Tensor hierarchy algebras and extended geometry II: Gauge structure and dynamics

TL;DR

This paper explores the gauge structure of extended geometry using tensor hierarchy algebras, revealing their role in describing gauge transformations and dynamics, especially when ancillary transformations are present.

Contribution

It introduces tensor hierarchy algebras as the algebraic framework for extended geometry gauge structures, extending previous work and analyzing their implications for finite-dimensional structure groups.

Findings

Tensor hierarchy algebras replace Borcherds superalgebras in describing gauge structures.

Partial analysis of gauge structure via $L_$ algebra for finite-dimensional groups.

Invariant pseudo-action formulated for certain extended geometries.

Abstract

The recent investigation of the gauge structure of extended geometry is generalised to situations when ancillary transformations appear in the commutator of two generalised diffeomorphisms. The relevant underlying algebraic structure turns out to be a tensor hierarchy algebra rather than a Borcherds superalgebra. This tensor hierarchy algebra is a non-contragredient superalgebra, generically infinite-dimensional, which is a double extension of the structure algebra of the extended geometry. We use it to perform a (partial) analysis of the gauge structure in terms of an $L_\infty$ algebra for extended geometries based on finite-dimensional structure groups. An invariant pseudo-action is also given in these cases. We comment on the continuation to infinite-dimensional structure groups. An accompanying paper deals with the mathematical construction of the tensor hierarchy algebras.