Normal forms generalizing action-angle coordinates for Hamiltonian actions of Lie groups

Let (M, Ω) be a symplectic manifold on which a Lie group G acts by a Hamiltonian action. Under some restrictive assumptions, we show that there exists a symplectic diffeomorphism ψ of a G-invariant open neighbourhood U of a given G-orbit in M, onto an open subset ψ(U) of a vector bundle F ^*, with base space G. Explicit expressions are given for the symplectic 2-form, for the momentum map and for a Hamiltonian vector field whose Hamiltonian function is G-invariant, on the model symplectic manifold ψ(U).     

Group Actions on DG-Manifolds and Exact Courant Algebroids

Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a ${\mathbb{R}[n]}$ -bundle over T[1]M is measured by the G equivariant cohomology of M. We explicitly calculate the differential graded Lie algebra of the symmetries of the ${\mathbb{R}[n]}$ -bundle over T[1]M and we use this differential graded Lie algebra to understand which actions are hamiltonian. We show how split Exact Courant algebroids could be obtained as the derived Leibniz algebra of the symmetries of ${\mathbb{R}[2]}$ -bundles over T[1]M, and we use this construction to propose that the infinitesimal symmetries of a split Exact Courant algebroid should be encoded in the differential graded Lie algebra of symmetries of a ${\mathbb{R}[2]}$ -bundle over T[1]M. With this setup at hand, we propose a definition for an action of a Lie group on an Exact Courant algebroid and we propose conditions for the action to be hamiltonian.     

A grand superspace for unified field theories

Agrand superspace is proposed as the phase space for gauge field theories with a fixed structure groupG over a fixed space-time manifoldM. This superspace incorporatesall principal fiber bundles with these data. This phase space is the space of isomorphism classes ofall connections onall G-principal fiber bundles overM (fixedG andM). The justification for choosing this grand superspace for the phase space is that the space-time and the structure group are determinants of the physical theory, but the principal fiber bundle with the givenG andM is not. Grand superspace is studied in terms of a natural universal principal fiber bundle overM, canonically associated withM alone, and with a natural universal connection on this bundle. This bundle and its connection are universal in the sense that all connections on allG-principal fiber bundles (anyG) overM can be recovered from this universal bundle and its universal connection by a canonical construction. WhenG is Abelian, grand superspace is shown to be an Abelian group. Various subspaces of grand superspace consisting of the isomorphism classes of flat connections and of Yang-Mills connections are also discussed.     

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

Nonlinear realizations of symmetry groups in terms of differential geometry

Nonlinear realizations of a symmetry group G, which become linear when restricted to a subgroup H are described in terms of fibre bundles. It is shown that so-called “covariant derivatives” occuring in nonlinear Lagrangians are equivalent to the covariant derivatives of the canonical connection in the principal bundle (G, G/H, H, δ). After the specification of a cross-section of the bundle, our formulae for the covariant derivatives coincide with those obtained by other authors in a group-theoretical way. In a special case where G is a chiral group and H is its diagonal subgroup, the canonical connection induces the Riemannian connection in the tangent bundle over G/H. For G = SU(2) × SU(2) and H = SU(2) this connection coincides with the Riemannian connection on the three-dimensional sphere introduced by K. Meetz.     

Infinitesimal Deformations of a Formal Symplectic Groupoid

Given a formal symplectic groupoid G over a Poisson manifold (M, π ₀), we define a new object, an infinitesimal deformation of G, which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation \({\pi_0 + \varepsilon \pi_1}\) of the Poisson bivector field π ₀. To any pair of natural star products \({(\ast,\tilde\ast)}\) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G. We call it the deformation groupoid of the pair \({(\ast,\tilde\ast)}\) . To each star product with separation of variables \({\ast}\) on a Kähler–Poisson manifold M we relate another star product with separation of variables \({\hat\ast}\) on M. We build an algorithm for calculating the principal symbols of the components of the logarithm of the formal Berezin transform of a star product with separation of variables \({\ast}\) . This algorithm is based upon the deformation groupoid of the pair \({(\ast,\hat\ast)}\) .     

Relativity groups in the presence of matter

We show that the action of the boosts on an infinite system can be described continuously by bundle maps of Hilbert bundles based on the manifoldsG/G ₀, whereG is the full relativity group andG ₀ its closed subgroup which can be unitarily implemented on the fibre, which is a Hilbert space. We then develop a general theory of group representations on product bundlesM × ?, whereM is a manifold and ? a Hilbert space. We construct certain bundle representations of the Galilei and the Poincaré group and find that they correspond to various classes of elementary excitations. In particular, we define nonrelativistic zero-mass systems and obtain an analogue of the Faraday effect for the passage of hot electrons through matter. Our construction remains valid for the case whenG ₀ is the product of a lattice translation group and the time translations. We conclude that many qualitative features of the physics of condensed matter can be directly understood in terms of relativity group action on a bundle space as state space, which also suggests some avenues for further work.     

Unitarity in “Quantization Commutes with Reduction”

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M // G. Guillemin and Sternberg [Invent. Math. 67, 515–538 (1982)] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M //G and the G-invariant subspace of the quantum Hilbert space over M.Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction” and show that in this setting there is still a natural invertible map between the Hilbert space over M // G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M // G.     

Lifting classes of principal bundles

Given a principal G-bundle P over X, we define a particularly suitable equivalence relation between liftings of P with respect to a group morphism σ:M→G. Supposing that σ has a central kernel C, we obtain a free operation of H¹(X$\text{C}$) (with coefficients in the sheaf of C-valued functions) on the set of lifting classes of P, which is natural under change of groups and base spaces. It is simply transitive, if in addition σ is an epimorphism; otherwise we classify its orbits by sections in the associated bundle P × _G(G/σM).For $\text{C=}\text{Z}{}_{\mathrm{n}}$ we relate the lifting classes to similar classes of n-th roots of associated line bundles. In the differentiable case and for an epimorphism σ with discrete kernel, there is a natural lifting of partial principal connections in each of these lifting classes. Finally, we indicate some applications to geometric quantization.     

A universal phase space for particles in Yang-Mills fields

Given a principal G-bundle P over M and a Hamiltonian G-space Q, one may construct the reduced symplectic manifold (T*P x Q)₀. When a connection on P is chosen, this manifold becomes a bundle over T*M with fibre Q. It is shown that this bundle is precisely the phase space constructed by Sternberg for a classical particle in a Yang-Mills field.Research partially supported by NSF Grant MCS 74-23180.A01.     

Conformal Geometry of the Supercotangent and Spinor Bundles

We study the actions of local conformal vector fields \({X \in {\rm conf}(M,g)}\) on the spinor bundle of (M, g) and on its classical counterpart: the supercotangent bundle \({\mathcal{M}}\) of (M, g). We first deal with the classical framework and determine the Hamiltonian lift of conf (M, g) to \({\mathcal{M}}\) . We then perform the geometric quantization of the supercotangent bundle of (M, g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map.The quantum and classical actions of conf (M, g) turn, respectively, the space of differential operators acting on spinor densities and the space of their symbols into conf (M, g)-modules. They are filtered and admit a common associated graded module. In the conformally flat case, the latter helps us determine the conformal invariants of both conf (M, g)-modules, in particular the conformally odd powers of the Dirac operator.     

The horospheric approach to harmonic analysis on a semi-simple Lie group

We extend the Gelfand-Helgason theory of horospheric transformations to the regular representation of an arbitrary connected semi-simple Lie group. Let G=NAK be an Iwasawa decomposition of such a group G, V the subgroup contragredient to N, and M the centralizer of A in K: then the horospheric transform is a mapping of the function space over G onto one over the topological product N×V×M×A; and the Fourier transform over G in the principal most continuous series reduces to an expansion into multihomogeneous functions on A and irreducible representations of M, while the remaining principal series are characterised by associated homogeneity in one or more variables of A. The inversion formula for the Fourier transform is our principal result. The measure required for the most continuous series of a group of real-rank unity is the square of the Plancherel measure for G; and at the double poles of this the residues are precisely the contributions of the remaining principal series. This enables us to write the entire inversion formula for such a group as a single formal contour integral; analogous results hold for groups of higher real-rank, but the measure is slightly less simple.     

Loop and Path Spaces and¶Four-Dimensional BF Theories:¶Connections, Holonomies and Observables

We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non-topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M. Received: 28 March 1998 / Accepted: 12 September 1998     

Geometric quantization in the presence of an electromagnetic field

Some aspects of the formalism of geometric quantization are described emphasizing the role played by the symmetry group of the quantum system which, for the free particle, turns out to be a central extensionG(_m) of the Galilei groupG. The resulting formalism is then applied to the case of a particle interacting with the electromagnetic field, which appears as a necessary modification of the connection 1-form of the quantum bundle when its invariance group is generalized to alocal extension ofG. Finally, the quantization of the electric charge in the presence of a Dirac monopole is also briefly considered.     

On diagonalization in map(M,G)

Motivated by some questions in the path integral approach to (topological) gauge theories, we are led to address the following question: given a smooth map from a manifoldM to a compact groupG, is it possible to smoothly “diagonalize” it, i.e. conjugate it into a map to a maximal torusT ofG? We analyze the local and global obstructions and give a complete solution to the problem for regular maps. We establish that these can always be smoothly diagonalized locally and that the obstructions to doing this globally are non-trivial Weyl group and torus bundles onM. We explain the relation of the obstructions to winding numbers of maps intoG/T and restrictions of the structure group of a principalG bundle toT and examine the behaviour of gauge fields under this diagonalization. We also discuss the complications that arise in the presence of non-trivialG-bundles and for non-regular maps. We use these results to justify a Weyl integral formula for functional integrals which, as a novel feature not seen in the finite-dimensional case, contains a summation over all those topologicalT-sectors which arise as restrictions of a trivial principalG bundle and which was used previously to solve completely Yang-Mills theory and theG/G model in two dimensions.     

Fourier analysis on multiplier representations of locally compact Abelian groups

Generalising known results [2] for vector groups, it is shown that, for an arbitrary multiplier ω for an arbitrary locally compact Abelian group G, there is a faithful normal semifinite trace on the von Neuman algebra generated by the regular ω-representation of G which is translation invariant in a certain sense. Analogues of the Fourier transformation, the Plancherel identity and the Fourier inversion formula are obtained in which this trace replaces the Haar measure on the dual group.     

Geometry of the Aharonov–Bohm Effect

We show that the connection responsible for any Abelian or non-Abelian Aharonov–Bohm effect with n parallel “magnetic” flux lines in ℝ³, lies in a trivial G-principal bundle P→M, i.e. P is isomorphic to the product M×G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space , where path integrals are computed, and the associated bundle P× _G ℂ ^m →M, where the wave function and its covariant derivative are sections.