The tensor hierarchy algebra

TL;DR

This paper introduces an infinite-dimensional Lie superalgebra extending U-duality in maximal supergravity, revealing its structure, relation to Borcherds-Kac-Moody superalgebras, and implications for gauge deformations and supersymmetry constraints.

Contribution

It constructs a new tensor hierarchy algebra that extends known algebraic structures and clarifies their relation to supergravity gauge deformations and dualities.

Findings

The algebra extends U-duality Lie algebra for D=3 to 7.

It contains the tensor hierarchy representations at positive levels.

It relates negative levels to supersymmetry and closure constraints.

Abstract

We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for D between 3 and 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D-2-p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.