Cutkosky Rules and Unitarity (Violation) in D-instanton Amplitudes

TL;DR

This paper investigates the unitarity of D-instanton amplitudes in string theory, demonstrating that under certain conditions, these amplitudes satisfy modified cutting rules and unitarity constraints, with implications for the reality of the effective action.

Contribution

It establishes modified cutting rules for D-instanton amplitudes and shows their satisfaction in superstring theories using Picard-Lefschetz theory.

Findings

D-instanton amplitudes satisfy specific cutting rules if the effective action is real.

Violation of unitarity in 2D string theory is due to non-reality of the effective action.

Multi-instanton amplitudes in superstring theories satisfy the reality condition.

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.