On non-Abelian T-duality for non-semisimple groups

TL;DR

This paper analyzes non-Abelian T-duality for non-semisimple groups, showing that the resulting sigma-models are solutions to generalized supergravity equations when accounting for anomalies and the structure constants.

Contribution

It demonstrates that non-Abelian T-duals of non-semisimple groups satisfy generalized supergravity equations by incorporating anomaly variations and identifying the Killing vector with the trace of structure constants.

Findings

Non-Abelian T-duals with non-semisimple groups solve generalized supergravity.

Accounting for anomalies is crucial for correct equations of motion.

Illustrations provided with Bianchi spacetimes.

Abstract

We revisit non-Abelian T-duality for non-semisimple groups, where it is well-known that a mixed gravitational-gauge anomaly leads to $\sigma$-models that are scale, but not Weyl-invariant. Taking into account the variation of a non-local anomalous term in the T-dual $\sigma$-model of Elitzer, Giveon, Rabinovici, Schwimmer \& Veneziano, we show that the equations of motion of generalized supergravity follow from the $\sigma$-model once the Killing vector $I$ is identified with the trace of the structure constants. As a result, non-Abelian T-duals with respect to non-semisimple groups are solutions to generalized supergravity. We illustrate our findings with Bianchi spacetimes.