Abstract
We consider the effective action of the Heterotic Superstring to first order in α′ 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 in an arbitrary spinorial basis and corresponding arbitrary gamma matrices. 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 off-shell relations existing between 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.
Highlights
The construction and study of the classical solutions of a theory always provides a great deal of information about its properties and predictions
We consider the effective action of the Heterotic Superstring to first order in α 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 in an arbitrary spinorial basis and corresponding arbitrary gamma matrices
We find the off-shell relations existing between the bosonic equations of motion evaluated on supersymmetric field configurations using the Killing Spinor Identities instead of the integrability conditions of the Killing Spinor Equations as it is common in the literature
Summary
The construction and study of the classical solutions of a theory always provides a great deal of information about its properties and predictions. We are not including the Bianchi identity of the Yang-Mills gauge field strength because in order to write it one needs to know the gauge connection, which completely determines and trivializes the Bianchi identity After considering all these difficulties and the solutions found for some of them, it is clear that, being conservative, we will have to content ourselves with carrying our program to first order in α only: we will determine necessary and sufficient conditions for unbroken supersymmetry valid to O(α ) and relations between equations of motion evaluated on supersymmetric configurations valid to the same order in α. We explain how the equations of motion are obtained and simplified at first order in α and the derivation of the KSIs involving the Bianchi identity of the Kalb-Ramond 3-form field strength.
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