To Örebro University

The pure spinor superfield formalism reveals that, in any dimension and with any amount of supersymmetry, one particular supermultiplet is distinguished from all others. This “canonical supermultiplet” is equipped with an additional structure that is not apparent in any component-field formalism: a (homotopy) commutative algebra structure on the space of fields. The structure is physically relevant in several ways; it is responsible for the interactions in ten-dimensional super Yang–Mills theory, as well as crucial to any first-quantised interpretation. We study the L∞ algebra structure that is Koszul dual to this commutative algebra, both in general and in numerous examples, and prove that it is equivalent to the subalgebra of the Koszul dual to functions on the space of generalised pure spinors in internal degree greater than or equal to three. In many examples, the latter is the positive part of a Borcherds–Kac–Moody superalgebra. Using this result, we can interpret the canonical multiplet as the homotopy fiber of the map from generalised pure spinor space to its derived replacement. This generalises and extends work of Movshev–Schwarz and Gálvez–Gorbounov–Shaikh–Tonks in the same spirit. We also comment on some issues with physical interpretations of the canonical multiplet, which are illustrated by an example related to the complex Cayley plane, and on possible extensions of our construction, which appear relevant in an example with symmetry type G2 × A1 .

We construct the teleparallel dynamics for extended geometry where the structure algebra is (an extension of) an untwisted affine Kac-Moody algebra. This provides a geometrisation of the Geroch symmetry appearing on dimensional reduction of a gravitational theory to two dimensions. The formalism is adapted to the underlying tensor hierarchy algebra, and will serve as a stepping stone towards the geometrisation of other infinite-dimensional, e.g. hyperbolic, symmetries.

We provide, through the framework of extended geometry, a geometrisation of the duality symmetries appearing in magical supergravities. A new ingredient is the general formulation of extended geometry with structure group of non-split real form. A simple diagrammatic rule for solving the section constraint by inspection of the Satake diagram is derived.

We formalise the teleparallel version of extended geometry (including gravity) by the introduction of a complex, the differential of which provides the linearised dynamics. The main point is the natural replacement of the two-derivative equations of motion by a differential which only contains terms of order 0 and 1 in derivatives. Second derivatives arise from homotopy transfer (elimination of fields with algebraic equations of motion). The formalism has the advantage of providing a clear consistency relation for the algebraic part of the differential, the "dualisation", which then defines the dynamics of physical fields. It remains unmodified in the interacting BV theory, and the full non-linear models arise from covariantisation. A consequence of the use of the complex is that symmetry under local rotations becomes as good as manifest, instead of arising for a specific combination of tensorial terms, for less obvious reasons. We illustrate with a derivation of teleparallel Ehlers geometry, where the extended coordinate module is the adjoint module of a finite-dimensional simple Lie group.

For any decomposition of a Lie superalgebra G into a direct sum G=H circle plus E of a subalgebra H and a subspace E, without any further resctrictions on H and E, we construct a nonlinear realization of G on E. The result generalizes a theorem by Kantor from Lie algebras to Lie superalgebras. When G is a differential graded Lie algebra, we show that it gives a construction of an associated L-infinity-algebra.

Extended geometry is based on an underlying tensor hierarchy algebra. We extend the previously considered L-infinity structure of the local symmetries (the diffeomorphisms and their reducibility) to incorporate physical fields, field strengths and Bianchi identities, and identify these as elements of the tensor hierarchy algebra. The field strengths arise as generalised torsion, so the naturally occurring complex in the L-infinity algebra is

... <- torsion BI's <- torsion <- vielbein <- diffeomorphism parameters <- ...

In order to obtain equations of motion, which are not in this complex, (pseudo-)actions, quadratic in torsion, are given for a large class of models. This requires considering the dual complex. We show how local invariance under the compact subgroup locally defined by a generalised metric arises as a "dual gauge symmetry" associated with a certain torsion Bianchi identity, generalising Lorentz invariance in the teleparallel formulation of gravity. The analysis is performed for a large class of finite-dimensional structure groups, with E-5 as a detailed example. The continuation to infinite-dimensional cases is discussed.

Tensor hierarchy algebras are infinite-dimensional generalisations of Cartantype Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental role they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac-Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac-Moody algebra by a Virasoro derivation L-1. A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.