Supersymmetry Restoration in Superstring Perturbation Theory

TL;DR

This paper proves that supersymmetry can be restored in superstring perturbation theory by analyzing Ward identities, mass relations, and vacuum shifts, with explicit checks in heterotic string models.

Contribution

It derives Ward identities for supersymmetry in superstring perturbation theory and demonstrates supersymmetry restoration in shifted vacua, including explicit mass equality proofs.

Findings

Ward identities confirm supersymmetry in shifted vacua

Bosonic and fermionic masses are equal at one loop in the shifted vacuum

Two-loop dilaton tadpole appears in the perturbative vacuum

Abstract

Superstring perturbation theory based on the 1PI effective theory approach has been useful for addressing the problem of mass renormalization and vacuum shift. We derive Ward identities associated with space-time supersymmetry transformation in this approach. This leads to a proof of the equality of renormalized masses of bosons and fermions and identities relating fermionic amplitudes to bosonic amplitudes after taking into account the effect of mass renormalization. This also relates unbroken supersymmetry to a given order in perturbation theory to absence of tadpoles of massless scalars to higher order. The results are valid at the perturbative vacuum as well as in the shifted vacuum when the latter describes the correct ground state of the theory. We apply this to SO(32) heterotic string theory on Calabi-Yau 3-folds where a one loop Fayet-Iliopoulos term apparently breaks supersymmetry at one loop, but analysis of the low energy effective field theory indicates that there is a nearby vacuum where supersymmetry is restored. We explicitly prove that the perturbative amplitudes of this theory around the shifted vacuum indeed satisfy the Ward identities associated with unbroken supersymmetry. We also test the general arguments by explicitly verifying the equality of bosonic and fermionic masses at one loop order in the shifted vacuum, and the appearance of two loop dilaton tadpole in the perturbative vacuum where supersymmetry is expected to be broken.