arXiv subjects

Paul Schwahn

Publications and source records attributed to Paul Schwahn.

12 recordsLinked to original sources

arXiv2026

Einstein metrics, their moduli spaces and stability

This survey deals with two closely connected topics: first, the stability of Einstein metrics under the Einstein-Hilbert functional, and second, their deformation theory and the study of the moduli space of Einstein metrics on a compact manifold. To first order, both problems reduce to studying the spectrum and eigentensors of the Lichnerowicz Laplacian. We give an introduction to the classical theory and survey recent results and advances.

math.DG↗

arXiv2026

The Standard Model Gauge Group from the Exceptional Jordan Algebra

We construct the Standard Model gauge group using the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$ and its automorphism group $\text{F}_4$. The group $\text{F}_4$ acts on pairs of Jordan subalgebras $X \subset B \subset \mathfrak{h}_3(\mathbb{O})$ with $X \cong \mathfrak{h}_2(\mathbb{C})$ and $B \cong \mathfrak{h}_3(\mathbb{C})$, and for any such pair the stabilizer of $X$ intersected with the identity component of the stabilizer of $B$ is isomorphic to the Standard Model gauge group. Since $\mathfrak{h}_2(\mathbb{C})$ is the Jordan algebra of observables of a qubit and $\mathfrak{h}_3(\mathbb{C})$ is the Jordan algebra of observables of a qutrit, we could say $\mathfrak{h}_3(\mathbb{O})$ is the Jordan algebra of observables of an 'octonionic qutrit'. In this language our result says roughly that the Standard Model gauge group is the group of symmetries of an octonionic qutrit that restrict to act as unitary operators on an ordinary qutrit and, within that, a qubit.

math-ph↗

arXiv2026

Geometries with parallel, skew-symmetric and closed torsion

We study Riemannian manifolds carrying a metric connection with parallel, skew-symmetric and closed torsion, which we call in short PSCT manifolds. We prove that PSCT manifolds always locally split into a product of well-understood factors, allowing a complete local classification. Further, we investigate various $G$-structures of PSCT type, with a focus on almost Hermitian structures and their possible Gray--Hervella classes.

math.DG↗

arXiv2026

Balanced subsets in root systems

Balanced and well-balanced subsets of the set of positive roots of compact Lie algebras arise naturally in problems related to Hermitian and spin geometry. In this paper we compute the maximal and minimal size of well-balanced subsets in all simple root systems.

math.RT↗

arXiv2025

Formalizing a classification theorem for low-dimensional solvable Lie algebras in Lean

We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such ubiquitous in geometry and physics. Our work involves explicit calculations on the level of the underlying vector spaces and provides a use case for the linear algebra and Lie theory routines in Lean's mathematical library mathlib. Along the way we formalize results about Lie algebras, define the semidirect product within this setting and add API for bases of vector spaces. In a wider context, this project aims to provide a complete mechanization of a classification theorem, covering both the statement and its full formal proof, and contribute to the development and broader adoption of such results in formalized mathematics.

cs.LO↗

arXiv2025

Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras

We study the deformability of the symmetric Einstein metrics on the spaces $\mathrm{SU}(n)/\mathrm{SO}(n)$ and $\mathrm{SU}(2n)/\mathrm{Sp}(n)$, thereby concluding the problem to second order for all irreducible symmetric spaces. The obstruction integrals are calculated from invariant polynomials on certain Lie algebra representations. To aid the computation, we develop so-called sandwich operators for compact Lie algebras and relate them to quadratic Casimir operators. We also explain the source of the infinitesimal Einstein deformations on irreducible symmetric spaces, except for the complex Grassmannians, by exploring their relation to simple Jordan algebras. As an application we prove the nonlinear instability of most of the infinitesimally deformable irreducible compact symmetric spaces.

math.DG↗

arXiv2024

Submersion constructions for geometries with parallel skew torsion

In the absence of a de Rham decomposition theorem for geometries with torsion, we develop and unify ways to view a geometry with parallel skew torsion as the total space of a locally defined, not necessarily unique Riemannian submersion with totally geodesic fibers. We complete and extend the Cleyton-Swann classification of irreducible such geometries and characterize the cases where the stabilizer of the torsion is larger than the holonomy. As a byproduct, we obtain structure results on Gray manifolds, nearly parallel G$_2$-manifolds and Sasaki manifolds with reducible holonomy.

math.DG↗

arXiv2024

The Lichnerowicz Laplacian on normal homogeneous spaces

We give a new formula for the Lichnerowicz Laplacian on normal homogeneous spaces in terms of Casimir operators. We derive some practical estimates and apply them to the known list of non-symmetric, compact, simply connected homogeneous spaces $G/H$ with $G$ simple whose standard metric is Einstein. This yields many new examples of Einstein metrics which are stable in the Einstein-Hilbert sense, which have long been lacking in the positive scalar curvature setting.

math.DG↗

arXiv2024

On the rigidity of the complex Grassmannians

We study the integrability to second order of the infinitesimal Einstein deformations of the symmetric metric $g$ on the complex Grassmannian of $k$-planes inside $\mathbb{C}^n$. By showing the nonvanishing of Koiso's obstruction polynomial, we characterize the infinitesimal deformations that are integrable to second order as an explicit variety inside $\mathfrak{su}(n)$. In particular we show that $g$ is isolated in the moduli space of Einstein metrics if $n$ is odd.

math.DG↗

arXiv2023

Stability of the Non-Symmetric Space $E_7/\mathrm{PSO}(8)$

We prove that the normal metric on the homogeneous space $E_7/\mathrm{PSO}(8)$ is stable with respect to the Einstein-Hilbert action, thereby exhibiting the first known example of a non-symmetric metric of positive scalar curvature with this property.

math.DG↗

arXiv2022

Coindex and rigidity of Einstein metrics on homogeneous Gray manifolds

Any $6$-dimensional strict nearly Kähler manifold is Einstein with positive scalar curvature. We compute the coindex of the metric with respect to the Einstein-Hilbert functional on each of the compact homogeneous examples. Moreover, we show that the infinitesimal Einstein deformations on $F_{1,2}=\mathrm{SU}(3)/T^2$ are not integrable into a curve of Einstein metrics.

math.DG↗

arXiv2021

Stability of Einstein metrics on symmetric spaces of compact type

We prove the linear stability with respect to the Einstein-Hilbert action of the symmetric spaces $\mathrm{SU}(n)$, $n\geq3$, and $E_6/F_4$. Combined with earlier results, this resolves the stability problem for irreducible symmetric spaces of compact type.

math.DG↗