On the supersymmetric solutions of the Heterotic Superstring effective action

TL;DR

This paper derives the conditions for supersymmetric solutions in the Heterotic Superstring effective action, using spinor bilinear methods and Killing spinor identities, providing a detailed algebraic and geometric analysis.

Contribution

It introduces a systematic derivation of supersymmetric conditions employing spinor bilinear algebra and Killing spinor identities, including the Bianchi identity, for the first-order heterotic superstring effective action.

Findings

Derived necessary and sufficient conditions for supersymmetry.

Computed the algebra of spinor bilinears and Spin(7) structures.

Connected equations of motion with Killing spinor identities.

Abstract

We consider the effective action of the Heterotic Superstring to first order in alpha prime and derive the necessary and sufficient conditions that a field configuration has to satisfy in order to admit at least one Killing spinor using the spinor bilinear method and making minimal coordinate and frame choices. As a previous step in this derivation, we compute the complete spinor bilinear algebra using the Fierz identities, obtaining as a by-product the algebra satisfied by the Spin(7) structure contained in the bilinears in an arbitrary basis. We find the relations existing between the left-hand-sides of the bosonic equations of motion evaluated on supersymmetric field configurations using the Killing Spinor Identities instead of the (far more complicated) integrability conditions of the Killing Spinor Equations as it is common in the literature. We show how to include the Kalb-Ramond's Bianchi identity in the Killing Spinor Identities.