Tensor hierarchy algebras and extended geometry. Part II. Gauge structure and dynamics

Martin Cederwall,Jakob Palmkvist

doi:10.1007/jhep02(2020)145

Martin Cederwall, Jakob Palmkvist

Open Access

https://doi.org/10.1007/jhep02(2020)145

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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∞ 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 [1] deals with the mathematical construction of the tensor hierarchy algebras.

Highlights

Diffeomorphisms do not close into themselves, but only up to “ancillary” g transformations [39, 40, 42, 47]
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
We extend the analysis and include the cases where ancillary transformations are present by using a THA S(g+) rather than B(g+)

Summary

Review of extended geometry

The input for extended geometry [2] is a structure group G (with Lie algebra g) together with a lowest weight “coordinate representation” (i.e., the representation of a generalised tangent vector) R(−λ). The structure algebra can in principle be any Kac-Moody algebra, but we will, for simplicity, take g to be laced (or at least that λi = 0 when αi is a short root) and normalise the simple roots to have length squared 2. We often use an extension of g to a Lie algebra g+, which is obtained by adjoining one node, corresponding to a simple root with length squared 2, to the Dynkin diagram of g, with lines corresponding to the coefficients of λ expressed in the basis of fundamental weights, see figure 1

Generalised diffeomorphisms and ancillary transformations

Dynamics

The tensor hierarchy algebra S

Ancillary transformations from S

Definitions and identities

Discussion