Abstract
We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin–Vilkovisky framework, or equivalently, an L_infty algebra. The L_infty brackets are given as derived brackets constructed using an underlying Borcherds superalgebra {mathscr {B}}({{mathfrak {g}}}_{r+1}), which is a double extension of the structure algebra {{mathfrak {g}}}_r. The construction includes a set of “ancillary” ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra.
Highlights
The ghosts in exceptional field theory [1], and generally in extended field theory with an extended structure algebra gr [2], are known to fall into B+(gr ), the positive levels of a Borcherds superalgebra B(gr ) [3,4]
It was shown in Ref. [3] how generalised diffeomorphisms for Er have a natural formulation in terms of the structure constants of the Borcherds superalgebra B(Er+1)
The gauge structure of extended geometry will be described as an L∞ algebra, governed by an underlying Borcherds superalgebra B(gr+1)
Summary
The ghosts in exceptional field theory [1], and generally in extended field theory with an extended structure algebra gr [2], are known to fall into B+(gr ), the positive levels of a Borcherds superalgebra B(gr ) [3,4]. We will not treat the situation where ancillary transformations arise in the commutator of two generalised diffeomorphisms, but we will extend the concept of ancillary ghosts to higher ghost number It will become clear from the structure of the doubly extended Borcherds superalgebra B(gr+1) why and when such extra restricted ghosts appear, and what their precise connection to e.g. the loss of covariance is. We note that the gauge symmetries of exceptional generalised geometry have been dealt with in the L∞ algebra framework earlier [42] This was done in terms of a formalism where ghosts are not collected into modules of Er , but consist of the diffeomorphism parameter together with forms for the ghosts of the tensor gauge transformations (i.e., in generalised geometry, not in extended geometry). We conclude with a discussion, with focus on the extension of the present construction to situations where ancillary transformations are present already in the commutator of two generalised diffeomorphisms
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