Abstract
Extended geometry provides a unified framework for double geometry, exceptional geometry, etc., i.e., for the geometrisations of the string theory and M-theory dualities. In this talk, we will explain the structure of gauge transformations (generalised diffeomorphisms) in these models. They are generically infinitely reducible, and arise as derived brackets from an underlying Borcherds superalgebra or tensor hierarchy algebra. The infinite reducibility gives rise to an L∞ structure, the brackets of which have universal expressions in terms of the underlying superalgebra.
Highlights
The motivation for the investigation lies in the dualities appearing in string/M-theory, and the possibility to “geometrise” them
When M-theory is compactified on an n-torus, the U-duality group is En(n)(Z)
It turns out that it can be geometrised, so that the duality group derives from an “extended geometry” like the mapping class group from geometry
Summary
The motivation for the investigation lies in the dualities appearing in string/M-theory, and the possibility to “geometrise” them. Given a Kac–Moody algebra g and a lowest weight coordinate representation R(−λ) (we use conventions where extended tangent space vectors are in lowest weight modules and cotangent vectors in highest weight modules; R(−λ) denotes the lowest weight representation with lowest weight −λ, for dominant λ), they are expressed in terms of a generalised Lie derivative as δU φ = LU φ, for any field φ transforming covariantly, where
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