
We can summarize the general motivations for studying SFT:
• field theory (second-quantization);
• more rigorous and constructive formulation;
• make gauge invariance explicit (L ∞ algebras et al.);
• use standard QFT techniques (renormalization, analyticity. . . )
→ remove IR divergences, prove consistency (Cutkosky rules, unitarity, soft
theorems, background independence. . . );
• worldvolume theory ill-defined for (p > 1)-branes.
Beyond these general ideas, SFT has been developed in order to address different
questions:
• worldsheet scattering amplitudes;
• effective actions;
• map of the consistent backgrounds (classical solutions, marginal deformations,
RR fluxes. . . );
• collective, non-perturbative, thermal, dynamical effects;
• symmetry breaking effects;
• dynamics of compactification;
• proof of dualities;
• proof of the AdS/CFT correspondence.
The last series of points is still out of reach within the current formulation of SFT.
However, the last two decades have seen many important developments:
• construction of the open, closed and open-closed superstring field theories:
– 1PI and BV actions and general properties [26, 67, 68, 79, 80, 82, 84, 86, 88, 91,
92, 96],
– dressing of bosonic products using the WZW construction and homotopy
algebra [8, 9, 20–23, 29–33, 35, 38, 45, 46, 50–56, 70],
– light-cone super-SFT [39–42],
– supermoduli space [69, 97];
1 Introduction
Another reason to use SFT is gauge invariance: it is always easier to describe a
system when its gauge invariance is manifest. We have explained that string theory
contains Yang–Mills and graviton fields with the corresponding (spacetime) gauge
invariances (non-Abelian gauge symmetry and diffeomorphisms). In fact, these
symmetries are enhanced to an enormous gauge invariance when taking into account
the higher-spin fields. This invariance is hidden in the standard formulation and
cannot be exploited fully. On the other hand, the full gauge symmetry is manifest in
string field theory.
Finally, the worldvolume description of p-brane is difficult because there is no
analogue of the Polyakov action. If one could find a first-principle description of
SFT which does not rely on CFT and first-quantization, then one may hope to
generalize it to build a brane field theory.
We can summarize the general motivations for studying SFT:
• field theory (second-quantization);
• more rigorous and constructive formulation;
• make gauge invariance explicit (L ∞ algebras et al.);
• use standard QFT techniques (renormalization, analyticity. . . )
→ remove IR divergences, prove consistency (Cutkosky rules, unitarity, soft
theorems, background independence. . . );
• worldvolume theory ill-defined for (p > 1)-branes.
Beyond these general ideas, SFT has been developed in order to address different
questions:
• worldsheet scattering amplitudes;
• effective actions;
• map of the consistent backgrounds (classical solutions, marginal deformations,
RR fluxes. . . );
• collective, non-perturbative, thermal, dynamical effects;
• symmetry breaking effects;
• dynamics of compactification;
• proof of dualities;
• proof of the AdS/CFT correspondence.
The last series of points is still out of reach within the current formulation of SFT.
However, the last two decades have seen many important developments:
• construction of the open, closed and open-closed superstring field theories:
– 1PI and BV actions and general properties [26, 67, 68, 79, 80, 82, 84, 86, 88, 91,
92, 96],
– dressing of bosonic products using the WZW construction and homotopy
algebra [8, 9, 20–23, 29–33, 35, 38, 45, 46, 50–56, 70],
– light-cone super-SFT [39–42],
– supermoduli space [69, 97];