Normalization of the Boundary Superstring Field Theory Action

The normalization of the boundary superstring field theory action is determined by computing the near on-shell amplitude involving three gauge fields. Contribution to the proceedings of Strings 2001, Mumbai, India. [email protected] formerly Mehta Reasearch Institute of Mathematics & Mathematical Physics. In the recent past there has been some progress in understanding the behaviour of the tachyonic scalar field in open string theory. In the bosonic theory, this scalar is the lowest mode on the D-p-branes. In superstring it appears on the non-BPS D-p-branes as well as on non-supersymmetric Dp-Dp̄ systems. Sen has conjectured[1] that there is a (local) minimum of the tachyon potential which describes the closed string vacuum without any D-brane. At this minimum the negative contribution from the tachyon potential exactly cancels the tension of the D-brane(s). Further, a codimension one lump solution in the bosonic case describes a D-(p − 1)-brane, while in the superstring case, a kink solution represents a BPS D-(p− 1)-brane (respectively a non-BPS D-(p− 1)-brane) in the same theory. This picture generalizes to solitons of higher codimension. While support for these conjectures come from the analysis of the worldsheet theory[2, 1], various toy models of tachyon condensation[3, 4], as well as noncommutative limit of the effective field theory of the tachyon[5], the natural setting to prove these is off-shell string field theory (SFT). The most well understood SFT is the covariant open bosonic string field theory[6]. There has been numerical investigation in the level truncation scheme of this theory[7] that provides very good evidence in the favour of the conjectures. Similar analysis hass also been done for superstrings in a covariant open string field theory proposed by Berkovits[8]. There is also a formulation of bosonic open string field theory based on the boundary perturbation of the worldsheet theory on the disc[9, 10]. It has been shown[11, 12] that it is possible to prove the conjectures for the bosonic open string in the framework of this so called boundary string field theory (B-SFT). A crucial ingredient in the exact verification of this idea is the determination of the normalization of the B-SFT action. This was computed in Ref.[13]. Soon afterwards, (inspired by earlier works[14]), a proposal for the boundary superstring field theory (B-SSFT) was made and the behaviour of the tachyon potential analyzed[15]. More recently there have been attempts[16] to derive this action along the lines of Refs.[9]. In this note we shall compute the normalization of this B-SSFT. In order to define boundary superstring field theory, we take the worldsheet to be a disc of unit radius with a flat metric on it. A field configuration of the string is now associated with a boundary operator of ghost number +1 picture number −1 in the worldsheet field theory of matter and ghost system. For definiteness, let us consider the boundary superstring theory of a non-BPS D-p-brane, and restrict to operators of the form O = ceV , where c is the ghost field, φ comes from the ‘bosonization’ of the βγ superghost system, (β = (∂ξ)e, γ = ηe), and V = ∑ α λ Vα is a boundary operator in the matter theory. We assume that the ghosts and matter are decoupled, and that {Vα} denotes a complete set of boundary vertex operators. The theory has superconformal invariance which is in general broken on the boundary by boundary perturbations. If we use the Hilbert space of the SCFT in the ab-