N=(4,2) Chiral Supergravity in Six Dimensions and Solvable Lie Algebras

TL;DR

This paper explores the theoretical structure of an N=(4,2) chiral supergravity in six dimensions, its relation to known supergravities, and the challenges posed by gravitational anomalies, suggesting it extends to N=(4,4) supergravity.

Contribution

It identifies the algebraic and geometric foundations of N=(4,2) supergravity in D=6 and analyzes its anomaly issues, proposing an extension to N=(4,4) supergravity.

Findings

N=(4,2) supergravity has a gravitational anomaly of 4/7 that of N=(4,0).

The theory likely cannot be consistent without matter to cancel the anomaly.

The N=(4,2) theory extends to N=(4,4) supergravity due to anomaly-related constraints.

Abstract

Decomposition of the solvable Lie algebras of maximal supergravities in D=4, 5 and 6 indicates, at least at the geometrical level, the existence of an N=(4,2) chiral supergravity theory in D=6 dimensions. This theory, with 24 supercharges, reduces to the known N=6 supergravity after a toroidal compactification to D=5 and D=4. Evidence for this theory was given long ago by B. Julia. We show that this theory suffers from a gravitational anomaly equal to 4/7 of the pure N=(4,0) supergravity anomaly. However, unlike the latter, the absence of N=(4,2) matter to cancel the anomaly presumably makes this theory inconsistent. We discuss the obstruction in defining this theory in D=6, starting from an N=6 five-dimensional string model in the decompactification limit. The set of massless states necessary for the anomaly cancellation appears in this limit; as a result the N=(4,2) supergravity in D=6 is extended to N=(4,4) maximal supergravity theory.