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Group-Valued Momentum Maps for Actions of Automorphism Groups

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Group-Valued Momentum Maps for Actions of Automorphism Groups Group-valued momentum maps for actions of automorphism groups Tobias Dieza and Tudor S. Ratiub February 5, 2020 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.¹ º 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. Keywords: symplectic structure on moduli spaces of geometric structures, infinite- dimensional symplectic geometry, momentum maps, symplectic fiber bundles, differential characters, Kähler geometry, Teichmüller space MSC 2010: 53D20, (58D27, 53C08, 53C10, 58B99, 32G15) a Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany and Institut für Theoretische Physik, Universität Leipzig, 04009 Leipzig, Germany and Institute of Ap- arXiv:2002.01273v1 [math.DG] 4 Feb 2020 plied Mathematics, Delft University of Technology, 2628 XE Delft, Netherlands. Partially supported by the Max Planck Institute for Mathematics in the Sciences (Leipzig), by the German National Academic Foundation (DAAD) and by the above NWO grant 639.032.734. [email protected] b School of Mathematical Sciences and Ministry of Education Laboratory of Scientific Computing (MOE-LSC), Shanghai Jiao Tong University, Minhang District, 800 Dongchuan Road, 200240 China, Section de Mathématiques, Université de Genève, 2–4 rue du Lièvre, case postale 64, 1211 Genève 4, and Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. Partially supported by by the National Natural Science Foundation of China (No. 11871334) and by NCCR SwissMAP grant of the Swiss National Science Foundation. [email protected], [email protected] Contents 2 Contents 1 Introduction 2 2 Group-Valued Momentum Maps 8 2.1 Poisson momentum maps 8 2.2 Dual pairs of Lie algebras 10 2.3 Group-valued momentum maps 13 2.4 Existence and uniqueness 16 2.5 Equivariance and Poisson property 19 2.6 Momentum maps for group extensions 35 3 Global Analysis of Symplectic Fiber Bundles 38 3.1 Symplectic form 38 3.2 Action of the automorphism group 39 3.2.1 Momentum map for the gauge group 41 3.2.2 Momentum map for the diffeomorphism group 44 3.2.3 Momentum map for volume-preserving diffeomorphisms 44 3.2.4 Momentum map for symplectomorphisms 49 4 Applications 52 4.1 Trivial bundle case 52 4.1.1 Hydrodynamics 53 4.1.2 Lagrangian embeddings 59 4.2 Reduction of structure group 60 4.2.1 Pull-back of prequantum bundles as associated bundles 61 4.2.2 Kähler geometry 64 4.2.3 Teichmüller space and weighted Lagrangian subbundles 70 4.3 Action of the quantomorphism group 77 4.4 Gauge theory 79 Appendices 87 a Notations and conventions 87 b Fiber integration 90 c Hat product for fiber bundles 92 d Cheeger-Simons differential characters 95 d.1 Motivation: circle bundles 96 d.2 Differential characters 97 d.3 Hat product of differential characters 102 1 Introduction Noether’s theorem states that every symmetry of a given system has a corre- sponding conservation law. In terms of symplectic geometry, these conserved quantities are encoded in the momentum map. It was, however, quickly realized Introduction 3 that the momentum map geometry not only plays an important role in dynamical systems but is also a valuable tool in the study of differential geometric questions. In their seminal work, Atiyah and Bott [AB83] showed that the curvature of a connection on a principal bundle over a Riemannian surface is the momentum map for the action of the group of gauge transformations. They applied Morse theory to the norm-squared of the momentum map (the Yang–Mills functional) in order to obtain the cohomology of the moduli space of Yang–Mills solutions which, by the Narasimhan–Seshadri theorem, can be identified with the moduli space of stable holomorphic structures. Within the same circle of ideas, Fujiki [Fuj92] and Donaldson [Don97; Don00; Don03] provided a momentum map picture for the relationship between the existence of constant scalar curvature Kähler metrics and stability in the sense of geometric invariant theory. The first main aim of this paper is to provide a framework which encompasses the gauge theory setting of Atiyah and Bott and, at the same time, the action of diffeomorphism groups of Fujiki and Donaldson. Our starting point is a symplectic fiber bundle of the form F P ×G F for a principal G-bundle P ! M, where the typical fiber F is endowed with a G-invariant symplectic form. The fiberwise symplectic structure, combined with a volume form on the base M, induces a symplectic form Ω on the space F of sections of F ! M. The gauge group of P acts in a natural way on F, leaving the induced symplectic form Ω invariant. As we will see, this action possesses a momentum map which is completely determined by the momentum map of the G-action on the fiber F. Suppose the bundle P is natural, i.e., it comes with a lift of Diff¹Mº to bundle automorphisms (for example, this is the case when P is the frame bundle of M). In this case, every subgroup of the group of volume-preserving diffeomorphisms acts naturally on the space F of sections and leaves Ω invariant. Our first result is summarized in Theorems 3.3 and 3.7 which determine the momentum map for the group of volume-preserving diffeomorphisms and symplectomorphisms, respectively. There are, essentially, two contributions to the momentum map. The first term is the pull-back of the fiberwise symplectic structure. The second term involves the fiber momentum map and, morally speaking, captures how much the lift of diffeomorphisms to bundle automorphisms shifts in the vertical direction. The interesting point is that the momentum map for the automorphism group on the infinite-dimensional space of sections is canonically constructed from the finite-dimensional symplectic G-manifold F. In contrast to the case of the action of the gauge group, the momentum map for the symplectic action of the diffeomorphism group on the space of sections does not exist in full generality. This was already pointed out in [Don00; Don03]. The obstruction has a topological character, i.e., certain cohomology groups have to vanish. To remedy this situation, one restricts attention to a certain “exact” subgroup, e.g., the subgroup of Hamiltonian diffeomorphisms in the group of all Introduction 4 symplectomorphisms. The action of this subgroup then usually admits a classical momentum map. Working from a completely different point of view, similar observations were made by Gay-Balmaz and Vizman [GV12] in their study of the classical dual pair in hydrodynamics. In this case, the symplectic action of volume-preserving diffeomorphisms on a symplectic manifold of mappings only has a momentum map under certain topological conditions and one is forced to work with suitable central extensions of the group of exact volume-preserving diffeomorphisms. In this paper, we take the viewpoint that the above mentioned topological obstructions are not a bug but a feature of the theory. The action of the diffeo- morphism group interacts with, and is largely determined by, the topological structure of the bundle. Thus, one would expect to capture certain topological data (like characteristic classes) that are “conserved” by the action. Such “con- servation laws” should be encoded in the momentum map. Since the classical momentum map takes values in a continuous vector space, there is no space to “store” discrete topological information. Hence, whenever those classes do not vanish, a classical momentum map does not exists. Nonetheless, one could hope that a generalized momentum map exists and captures the conserved topological data. The second main aim of the paper is to translate these philosophical remarks into explicit mathematical statements. In order to do this, we generalize the notion of momentum maps allowing them to take values in groups. Our concept of a group-valued momentum map is inspired by the Lu momentum map [Lu90; LW90] in Poisson geometry. We emphasize right away that the group-valued momentum map we introduce in this paper is a vast generalization of many notions of momentum maps appearing in the literature including circle-valued, cylinder-valued, and Lie algebra-valued momentum maps. We show that our generalized group-valued momentum map always exists for the action of the diffeomorphism group, without any topological assumptions on the base but some integrability conditions on the fiber model. The resulting momentum map captures topological invariants of the geometry, exactly in (the dual of) those cohomology classes which prevented the existence of a classical momentum map. This approach of extending the definition of the momentum map, besides the situation described above in Poisson geometry, in order to capture conservation laws not available using the classical definition, has been used successfully before in the theory of the cylinder-valued and optimal momentum maps; see [OR03] for a detailed presentation.
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