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Martin Cederwall

Publications and source records attributed to Martin Cederwall.

At least 19 recordsLinked to original sources
arXiv2025

Classical BV cohomology of the $N=1$ spinning particle

We show that the classical Batalin--Vilkovisky cohomology at negative ghost number of the spinning particle, observed in ref. arXiv:1511.02135, is removed by a Koszul--Tate resolution involving saturation of Grassmann odd variables. The model thus satisfies the axioms of Felder and Kazhdan. The AKSZ formulation of the resolved model is described. We reveal partial information on the resolution of the constrained phase space, which involves resolving the parity-shifted tangent sheaf of the light-c\^one. Specialising to dimension one, we describe the full resolution.

math-ph↗
arXiv2025

Non-associative structures in extended geometry

We consider a generalisation of vector fields on a vector space, where the vector space is generalised to a highest-weight module over a Kac-Moody algebra. The generalised vector field is an element in a non-associative superalgebra defined by the module and the Kac-Moody algebra. Also the Lie derivative of a vector field parameterised by another is generalised and expressed in a simple way in terms of this superalgebra. It reproduces the generalised Lie derivative in the general framework of extended geometry, which in special cases reduces to the one in exceptional field theory, unifying diffeomorphisms with gauge transformations in supergravity theories.

hep-th↗
arXiv2025

Some remarks on invariants

The demand to know the structure of functionally independent invariants of tensor fields arises in many problems of theoretical and mathematical physics, for instance for the construction of interacting higher-order tensor field actions. In mathematical terms the problem can be formulated as follows. Given a semi-simple finite-dimensional Lie algebra $\mathfrak g$ and a $\mathfrak g$-module $V$, one may ask about the structure of the sub-ring of $\mathfrak g$-invariants inside the ring freely generated by the module. We point out how some information about the ring of invariants may be obtained by studying an extended Lie algebra. Numerous examples are given, with particular focus on the difficult problem of classifying invariants of a self-dual 5-form in 10 dimensions.

hep-th↗
arXiv2025

The octonionic phase space Hopf map

The octonionic Hopf map, expressing $S^{15}$ as an $S^7$ bundle over $S^8$, appears in the twistor transform in 10 dimensions, $S^8$ playing the r\^ole of the celestial sphere. A symplectic lift to twistor space manifests $Spin(2,10)$ symmetry. The 25-dimensional spinor orbit of $Spin(2,10)$ is an $S^7$ bundle over the phase space of a massless particle.

hep-th↗
arXiv2025

BV actions for extended geometry

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].

hep-th↗
arXiv2025

Gradient structures from extensions of over-extended Kac-Moody algebras

Over-extended Kac-Moody algebras contain so-called gradient structures - a gl(d)-covariant level decomposition of the algebra contains strings of modules at different levels that can be interpreted as spatial gradients. We present an algebraic origin for this phenomenon, based on the recently introduced Lie algebra extension of an over-extended Kac-Moody algebra by its fundamental module, appearing in tensor hierarchy algebra super-extensions of over-extended Kac-Moody algebras. The extensions are described in terms of Lie algebra cohomology, vanishing for finite-dimensional simple Lie algebras, but non-vanishing in relevant infinite-dimensional cases. The extension is described in a few different gradings, where it is given a covariant description with respect to different subalgebras. We expect the results to be important for the connection between extended geometry and cosmological billiards.

hep-th↗
arXiv2024

Curvature of an exotic 7-sphere

We study the geometry of the Gromoll-Meyer sphere, one of Milnor's exotic $7$-spheres. We focus on a Kaluza-Klein Ansatz, with a round $S^4$ as base space, unit $S^3$ as fibre, and $k=1,2$ $SU(2)$ instantons as gauge fields, where all quantities admit an elegant description in quaternionic language. The metric's moduli space coincides with the $k=1,2$ instantons' moduli space quotiented by the isometry of the base, plus an additional $\mathbb{R}^+$ factor corresponding to the radius of the base, $r$. We identify a "center" of the $k=2$ instanton moduli space with enhanced symmetry. This $k=2$ solution is used together with the maximally symmetric $k=1$ solution to obtain a metric of maximal isometry, $SO(3)\times O(2)$, and to explicitly compute its Ricci tensor. This allows us to put a bound on $r$ to ensure positive Ricci curvature, which implies various energy conditions for an $8$-dimensional static space-time. This construction then enables a concrete examination of the properties of the sectional curvature.

hep-th↗
arXiv2024

Teleparallel Geroch geometry

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.

hep-th↗
arXiv2023

Cartanification of contragredient Lie superalgebras

Let $B$ be a $\mathbb{Z}$-graded Lie superalgebra equipped with an invariant $\mathbb{Z}_2$-symmetric homogeneous bilinear form and containing a grading element. Its local part (in the terminology of Kac) $B_{-1} \oplus B_{0} \oplus B_{1}$ gives rise to another $\mathbb{Z}$-graded Lie superalgebra, recently constructed in arXiv:2207.12417, that we here denote $B^W$ and call the cartanification of $B^W$, since it is of Cartan type in the cases where it happens to finite-dimensional. In cases where $B$ is given by a generalised Cartan matrix, we compare $B^W$ to the tensor hierarchy algebra $W$ constructed from the same generalised Cartan matrix by a modification of the generators and relations. We generalise this construction and give conditions under which $W$ and $B^W$ are isomorphic, proving a conjecture in arXiv:2207.12417. We expect that the algebras with restricted associativity underlying the cartanifications will be useful in applications of tensor hierarchy algebras to the field of extended geometry in physics.

math.RT↗
arXiv2023

Canonical supermultiplets and their Koszul duals

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 $L_\infty$ 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 G\'alvez-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 $G_2 \times A_1$.

hep-th↗
arXiv2023

The teleparallel complex

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.

hep-th↗
arXiv2023

Extended geometry of magical supergravities

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.

hep-th↗
arXiv2022

A minimal b ghost

The $b$ ghost, or $b$ operator, used for fixing Siegel gauge in the pure spinor superfield formalism, is a composite operator of negative ghost number, satisfying $\{q,b\}=\square$, where $q$ is the pure spinor differential (BRST operator). It is traditionally constructed using non-minimal variables. However, since all cohomology has minimal representatives, it seems likely that there should be versions of physically meaningful operators, also with negative ghost number, using only minimal variables. The purpose of this letter is to demonstrate that this statement holds by providing a concrete construction in $D=10$ super-Yang-Mills theory, and to argue that it is a general feature in the pure spinor superfield formalism.

hep-th↗
arXiv2022

Pure spinors in classical and quantum supergravity

This is an overview of the method of pure spinor superfields, written for "Handbook of Quantum Gravity", eds. C. Bambi, L. Modesto and I. Shapiro. The main focus is on the use of the formalism in maximal supergravity on a flat background. The basics of pure spinor superfields, and their relation to standard superspace, is reviewed. The pure spinor superstring model of Berkovits is briefly discussed. Consequences for divergence properties of loop diagrams in maximal supergravity are restated. Some final remarks are made concerning desirable development of the theoretical framework.

hep-th↗
arXiv2022

Tensor hierarchy algebras and restricted associativity

We study local algebras, which are structures similar to $\mathbb{Z}$-graded algebras concentrated in degrees $-1,0,1$, but without a product defined for pairs of elements at the same degree $\pm1$. To any triple consisting of a Kac-Moody algebra $\mathfrak{g}$ with an invertible and symmetrisable Cartan matrix, a dominant integral weight of $\mathfrak{g}$ and an invariant symmetric bilinear form on $\mathfrak{g}$, we associate a local algebra satisfying a restricted version of associativity. From it, we derive a local Lie superalgebra by a commutator construction. Under certain conditions, we identify generators which we show satisfy the relations of the tensor hierarchy algebra $W$ previously defined from the same data. The result suggests that an underlying structure satisfying such a restricted associativity may be useful in applications of tensor hierarchy algebras to extended geometry.

math.RA↗
arXiv2021

Teleparallelism in the algebraic approach to extended geometry

Extended geometry is based on an underlying tensor hierarchy algebra. We extend the previously considered $L_\infty$ 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 $L_\infty$ algebra is $\ldots\leftarrow$ torsion BI's $\leftarrow$ torsion $\leftarrow$ vielbein $\leftarrow$ diffeomorphism parameters $\leftarrow\ldots$ 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.

hep-th↗
arXiv2021

SL(5) supersymmetry

We consider supersymmetry in five dimensions, where the fermionic parameters are a 2-form under SL(5). Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite-dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra ${\scr B}(E_4)$, the tensor hierarchy algebra $S(E_4)$ and the exceptional superalgebra $E(5,10)$. A theorem relating ${\scr B}(E_4)$ and $E(5,10)$ to all levels is given.

math.RT↗