Abstract
We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module. Generalised diffeomorphisms are constructed, as well as solutions to the section constraint. Generically, additional (“ancillary”) gauge transformations are present, and we give a concrete criterion determining when they appear. A universal form of the (pseudo-)action determines the dynamics in all cases without ancillary transformations, and also for a restricted set of cases based on the adjoint representation of a finite-dimensional simple Lie group. Our construction reproduces (the internal sector of) all previously considered cases of double and exceptional field theories.
Highlights
JHEP02(2018)071 the highest weight λ are non-negative integers) [52]
We present a unified and completely general formulation of extended geometry, characterised by a Kac-Moody algebra and a highest weight coordinate module
We have presented a unified formalism for dealing with extended geometry, only relying on the choice of structure group and coordinate module
Summary
From the coordinate module R(λ) as R1 and R1, and from R2 and R2 defined in (2.12), representations Rp and Rp can be defined for all positive integers p (possibly trivial for all but finitely many p) by extending the Lie algebra g in a certain way, and decomposing the adjoint representation of the extended algebra under g. In the construction of B from the Cartan matrix B one starts with the Lie superalgebra generated by two odd elements e0, f0 and 3r+1 even elements ei, fi, hI modulo the relations [hI , eJ ] = BIJ eJ ,. The additional row and column in the Cartan matrix correspond to an additional (odd) simple root β0 in an extended weight space with metric given by (DB)IJ = (βI , βJ ), where βi = αi. B, it can be decomposed as C = p∈Z Cp, where e0 ∈ C1 and f0 ∈ C−1 and all other generators belong to C0 This is not a consistent Z-grading; C±1 is the direct sum of an even and an odd subspace, which can be identified with A±1 and B±1, respectively, by EM = −[e−1, EM ] and FM = [f−1, FM ]. By performing an “odd reflection” with respect to the outermost gray node in the Dynkin diagram of C one can obtain an equivalent Dynkin diagram, where instead the embedding of A into C is manifest [1]
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