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Michel Rausch de Traubenberg

Publications and source records attributed to Michel Rausch de Traubenberg.

At least 19 recordsLinked to original sources
arXiv2026

On a new class of high-corank Kac-Moody algebras

We present recursive constructions of several families of generalized Cartan matrices associated with Kac-Moody algebras, whose sizes and coranks grow exponentially. The constructions are encoded by connected multigraphs and by block-doubling operations on their associated symmetric generalized Cartan matrices. Equivalently, the corank problem is translated into a spectral graph-theoretic problem: the corank of $2\mathrm{Id}-\operatorname{Adj}(G)$ is the multiplicity of the adjacency eigenvalue $2$. We give two explicit recursive families, compute their spectra and coranks, and emphasize the difference between absolute exponential growth and relative asymptotic density. The resulting algebras are typically indefinite and singular of corank larger than one, and therefore contain several independent central directions and several isotropic radical directions in the root lattice. We also discuss alternative constructions and possible applications to the algebraic structures appearing in gravity, supergravity, string/M-theory and related generalized symmetry problems.

math.RA↗
arXiv2025

New perspectives in Kac-Moody algebras associated to higher dimensional manifolds

In this review, we present a general framework for the construction of Kac-Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on the circle $\mathbb{S}^{1}$, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases and Killing vectors, we present the construction of generalized current algebras $\mathfrak{g}(\mathcal{M})$, their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and illustrate the framework through explicit realizations on $SU(2)$, $SU(2)/U(1)$, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications.

hep-th↗
arXiv2025

Kac-Moody Algebras on Soft Group Manifolds

Within the so-called group geometric approach to (super)gravity and (super)string theories, any compact Lie group manifold $G_{c}$ can be smoothly deformed into a group manifold $G_{c}^{μ}$ (locally diffeomorphic to $G_{c}$ itself), which is `soft', namely, based on a non-left-invariant, intrinsic one-form Vielbein $μ$, which violates the Maurer-Cartan equations and consequently has a non-vanishing associated curvature two-form. Within the framework based on the above deformation (`softening'), we show how to construct an infinite-dimensional (infinite-rank), generalized Kac-Moody (KM) algebra associated to $G_{c}^{μ}$, starting from the generalized KM algebras associated to $G_{c}$. As an application, we consider KM algebras associated to deformed manifolds such as the `soft' circle, the `soft' two-sphere and the `soft' three-sphere. While the generalized KM algebra associated to the deformed circle is trivially isomorphic to its undeformed analogue, and hence not new, the `softening' of the two- and three- sphere includes squashed manifolds (and in particular, the so-called Berger three-sphere) and yields to non-trivial results.

hep-th↗
arXiv2024

Kac-Moody and Virasoro algebras on the two-sphere and the two-torus

We particularise the construction of generalised Kac-Moody algebras associated to compact real manifolds to the case of the two-torus $\mathbb T_2$ and the two-sphere ${\mathbb S}^2$. It is shown that these algebras, as well as a Virasoro algebra associated to the two-torus and the two-sphere, can also be derived considering the usual Kac-Moody and Virasoro algebras.

math-ph↗
arXiv2024

An infinite-rank Lie algebra associated to SL$(2,\mathbb R)$ and SL$(2,\mathbb R)/U(1)$

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$ SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This construction is achieved through two equivalent ways: by means of the Plancherel Theorem and by identifying a Hilbert basis within $L^2(\mathcal{M})$. We analyse the existence of central extensions and identify those in duality with Hermitean operators on $\cal M$. By inspecting the Clebsch-Gordan coefficients of $\mathfrak{sl}(2,\mathbb{R})$, we derive the Lie brackets characterising the corresponding generalised Kac-Moody algebras. The root structure of these algebras is identified, and it is shown that an infinite number of simultaneously commuting operators can be defined. Furthermore, we briefly touch upon applications of these algebras within the realm of supergravity, particularly in scenarios where the scalar fields coordinatize the non-compact manifold $\text{SL}(2,\mathbb{R})/U(1)$.

math-ph↗
arXiv2023

Vertex operator for generalised Kac--Moody algebras associated to the two-sphere and the two-torus

We pursue our study of generalised Kac-Moody and Virasoro algebras defined on compact homogeneous manifolds. Extending the well-known Vertex operator in the case of the two-torus or the two-sphere, we obtain explicit bosonic realisations of the semi-direct product of the extension of Kac-Moody and Virasoro algebras on $\mathbb S^1 \times \mathbb S^1$ and $\mathbb S^2$, respectively. As for the fermionic realisation previously constructed, in order to have well defined algebras, we introduce, beyond the usual normal ordering prescription, a regulator and regularise infinite sums by means of the Riemann $ζ$-function.

math-ph↗
arXiv2023

Fermion realisations of generalised Kac--Moody and Virasoro algebras associated to the two-sphere and the two-torus

Using the notion of extension of Kac-Moody algebras for higher dimensional compact manifolds recently introduced in [1], we show that for the two-torus $\mathbb S^1 \times \mathbb S^1$ and the two-sphere $\mathbb S^2$, these extensions, as well as extensions of the Virasoro algebra can be obtained naturally from the usual Kac-Moody and Virasoro algebras. Explicit fermionic realisations are proposed. In order to have well defined generators, beyond the usual normal ordering prescription, we introduce a regulator and regularise infinite sums by means of Riemann $ζ-$function.

math-ph↗
arXiv2023

Group invariants for Feynman diagrams

It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved. Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics, here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.

math.RT↗
arXiv2022

Supergravity: Application in Particle Physics

We provide a pedagogical introduction to $N=1$ supergravity/supersymmetry in relation to particle physics. The various steps in the construction of a generic $N=1$ supergravity model are briefly described, and we focus on its low energy supersymmetric limit. The conditions for supersymmetry and supergravity breaking are investigated, and realistic mechanisms suitable for particle physics identified. We then study the model-building aspects of `softly-broken' supersymmetric extensions of the Standard Model and discuss several of their phenomenological features.

hep-th↗
arXiv2022

Commutation relations of $\mathfrak g\_2$ and the incidence geometry of the Fano plane

We continue our study and classification of structures on the Fano plane ${\cal F}$ and its dual ${\cal F}^\ast$ involved in the construction of octonions and the Lie algebra $\mathfrak g_2 (\mathbb F)$ over a field $\mathbb F$. These are a "composition factor": ${\cal F}\times {\cal F} \to\{-1, 1\}$, inducing an octonion multiplication, and a function $δ^\ast : Aut({\cal F}) \times {\cal F}^\ast \to \{-1, 1\}$ such that $g \in Aut({\cal F})$ can be lifted to an automorphism of the octonions iff $δ^\ast(g, \cdot)$ is the Radon transform of a function on ${\cal F}$. We lift the action of $Aut({\cal F})$ on ${\cal F}$ to the action of a non-trivial eight-fold covering $Aut({\cal F})$ on a twofold covering $\hat {\cal F}$ of ${\cal F}$ contained in the octonions. This extends tautologically to an action on the octonions by automorphism. Finally, we associate to incident point-line pairs a generating set of $\mathfrak g_2 (\mathbb F)$ and express brackets in terms of the incidence geometry of ${\cal F}$ and $ε$.

math.CO↗
arXiv2022

An overview of generalised Kac-Moody algebras on compact real manifolds

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold $\mathcal{M}$ to a finite-dimensional Lie group, by means of complete orthonormal bases for a Hermitian inner product on the manifold and a Fourier expansion. The Peter--Weyl theorem for the case of manifolds related to compact Lie groups and coset spaces is discussed, and appropriate Hilbert bases for the space $L^{2}(\mathcal{M})$ of square-integrable functions are constructed. It is shown that such bases are characterised by the representation theory of the compact Lie group, from which a complete set of labelling operator is obtained. The existence of central extensions of generalised Kac-Moody algebras is analysed using a duality property of Hermitian operators on the manifold, and the corresponding root systems are constructed. Several applications of physically relevant compact groups and coset spaces are discussed.

math-ph↗
arXiv2022

Incidence geometry of the Fano plane and Freudenthal's ansatz for the construction of (split) octonions

In this article we consider structures on a Fano plane ${\cal F}$ which allow a generalisation of Freudenthal's construction of a norm and a bilinear multiplication law on an eight-dimensional vector space ${\mathbb O_{\cal F}}$ canonically associated to ${\cal F}$. We first determine necessary and sufficient conditions in terms of the incidence geometry of ${\cal F}$ for these structures to give rise to division composition algebras, and classify the corresponding structures using a logarithmic version of the multiplication. We then show how these results can be used to deduce analogous results in the split composition algebra case.

math.CO↗
arXiv2021

A pedagogical discussion of N = 1 four-dimensional supergravity in superspace

A short introduction to N = 1 supergravity in four dimensions in the superspace approach is given emphasising on all steps to obtain the final Lagrangian. In particular starting from geometrical principles and the introduction of superfields in curved superspace, the action coupling matter and gauge fields to supergravity is derived. This review is based on the book -A supergravity Primer: From geometrical principles to the Final Lagrangian- and on several lectures given at the doctoral school of Strasbourg.

hep-th↗
arXiv2020

Multiloop QED in the Euler-Heisenberg approach

I summarize what is known about the Euler-Heisenberg Lagrangian and its multiloop corrections for scalar and spinor QED, in various types of constant fields, and in various dimensions. Particular attention is given to the asymptotic properties of the weak-field expansion of the Lagrangian, which via Borel summation is related to Schwinger pair-creation, and the status of the "exponentiation conjecture" for the imaginary part of the Euler-Heisenberg to all loop orders.

hep-ph↗
arXiv2019

Dihedral Invariant Polynomials in the effective Lagrangian of QED

We present a new group-theoretical technique to calculate weak field expansions for some Feynman diagrams using invariant polynomials of the dihedral group. In particular we show results obtained for the first coefficients of the three loop effective lagrangian of 1+1 QED in an external constant field, where the dihedral symmetry appears. Our results suggest that a closed form involving rational numbers and the Riemann zeta function might exist for these coefficients.

hep-th↗
arXiv2018

Three-loop Euler-Heisenberg Lagrangian in 1+1 QED, part 1: single fermion-loop part

We study the three-loop Euler-Heisenberg Lagrangian in spinor quantum electrodynamics in 1+1 dimensions. In this first part we calculate the one-fermion-loop contribution, applying both standard Feynman diagrams and the worldline formalism which leads to two different representations in terms of fourfold Schwinger-parameter integrals. Unlike the diagram calculation, the worldline approach allows one to combine the planar and the non-planar contributions to the Lagrangian. Our main interest is in the asymptotic behaviour of the weak-field expansion coefficients of this Lagrangian, for which a non-perturbative prediction has been obtained in previous work using worldline instantons and Borel analysis. We develop algorithms for the calculation of the weak-field expansions coefficients that, in principle, allow their calculation to arbitrary order. Here for the non-planar contribution we make essential use of the polynomial invariants of the dihedral group D4 in Schwinger parameter space to keep the expressions manageable. As expected on general grounds, the coefficients are of the form r1+r2*zeta(3) with rational numbers r1, r2. We compute the first two coefficients analytically, and four more by numerical integration.

hep-th↗