Tobias Diez

Let $M$ be a manifold with a closed, integral $(k+1)$-form $ω$, and let $G$ be a Fréchet-Lie group acting on $(M,ω)$. As a generalization of the Kostant-Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of $𝔤$ by $ℝ$, indexed by $H^{k-1}(M,ℝ)^*$. We show that the image of $H_{k-1}(M,ℤ)$ in $H^{k-1}(M,ℝ)^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group $𝕋$. The idea is to represent a class in $H_{k-1}(M,ℤ)$ by a weighted submanifold $(S,β)$, where $β$ is a closed, integral form on $S$. We use transgression of differential characters from $S$ and $ M $ to the mapping space $ C^∞(S, M) $, and apply the Kostant-Souriau construction on $ C^∞(S, M) $.

We initiate the study of the norm-squared of the momentum map as a rigorous tool in infinite dimensions. In particular, we calculate the Hessian at a critical point, show that it is positive semi-definite along the complexified orbit, and determine a decomposition of the stabilizer under the complexified action. We apply these results to the action of the group of symplectomorphisms on the spaces of compatible almost complex structures and of symplectic connections. In the former case, we extend results of Calabi to not necessarily integrable almost complex structures that are extremal in a relative sense. In both cases, the momentum map is not equivariant, which gives rise to new phenomena and opens up new avenues for interesting applications. For example, using the prequantization construction, we obtain new central extensions of the group of symplectomorphisms that are encoding geometric information of the underlying finite-dimensional manifold.

We develop a powerful framework to calculate expectation values of polynomials and moments on compact Lie groups based on elementary representation-theoretic arguments and an integration by parts formula. In the setting of lattice gauge theory, we generalize expectation value formulas for products of Wilson loops by Chatterjee and Jafarov to arbitrary compact Lie groups, and study explicit examples for many classical compact Lie groups and the exceptional Lie group \$G\_2\$. Extending classical results by Collins and Lࡱ'evy, we use our framework to derive expectation value formulas of polynomials of matrix coefficients under the Haar measure, Brownian motion, and the Wilson action. In particular, we construct Weingarten functions for general compact Lie groups by studying the underlying tensor invariants, and apply this to \$ࡱmathrm\{SU\}(N)\$ and \$G\_2\$.

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. The proof of these normal forms is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a Slice Theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg-Witten moduli space and the moduli space of pseudoholomorphic curves.

We prove a theorem on singular symplectic cotangent bundle reduction in the Fréchet setting and apply it to Yang-Mills-Higgs theory with special emphasis on the Higgs sector of the Glashow-Weinberg-Salam model. For the latter model, we give a detailed description of the reduced phase space and show that the singular structure is encoded in a finite-dimensional Lie group action.

Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from \( M \) to the nonlinear Graßmannians \( {Gr}^S(M) \) of submanifolds of \( M \) of a fixed type \( S \). In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden-Weinstein symplectic form. The associated Kostant-Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.

The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more general class of group-valued momentum maps which is inspired by the Poisson Lie setting. In this approach, the group-valued momentum map assigns to every section of the symplectic fiber bundle a principal circle-bundle with connection. The power of this general framework is illustrated in many examples: we construct generalized Clebsch variables for fluids with integral helicity; the anti-canonical bundle turns out to be the momentum map for the action of the group of symplectomorphisms on the space of compatible complex structures; the Teichmüller moduli space is realized as a symplectic orbit reduced space associated to a coadjoint orbit of $SL(2,ℝ)$ and spaces related to the other coadjoint orbits are identified and studied. Moreover, we show that the momentum map for the group of bundle automorphisms on the space of connections over a Riemann surface encodes, besides the curvature, also topological information of the bundle.

Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving diffeomorphisms by push-forward has a group-valued momentum map that assigns to a Riemannian metric the canonical bundle. We then deduce that the Teichmüller space and the moduli space of Riemann surfaces can be realized as symplectic orbit reduced spaces.

Inspired by the Clebsch optimal control problem, we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry. Its special feature is that the Lagrange multipliers couple to the configuration variables via the symmetry group action. The resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system. We discuss the Hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting. We show that the Yang–Mills–Higgs action and the Einstein–Hilbert action fit into this new framework after a (1 + 3)-splitting. Moreover, we recover the Gauß constraint of Yang–Mills–Higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints.

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.

Given a principal bundle on an orientable closed surface with compact connected structure group, we endow the space of based gauge equivalence classes of smooth connections relative to smooth based gauge transformations with the structure of a Fréchet manifold. Using Wilson loop holonomies and a certain characteristic class determined by the topology of the bundle, we then impose suitable constraints on that Fréchet manifold that single out the based gauge equivalence classes of central Yang-Mills connections but do not directly involve the Yang-Mills equation. We also explain how our theory yields the based and unbased gauge equivalence classes of all Yang-Mills connections and deduce the stratified symplectic structure on the space of unbased gauge equivalence classes of central Yang-Mills connections. The crucial new technical tool is a slice analysis in the Fréchet setting.

A local normal form theorem for smooth equivariant maps between Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence of this equivariant normal form theorem, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space. Moreover, the theory of singular symplectic reduction is developed in the infinite-dimensional Fréchet setting. By refining the above construction, a normal form for momentum maps similar to the classical Marle-Guillemin-Sternberg normal form is established. Analogous to the reasoning in finite dimensions, this normal form result is then used to show that the reduced phase space decomposes into smooth manifolds each carrying a natural symplectic structure. Finally, the singular symplectic reduction scheme is further investigated in the situation where the original phase space is an infinite-dimensional cotangent bundle. The fibered structure of the cotangent bundle yields a refinement of the usual orbit-momentum type strata into so-called seams. Using a suitable normal form theorem, it is shown that these seams are manifolds. Taking the harmonic oscillator as an example, the influence of the seams on dynamics is illustrated. The general results stated above are applied to various gauge theory models. The moduli spaces of anti-self-dual connections in four dimensions and of Yang-Mills connections in two dimensions is studied. Moreover, the stratified structure of the reduced phase space of the Yang-Mills-Higgs theory is investigated in a Hamiltonian formulation.

A general slice theorem for the action of a Fréchet Lie group on a Fréchet manifolds is established. The Nash-Moser theorem provides the fundamental tool to generalize the result of Palais to this infinite-dimensional setting. The presented slice theorem is illustrated by its application to gauge theories: the action of the gauge transformation group admits smooth slices at every point and thus the gauge orbit space is stratified by Fréchet manifolds. Furthermore, a covariant and symplectic formulation of classical field theory is proposed and extensively discussed. At the root of this novel framework is the incorporation of field degrees of freedom F and spacetime M into the product manifold F × M. The induced bigrading of differential forms is used in order to carry over the usual symplectic theory to this new setting. The examples of the Klein-Gordon field and general Yang-Mills theory illustrate that the presented approach conveniently handles the occurring symmetries.

Inspired by questions about the geometry of moduli spaces and symplectic quotients, I will present a general approach to normal forms of smooth equivariant maps. In the first part of the talk, I show that a smooth map between infinite-dimensional manifolds can be brought into a certain normal form. This result generalizes and unifies the level set theorem, the constant rank theorem and the immersion theorem. In the second part of the talk, I discuss how this normal form can be combined with the slice theorem to yield a normal form for an equivariant map. As an application, the equivariant normal form will be used to show that subquotients are endowed with a Kuranishi structure in a very general setting. The main results are obtained for Banach manifolds and in the tame Fréchet category using the Nash-Moser inverse function theorem, but the focus of the presentation will lie on the geometric ideas and not on the technicalities of the infinite-dimensional analysis.

In mathematical physics, some conserved quantities have a discrete nature, for example because they have a topological origin. These conservation laws cannot be captured by the usual momentum map. I will present a generalized notion of a momentum map taking values in a Lie group, which is able to include discrete conversed quantities. It is inspired by the Lu-Weinstein momentum map for Poisson Lie group actions, but the groups involved do not necessarily have to be Poisson Lie groups. The most interesting applications include momentum maps for diffeomorphism groups which take values in groups of Cheeger-Simons differential characters. As an important example, I will show that the Teichmüller space with the Weil-Petersson symplectic form can be realized as symplectic orbit reduced space.

Inspired by questions about the geometry of moduli spaces and general bifurcation problems, I will present a general approach to normal forms of smooth equivariant maps. In the first part of the talk, I will show that a smooth map between infinite-dimensional manifolds can be brought into a certain normal form. This result generalizes and unifies the level set theorem, the constant rank theorem and the immersion theorem. In the second part of the talk, I will show how this normal form can be combined with a slice theorem to yield a normal form for an equivariant map. As an application, the equivariant normal form will be used to show that subquotients are endowed with a Kuranishi structure in a very general setting. The main results are obtained for Banach manifolds and in the tame Fréchet category using the Nash-Moser inverse function theorem.

In mathematical physics, some conserved quantities have a discrete nature, for example because they have a topological origin. These conservation laws cannot be captured by the usual momentum map. I will present a generalized notion of a momentum map taking values in a Lie group, which is able to include discrete conversed quantities. It is inspired by the Lu-Weinstein momentum map for Poisson Lie group actions, but the groups involved do not necessarily have to be Poisson Lie groups. The most interesting applications include momentum maps for diffeomorphism groups which take values in groups of Cheeger-Simons differential characters. As an important example, I will show that the Teichmüller space with the Weil-Petersson symplectic form can be realized as symplectic orbit reduced space.

In mathematical physics, some conserved quantities have a discrete nature, such as topological information. These conservation laws cannot be captured by the usual momentum map. I will present a generalized notion of a momentum map taking values in a Lie group, which is able to include discrete conversed quantities. It is inspired by the Lu-Weinstein momentum map for Poisson Lie group actions, but the groups involved do not necessarily have to be Poisson Lie groups. The most interesting applications include momentum maps for various bundle automorphism groups which take values in groups of Cheeger-Simons differential characters. As an important example, I will show how one can obtain Clebsch variables for incompressible fluids with non-vanishing helicity.

Inspired by questions about the geometry of moduli spaces and symplectic quotients, I will present a general approach to normal forms of smooth equivariant maps. In the first part of the talk, I will show that a smooth map between infinite-dimensional manifolds can be brought into a certain normal form. This result generalizes and unifies the level set theorem, the constant rank theorem and the immersion theorem. In the second part of the talk, I will show how this normal form can be combined with the slice theorem to yield a normal form for an equivariant map. As an application, the equivariant normal form will be used to show that subquotients are endowed with a Kuranishi structure in a very general setting. The main results are obtained for Banach manifolds and in the tame Fréchet category using the Nash-Moser inverse function theorem.

In mathematical physics, some conserved quantities have a discrete nature, such as topological information. These conservation laws cannot be captured by the usual momentum map. I will present a generalized notion of a momentum map taking values in a Lie group, which is able to include discrete conversed quantities. It is inspired by the Lu-Weinstein momentum map for Poisson Lie group actions, but the groups involved do not necessarily have to be Poisson Lie groups. The most interesting applications include momentum maps for various bundle automorphism groups which take values in groups of Cheeger-Simons differential characters. As an important example, I will show that the Teichmüller space with the Weil-Petersson symplectic form can be realized as symplectic orbit reduced space.

Many PDE’s that come from physics or differential geometry can be formulated as Hamiltonian systems on infinite-dimensional symplectic manifolds. Frequently, the equations are invariant under diffeomorphisms or gauge transformations and thus one is actually interested in solutions modulo these symmetries. In terms of symplectic geometry, the solution space is identified with the symplectically reduced phase space. In this talk, I will show how the singular Marsden-Weinstein reduction generalizes to infinite-dimensional Fréchet manifolds. The main focus will be on the physical relevant case where the original symplectic manifold is a cotangent bundle.

I will discuss smoothness properties of the holonomy map of a smooth principal G-bundle with connection. For this, the space PM of piecewise smooth paths is endowed with a natural infinite-dimensional manifold structure, which turns PM into a Lie groupoid (up to reparameterization and retrace). With respect to the derived manifold structure on the space LM of piecewise smooth loops, the holonomy of a connection turns out to be smooth as a map Hol: LM → G. An explicit formula for the derivative of the holonomy map at a loop γ in terms of the curvature and the horizontal lift of γ will be given. Conversely, every smooth map H: LM → G satisfying certain natural morphism properties gives rise to a smooth principal G-bundle with a connection having H as its holonomy. This reconstruction theorem and Maurer-Cartan theory in the infinite-dimensional setting shines a new light on well-known facts about the topological classification of principal bundles and Chern-Weil theory.

Inspired by questions concerning the geometry of moduli spaces and symplectic quotients, I will present an axiomatic approach to normal forms of smooth equivariant maps. In the first part of the talk, I will show that a smooth map between infinite dimensional manifolds can be brought into a certain normal form. This result generalizes and unifies the level set theorem, the constant rank theorem and the immersion theorem. In the second part of the talk, I will show how this normal form can be combined with the slice theorem to yield a normal form for an equivariant map. As an application, the equivariant normal form will be used to show that subquotients can be endowed with a Kuranishi structure in a very general setting. The main results are obtained for Banach manifolds and in the tame Fréchet category using the Nash-Moser inverse function theorem.

I'll present a geometric approach to construct central extensions of the group of Hamiltonian symplectomorphisms and of the group of exact volume-preserving diffeomorphisms. The idea is to consider an infinite-dimensional symplectic manifold where those groups act and construct a prequantum bundle over it. The central extensions are then obtained as the pull-back of Kostant's prequantum central extension. We will use Cheeger-Simons differential characters as a tool to construct prequantum bundles over function spaces via a modified transgression map.

Recently, Terence Tao showed that any quadratic ODE obeying a certain cancellation condition can be embedded into the incompressible Euler equations for some manifold M. In this talk, I'll review the construction and show how it can be refined to yield an embedding of the Euler-Poisson equation on any Lie algebra into the incompressible Euler equations on the cotangent bundle of the corresponding Lie group.

In this talk, smoothness properties of the holonomy map of a smooth principal G-bundle with connection will be discussed. For a natural infinite-dimensional manifold structure on the space LM of piecewise smooth loops, the holonomy turns out to be smooth as a map Hol: LM → G. We will give an explicit formula for the derivative of the holonomy map at a loop γ in terms of the curvature and the horizontal lift of γ. Conversely, every smooth map H: LM → G satisfying certain natural morphism properties gives rise to a smooth principal G-bundle with a connection having H as its holonomy. This reconstruction theorem and Maurer-Cartan theory in the infinite-dimensional setting allows to redrive well-known facts about the topological classification of principal bundles and Chern-Weil theory. If time permits, an application of these results to the moduli space of Yang-Mills connections over a Riemannian surface is outlined.

Many PDE’s that come from physics or differential geometry can be reformulated as Hamiltonian systems on infinite-dimensional symplectic manifolds. Since the equations are often invariant under diffeomorphisms or gauge transformations, one is actually interested in solutions modulo these symmetries. In terms of symplectic geometry, the solution space is identified with the symplectically reduced phase space. In this talk, we will show how the singular Marsden-Weinstein reduction can be generalized to infinite-dimensional Fréchet manifolds. As the main step, we will use the Nash-Moser inverse function theorem to prove a normal form theorem for equivariant maps. Formally, the normal form corresponds to an equivariant Lyapunov–Schmidt construction. Examples from gauge theory will illustrate the general theory.

Many PDE’s that come from physics or differential geometry can be reformulated as Hamiltonian systems on infinite-dimensional symplectic manifolds. Since the equations are often invariant under diffeomorphisms or gauge transformations, one is actually interested in solutions modulo these symmetries. In terms of symplectic geometry, the solution space is identified with the symplectically reduced phase space. In this series of talks, we will show how the singular Marsden-Weinstein reduction can be generalized to infinite-dimensional Fréchet manifolds. As the main step, we will use the Nash-Moser inverse function theorem to prove a normal form theorem for equivariant maps. Formally, the normal form corresponds to an equivariant Lyapunov–Schmidt construction. Examples from gauge theory will illustrate the general theory.

Many important PDE's with geometric or physical origin can be recast in terms of a momentum map for gauge or diffeomorphism groups. In this talk, I will present a general framework for constructing an infinite-dimensional symplectic system as the section space of a symplectic fibre bundle. The momentum map for the automorphism group of the bundle will be determined. As it turns out, a classical momentum map only exists for "exact" diffeomorphisms. We thus generalize the notion and consider certain group-valued momentum maps which are able to capture also topological invariants. Examples coming from fluid mechanics, Kähler geometry and Teichmüller theory will be discussed.

In Yang-Mills theory, many important properties of the physical system are encoded in the structure of the moduli space of connections with respect to the group of gauge transformations. Atiyah and Bott studied this moduli space in their seminal paper by using symplectic reduction adapted to the special case of a Riemann surface as the base manifold. The functional analytic problems were approached using Sobolev space techniques. In this talk, I will show how these results can be extended and reformulated in the framework of Fréchet manifolds. In particular, central Yang-Mills connections are represented as a subset of the Fréchet manifold of based gauge equivalence classes of connections and are identified as the inverse image under the Wilson loop map.

A general slice theorem for the action of a Fréchet Lie group on a Fréchet manifolds is presented. The Nash-Moser theorem provides the fundamental tool to generalize the seminal result of Palais to this infinite-dimensional setting. The Fréchet slice theorem is illustrated by its application to gauge theories: the action of the gauge transformation group admits smooth slices at every point and thus the gauge orbit space is stratified by Fréchet manifolds.