The 1.5 Order Formalism does not Generate a Valid BRS Transformation for Supergravity

TL;DR

The paper demonstrates that the 1.5 order formalism in supergravity fails to produce a valid, nilpotent BRS transformation, indicating the need for revised first-order transformations with additional auxiliary fields.

Contribution

It shows that the 1.5 order formalism does not generate a valid BRS operator in supergravity, challenging its widespread use and proposing a way to fix it.

Findings

The 1.5 order formalism produces an invalid BRS operator.

The algebra appears to close naively but is actually flawed.

Adding auxiliary fields can restore a valid, nilpotent BRS transformation.

Abstract

The 1.5 order formalism (sometimes called a `trick') is the cornerstone of modern supergravity. In this paper, the free massive Wess--Zumino theory is used as a simple toy model to look at the BRS symmetry of the first, second and 1.5 order formalisms. This easily shows that the 1.5 order formalism is flawed for all theories. The 1.5 algebra naively appears to close. However, when it is analyzed in detail, in a simple model, where easy calculations are available, the 1.5 formalism always generates an invalid BRS operator, which is not even nilpotent. This clearly is also the case for supergravity. It follows that a revised and completed set of nilpotent first order supergravity transformations is needed to properly understand 3+1 dimensional supergravity. Such a set seems easy to write down, by simply adding two more auxiliary fields so that the spin connection becomes part of a super--YM multiplet.