Abstract
In perturbative amplitudes in quantum field theory and string field theory, Cutkosky rule expresses the anti-hermitian part of a Feynman diagram in terms of sum over all its cut diagrams, and this in turn is used to prove unitarity of the theory. For D-instanton contribution to a string theory amplitude, the cutting rule needed for the proof of unitarity is somewhat different; we need to sum over only those cut diagrams for which all the world-sheet boundaries ending on some particular D-instanton lie on the same side of the cut. By working with the closed string effective action, obtained after integrating out the open string modes, we prove that the D-instanton amplitudes actually satisfy these cutting rules, provided the effective action is real. The violation of unitarity in the closed string sector of two dimensional string theory can be traced to the failure of this reality condition. In the critical superstring theory, multi-instanton and multi anti-instanton amplitudes satisfy the reality condition. Contribution to the amplitudes from the instanton anti-instanton sector satisfies the reality condition if we make a specific choice of integration cycle over the configuration space of string fields, whereas contribution due to the non-BPS D-instantons will need to either vanish or have an overall real normalization in order for it to give real contribution. We use Picard-Lefschetz theory to argue that these conditions are indeed satisfied in superstring theories.
Highlights
Introduction and summaryCutkosky rule in quantum field theory expresses the anti-hermitian part of a Feynman diagram in terms of sum of its cut diagrams, and is used to prove perturbative unitarity of the theory [1,2,3,4,5]
For a given Feynman diagram in this open-closed string field theory, we introduce an auxiliary diagram, in which we fuse each pair of interaction vertices of the original diagram, that share a label, into a composite vertex, and fuse all the open string propagators connected to the original vertices into the composite vertex
The main results of this paper may be summarized as follows: 1. We have described a systematic procedure for constructing a novel gauge invariant effective action obtained by integrating out the open string modes on the D-instanton
Summary
Cutkosky rule in quantum field theory expresses the anti-hermitian part of a Feynman diagram in terms of sum of its cut diagrams, and is used to prove perturbative unitarity of the theory [1,2,3,4,5]. For D-instanton induced amplitudes, one of the assumptions that fails in the proof of cutting rules is momentum conservation at the interaction vertices of open-closed string field theory relevant for computing the amplitudes As already discussed, this failure is due to the breaking of translational invariance by the D-instanton boundary condition. In the analysis of [6], the constraints on internal momenta due to momentum conservation at the interaction vertices were used to determine what kind of contour deformations are allowed in the complex plane of the internal energies This in turn was used to determine the regions of integration that contribute to the anti-hermitian part of an amplitude, leading to the cutting rules in the form described earlier. Rules to the original open closed string field theory, and show that these cutting rules precisely correspond to the ones that are needed for the unitarity of the theory
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