BV actions for extended geometry

TL;DR

The paper develops a BV-action framework for extended geometry by leveraging tensor hierarchy algebras to organize fields, ghosts, and Bianchi identities into a graded structure. Generalized diffeomorphisms arise as derived brackets and are controlled by a section constraint that locally localizes extended coordinates. The central algebraic input is the map sigma, whose inverse contracts torsion in a teleparallel-like action, enabling a first-order BV action via a two-line complex, with nilpotency fixing sigma and covariantisation yielding non-linear dynamics. Ancillary ghosts and unfolding to larger gauge algebras are discussed, highlighting potential infinite towers for certain symmetry algebras and connections to affine or E11-type structures for achieving a more geometric formulation of extended geometry.

Abstract

I review the construction of actions for extended geometry from the grading of an underlying tensor hierarchy algebra, which provides the full set of Batalin-Vilkovisky fields. The dynamics is neatly encoded in a complex. This talk, presented at the Corfu Summer Institute 2024, is mainly based on joint work with J. Palmkvist, in particular ref. [1].

Figures (3)

• Figure 1: Dynkin diagram for $E_n$.
• Figure 2: Dynkin diagram for $A_1^+$.
• Figure 3: Dynkin diagram for $A_1^{++}$.