arXiv subjects

Tilmann Wurzbacher

Publications and source records attributed to Tilmann Wurzbacher.

14 recordsLinked to original sources

arXiv2026

Actions of Lie 2-algebras and comomentum maps

In this paper we introduce the notion of a 2-action of a Lie 2-algebra on an arbitrary manifold M. Furthermore, in [Rog12], given a n-plectic manifold (M, $ω$), the authors consider a Lie Infinity-algebra L$\infty$ (M, $ω$), which is a higher analogue of the Poisson algebra of observables associated to a symplectic manifold. This Lie Infinity-algebra reduces to a Lie 2-algebra L^2 (M, $ω$) when (M, $ω$) is 2-plectic. Following ideas of N.L. Delgado [Del18], we introduce the Lie 2-algebra D^2 (M, $ω$), which generalises the Lie 2-algebra L^2 (M, $ω$) and its extension containing Hamiltonian pairs. Given a two-plectic manifold (M, $ω$) and a Lie 2-algebra g_1 $\oplus$ g_0 acting on M we define a comomentum map as a lift of the action, i.e., as a Lie 2-algebra morphism from g_1 $\oplus$ g_0 to the extension of the Lie 2-algebra D^2 (M, $ω$). In an appendix, we discuss very explicitly numerous examples, classified according to their algebraic properties.

math-ph↗

arXiv2025

Hamiltonian dynamics and geometry on the two-plectic six-sphere

We study the two-plectic geometry of the six-sphere induced by pulling back a canonical $G_2$-invariant three-form from $\mathbb{R}^7$ . Notably we explicitly prove non-flatness of this structure and show that its infinitesimal automorphisms are given by the exceptional Lie algebra $\mathfrak{g}_2$. Several interesting classes of solutions of the dynamical Hamilton-de Donder-Weyl equations with one- and two-dimensional sources are exhibited.

math.DG↗

arXiv2024

Notes on equivalent formulations of Hamiltonian dynamics on multicotangent bundles

We show the equivalence of five different conditions on a classical field $ψ$ with values in a restricted multicotangent bundle to be a solution of the field equations, notably in terms of the Hamilton-Volterra equations, the principle of least action and several conditions based on the contraction of the multi-vector tangent to $ψ$ with canonical differential forms. Most prominently, we have equivalence to the "dynamical Hamilton-de Donder-Weyl equation", that can be vastly generalized to define Hamiltonian dynamics on multisymplectic manifolds, defined for sources of different dimensions.

math.SG↗

arXiv2021

On the prequantisation map for 2-plectic manifolds

For a manifold $M$ with an integral closed 3-form $ω$, we construct a $PU(H)$-bundle and a Lie groupoid over its total space, together with a curving in the sense of gerbes. If the form is non-degenerate, we furthermore give a natural Lie 2-algebra quasi-isomorphism from the observables of $(M,ω)$ to the weak symmetries of the above geometric structure, generalising the prequantisation map of Kostant and Souriau.

math.DG↗

arXiv2018

Lagrangian submanifolds of standard multisymplectic manifolds

We give a detailed, self-contained proof of Geoffrey Martin's normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein's famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential topology, i.e., a foliated tubular neighborhood theorem and a foliated relative Poincaré lemma.

math.DG↗

arXiv2018

An invitation to multisymplectic geometry

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we reformulate the latter in multisymplectic terms. Furthermore, we investigate basic questions on normal forms of multisymplectic manifolds, notably the questions wether and when Darboux-type theorems hold, and how many diffeomorphisms certain, important classes of multisymplectic manifolds possess. Finally, we survey recent advances in the area of symmetries and conserved quantities on multisymplectic manifolds.

math.DG↗

arXiv2016

Conserved quantities on multisymplectic manifolds

Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well-behaved under transgression to spaces of maps into M. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. We show that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.

math.SG↗

arXiv2016

Infinite-dimensional manifolds as ringed spaces

We analyze the possibility of defining infinite-dimensional manifolds as ringed spaces. More precisely, we consider three definitions of manifolds modeled on locally convex spaces: in terms of charts and atlases, in terms of ringed spaces, and in terms of functored spaces, as introduced by Douady in his thesis. It is shown that for large classes of locally convex model spaces (containing Fréchet spaces and duals of Fréchet-Schwartz spaces), the three definitions are actually equivalent. The equivalence of the definition via charts with the definition via ringed spaces is based on the fact that for the classes of model spaces under consideration, smoothness of maps turns out to be equivalent to their scalarwise smoothness (that is, the smoothness of their composition with smooth real-valued functions).

math.DG↗

arXiv2016

Superorbits

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy (or stabiliser) supergroups and orbits through general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov-Kostant-Souriau type. Applying a family version of Kirillov's orbit method, we decompose the regular representation of an odd Abelian supergroup into an odd direct integral of characters and construct universal families of representations, parametrised by a supermanifold, for two different super variants of the Heisenberg group.

math.DG↗

arXiv2014

Existence and unicity of co-moments in multisymplectic geometry

Given a multisymplectic manifold $(M,ω)$ and a Lie algebra $\frak{g}$ acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to the observable algebra $L(M,ω)$ associated to $(M,ω)$, in analogy with and generalizing the notion of a co-moment map in symplectic geometry. We give a cohomological characterization of existence and unicity for homotopy co-moment maps and show its utility in multisymplectic geometry by applying it to special cases as exact multisymplectic manifolds and simple Lie groups and by deriving from it existence results concerning partial co-moment maps, as e.g. covariant multimomentum maps and multi-moment maps.

math.DG↗

arXiv2014

Singular superspaces

We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in this category, morphisms with values in a supermanifold are still given in terms of coordinates. This framework gives a natural notion of relative supermanifolds over a locally finitely generated base. Moreover, the existence of inner homs, whose source is the spectrum of a Weil superalgebra, is established; they are generalisations of the Weil functors defined for smooth manifolds.

math.DG↗

arXiv2012

Integration of vector fields on smooth and holomorphic supermanifolds

We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on complex supermanifolds. Furthermore we discuss local actions associated to super vector fields, and give several examples and applications, as, e.g., the construction of an exponential morphism for an arbitrary finite dimensional Lie supergroup. In the 2nd version we corrected typos, presentation of figures and other small points. We added references in Sections 3 and 4. More details are given in part (4.2) of Section 4, treating the exponential morphism of a Lie supergroup. Notably we give now two constructions of the vector field leading to the exponential morphism and we show its completeness; furthermore we characterize the exponential morphism in terms of the exponential maps of the Lie groups of morphisms from superpoints to the given Lie supergroup.

math.DG↗

arXiv2011

The geodesic flow on a Riemannian supermanifold

We give a natural definition of geodesics on a Riemannian supermanifold and extend the usual geodesic flow defined on the cotangent bundle of the body of the supermanifold, associated to the induced Riemannian structure on the body, to a geodesic "superflow" on the cotangent bundle of the supermanifold. Integral curves of this flow turn out to be in natural bijection with geodesics on the Riemannian supermanifold. We also construct the corresponding exponential map and generalize the well-known faithful linearization of isometries to Riemannian supermanifolds.

math.DG↗

arXiv2009

On the geometry and quantization of symplectic Howe pairs

We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical correspondence between the orbit spaces of the respective moment images. Furthermore, we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group. In the Kaehler case we show that the linear representation of a pair of compact Lie groups on the geometric quantization of the manifold is then equipped with a representation-theoretic Howe duality.

math.SG↗