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 quantum bundles

Suppose we are given a group G acting through canonical transformations on a symplectic manifold (M, ω). If there is a quantum bundle over (M, ω), a carrier for wave functions in the geometric quantization theory, then G acts infinitesimally on the bundle in a natural way. We give a necessary and sufficient condition for the infinitesimal G-action to integrate up to a global G-action. This is used for an investigation how the choice of the quantum bundle over (M, ω) influences the integrability of the corresponding infinitesimal G-action. The relationship to group representations is briefly mentioned.     

Pseudoholomorphic Curves on Nearly Kähler Manifolds

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kähler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a 6-dimensional sphere with the standard (G ₂-invariant) almost complex structure.     

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 \({(\ast,\hat\ast)}\) .     

Sasakian and Parabolic Higgs Bundles

Let M be a quasi-regular compact connected Sasakian manifold, and let N = M/S ¹ be the base projective variety. We establish an equivalence between the class of Sasakian G–Higgs bundles over M and the class of parabolic (or equivalently, ramified) G–Higgs bundles over the base N.     

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.     

Non-holonomic and semi-holonomic frames in terms of Stiefel and Grassmann tangent bundles

We prove that the bundles of non-holonomic and semi-holonomic second-order frames of a real or complex manifold M can be obtained as extensions of the bundle F²(M) of second-order jets of (holomorphic) diffeomorphisms of into M, where or . If and is the bundle of -linear frames of M we will associate to the tangent bundle two new bundles and with fibers of type the Stiefel manifold and the Grassmann manifold , respectively, where . The natural projection of onto defines a -principal bundle. We have found that the subset of given by the horizontal n-planes is an open sub-bundle isomorphic to the bundle of semi-holonomic frames of second-order of M. Analogously, the subset of given by the horizontal n-bases is an open sub-bundle which is isomorphic to the bundle of non-holonomic frames of second-order of M. Moreover the restriction of the former projection still defines a -principal bundle. Since a linear connection is a horizontal distribution of n-planes invariant under the action of it therefore determines a -reduction of the bundle , in a bijective way. This is a new proof of a theorem of Libermann.     

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 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.     

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.     

Quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and Jacobi fields

We characterize the complexity of geodesic paths on a curved statistical manifold M_s through the asymptotic computation of the information geometric complexity V_Ms and the Jacobi vector field intensity J_Ms. The manifold M_s is a 2l-dimensional Gaussian model reproduced by an appropriate embedding in a larger 4l-dimensional Gaussian manifold and endowed with a Fisher-Rao information metric g_μν(Θ) with non-trivial off-diagonal terms. These terms emerge due to the presence of a correlational structure (embedding constraints) among the statistical variables on the larger manifold and are characterized by macroscopic correlational coefficients r_k. First, we observe a power law decay of the information geometric complexity at a rate determined by the coefficients r_k and conclude that the non-trivial off-diagonal terms lead to the emergence of an asymptotic information geometric compression of the explored macrostates Θ on M_s. Finally, we observe that the presence of such embedding constraints leads to an attenuation of the asymptotic exponential divergence of the Jacobi vector field intensity.     

The moment map and collective motion

Given a Hamiltonian action of a Lie group G on a symplectic manifold M there is an induced map Φ: $\text{M → g}{}^1$ where $\text{g}{}^1$ is the dual space to the Lie algebra, g, of G. The map Φ is called the moment map. Any function P on $\text{g}{}^1$ then gives rise to a function F = P ° Φ on M which is a “collective Hamiltonian” associated to the group action G. We show how the rigid rotor, liquid drop, and other collective models of the nucleus fit into this framework. We describe the steps involved in integrating collective equations of motion and indicate some principles involved in the choice of collective Hamiltonians, i.e., the functions P. We discuss these constructions in some detail for the case that G is a semidirect product.     

On Kaluza-Klein theory

Assuming the compactification of 4 + K-dimensional space-time implied in Kaluza-Kleintype theories, we consider the case in which the internal manifold is a quotient space, $\text{G}\text{H}$. We develop normal mode expansions on the internal manifold and show that the conventional gravitational plus Yang-Mills theory (realizing local G symmetry) is obtained in the leading approximation. The higher terms in the expansions give rise to field theories of massive particles. In particular, for the original Kaluza-Klein 4 + 1-dimensional theory, the higher excitations describe massive, charged, purely spin-2 particles. These belong to infinite dimensional representations of an O(1,2).     

Compactification and supersymmetry in d = 11 supergravity

We examine the conditions under which the ground state of d = 11 supergravity can be supersymmetric and be of the form M⁴ ? B⁷ with M⁴ Minkowski spacetime and B⁷ a compact seven-dimensional manifold. Since we have in mind a background that renders the effective action stationary we make no use of the classical field equations. We find that the requirement that the four-space be flat is very restrictive. It requires all components of the background four-index field to vanish and the compact manifold to be Ricci-flat and hence to have at most the abelian symmetries associated with tori.     

Invariant star-products on symplectic manifolds

Let (M,F) be a symplectic manifold and consider a Lie subalgebra $\text{G}$ of its Lie algebra of symplectic vector fields. We prove that every one-differentiable deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to $\text{G}$, extends to an invariant one-differentiable deformation of infinite order. If M admits a $\text{G}$-invariant linear connection, a similar result holds true for differentiable deformations and for star-products. In particular, if M admits a $\text{G}$- -invariant linear connection, there always exists a $\text{G}$-invariant star-product.     

Formal structures,the concepts of covariance,invariance, equivalent reference frames,and the principle Relativity

In this paper a given spacetime theoryT is characterized as the theory of a certainspecies of structure in the sense of Bourbaki [1]. It is then possible to clarify in a rigorous way the concepts ofpassive andactive covariance ofT under the action of the manifold mapping groupG _M . For eachT, we define also aninvariance groupG _I T and, in general,G _I T ≠G _M . This group is defined once we realize that, for eachτ ∈ModT, each explicit geometrical object defining the structure can be classified as absolute or dynamical [2]. All spacetime theories possess alsoimplicit geometrical objects that do not appear explicitly in the structure. These implicit objects are not absolute nor dynamical. Among them there are thereference frame fields, i.e., “timelike” vector fieldsX ∈TU, \(U \subseteq M\) M, whereM is a manifold which is part ofST, a substructure for eachτ ∈ModT, called spacetime. We give a physically motivated definition of equivalent reference frames and introduce the concept of theequivalence group of a class of reference frames of kind X according to T, G _X T. We define thatT admits aweak principle of relativity (WPR) only ifG _X T ≠ identity for someX. IfG _X T =G _I T for someX, we say thatT admits a strong principle of relativity (PR). The results of this paper generalize and clarify several results obtained by Anderson [2], Scheibe [3], Hiskes [4], Recami and Rodrigues [5], Friedman [6], Fock [7], and Scanavini [8]. Among the novelties here, there is the realization that the definitions ofG _I T andG _X T can be given only when certain boundary conditions for the equations of motion ofT can be physically realizable in the domainU \(U \subseteq M\) M, where a given reference frame is defined. The existence ofphysically realizable boundary conditions for eachτ ∈ModT (in ?U), in contrast with the mathematically possible boundary condition, is then seen to be essential for the validity of a principle of relativity forT. The methodology of the present paper has been applied to several topics of spacetime physics with very interesting results. Here we mention:

1. The Newtonian concepts of absolute space and absolute time can be presented in a very elegant way as “species of structure”. One of the surprising results is that we succeeded in finding a Lorentzian structure [9] in Newtonian spacetime without introducing any new explict geometrical object in the original structure. The Newtonian spacetime structure and its relation to the relativistic spacetime structure and to the structure of the spacetime of the so-called Lorentz aether theories [11,12] is fully discussed in [13].
2. It is possible to present in a novel and unified way the question concerning experiments designed to detect a possible breakdown of Lorentz invariance, a subject we already dedicated attention to in Rodrigues and Tiomno [11,12] and Rodrigues [14,15]. A full account of this subject will be published elsewhere.
3. In Rodrigues and Scanavini [16], we proved that there are models of General Relativity that contain a canonical privileged locally inertial reference frame that can be physically distinguished from any other frame by experiments doneinside the frame.

Although the formalism of this paper may at first sight look very abstract, actually it is easy to aplly it to specific theories. We present an example at the end of the paper which is sufficiently general to show “in action” almost all concepts introduced in this paper.     

Local Geometry of the <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> Moduli Space

We consider deformations of torsion-free G ₂ structures, defined by the G ₂-invariant 3-form φ and compute the expansion of \({\ast \varphi }\) to fourth order in the deformations of φ. By considering M-theory compactified on a G ₂ manifold, the G ₂ moduli space is naturally complexified, and we get a Kähler metric on it. Using the expansion of \({\ast \varphi }\), we work out the full curvature of this metric and relate it to the Yukawa coupling.     

Locally supersymmetric σ-model with Wess-Zumino term in two dimensions and critical dimensions for strings

We construct an N = 1 locally supersymmetric σ-model with a Wess-Zumino term coupled to supergravity in two dimensions. If one takes the σ-model manifold to be the product of d-dimensional Minkowski space M^d and a group manifold G, and if the radius of G is quantized in appropriate units of the string tension, then the model describes a Neveu-Schwarz-Ramond (NSR)-type string moving on M_d × G. (Our model generalizes earlier work of refs. [1,2] which do not contain a Wess-Zumino term and that of refs. [5,6] which is not locally supersymmetric.) The zweibein and the gravitino field equations yield constraints which generalize those of the NSR model to the case of a non-abelian group manifold. In particular, the fermionic constraint contains a new term trilinear in the fermionic fields. We quantize the theory in the light-cone gauge and derive the critical dimensions. We compute the mass spectrum of a closed string moving on M_d × G and show that massless fermions do not arise for non-abelian G for the spinning string, in agreement with the result of Friedan and Shenker [22].     

On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M,g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p,q∈M that are non-conjugate in a suitable sense, under the assumption that (M,g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.