Geometric BRST quantization,I: Prequantization

This is the first part of a two-part paper dedicated to the definition of BRST quantization in the framework of geometric quantization. After recognizing prequantization as a manifestation of the Poisson module structure of the sections of the prequantum line bundle, we define BRST prequantization and show that it is the homological analog of the symplectic reduction of prequantum data. We define a prequantum BRST cohomology theory and interpret it in terms of geometric objects. We then show that all Poisson structures correspond under homological reduction. This allows to prove, in the BRST context, that prequantization and reduction commute.     

Condition of Prequantization to Two—Dimensional Manifolds with Boundary

The Weil‘s integrality condition of prequantization is generalized to two-dimensional phase space with boundaries.It is shown that in the prequantization condition a term related to the symplectic potential on the boundary appears.The necessity of the generalized condition is proved by analyzing the isolated singularities of the Hermitian bundle while the sufficiency of the condition is proved via geometric construction on the space of equivalence class.     

Geometric Quantization of Relativistic Hamiltonian Mechanics

A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.     

Gravitational Descendants in Symplectic Field Theory

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown using the ideas in Okounkov and Pandharipande (Ann Math 163(2):517–560, 2006) that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we follow the ideas in Cieliebak and Latschev ( [math.s6], 2007) to compute the corresponding sequence of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold.     

Geometric quantization of the heavy top

The geometric quantization of the heavy top is described and its relation with the geometric quantization of the cotangent bundle of the orthogonal group is pointed out.     

On the geometric prequantization of Maxwell equations

The geometric prequantization of Maxwell equations in a vacuum is described and its relation with geometric prequantization of the extended phase space is pointed out.     

Lifting geometric objects to a cotangent bundle, and the geometry of the cotangent bundle of a tangent bundle

We show that the cotangent bundle T^*T of the tangent bundle of any differentiable manifold carries an integrable almost tangent structure which is generated by a natural lifting procedure from the canonical almost tangent structure (vertical endomorphism) of T . Using this almost tangent structure we show that T^*T is diffeomorphic to a tangent bundle, namely TT^* . This provides a new and geometrically instructive proof of a result of Tulczyjew, which has applications in Lagrangian and Hamiltonian dynamics and in field theory The requisite general definitions and results concerning liftings of geometric objects from a manifold to its cotangent bundle are given. As an application, we shed new light on the meaning of so-called adjoint symmetries of second-order differential equations.     

On the geometric prequantization of Poisson manifolds

The geometric prequantization of Poisson manifolds is described using the Weinstein theory of local symplectic groupoids.     

On symplectic submanifolds of cotangent bundles

Necessary and sufficient conditions are given for a symplectic submanifold of a cotangent bundle to itself be a cotangent bundle.Partially supported by NSF grant DMS-9222241.     

Prequantum bundles and projective Hilbert geometries

Principal circle bundles with connection and symplectic curvature over Banach manifolds are investigated. Using results on contact manifolds alternate proofs for some results of B. Kostant are given and a symplectic structure for the total space of the corresponding principal \ {0} bundle is constructed. As an example, these results are applied to the projective fibration of a complex Hilbert space. This gives close relations between the geometric formulation of classical and quantum dynamical systems. As another application, a functorial construction of the prequantization procedure of B. Kostant is given.     

The falling cat as a port-controlled Hamiltonian system

In this article, the falling cat is modeled as two jointed axial symmetric cylinders with arbitrary twist under the constraint of the vanishing total angular momentum. As a control system with the constraint taken into account, this model is formulated as a port-controlled Hamiltonian system defined on the cotangent bundle of the shape space for the jointed cylinders. A control is then designed as a function on the cotangent bundle, according to a standard procedure. Thus, the equations of motion are determined on the cotangent bundle together with the control. The whole motion as a vibrational motion of the falling cat is obtained after integrating the constraint equation of the vanishing total angular momentum. An example of the falling cat is given in which the model turns a somersault to approach a target state in equilibrium with an expected rotation after finishing a vibrational motion.     

Quantization and commutative superalgebras

The construction of the quantization on the cotangent superalgebra over the commutative superalgebra is presented and its connection with the quantization on the cotangent bundle of a supermanifold is discussed. It is shown that the famous quantizations, such as Weyl and Wick quantizations of Bose- and Fermi-systems, are special cases of this construction.     

On the geometry of reduced cotangent bundles at zero momentum

We consider the problem of cotangent bundle reduction for proper non-free group actions at zero momentum. We show that in this context the symplectic stratification obtained by Sjamaar and Lerman refines in two ways: (i) each symplectic stratum admits a stratification which we call the secondary stratification with two distinct types of pieces, one of which is open and dense and symplectomorphic to a cotangent bundle; (ii) the reduced space at zero momentum admits a finer stratification than the symplectic one into pieces that are coisotropic in their respective symplectic strata.     

Noncommutative geometry of phase space

A version of noncommutative geometry is proposed which is based on phase-space rather than position space. The momenta encode the information contained in the algebra of forms by a map which is the noncommutative extension of the duality between the tangent bundle and the cotangent bundle.     

Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of Poisson structures with background'.     

Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of ‘Poisson structures with background’.     

Systems of Hamilton-Jacobi equations

In this article we develop a generalization of the Hamilton-Jacobi theory, by considering in the cotangent bundle an involutive system of dynamical equations.     

Closed 2-Form of 2D Black Holes from Geometric Prequantization Method

In this paper, we consider effective Hamiltonian of 2D dilatonic black hole adding Axion field and calculate closed 2-form by using geometric prequantization method. It yields to the Schrödinger equation which may be solved to obtain wave function. We obtained a condition on the cosmological constant to obtain appropriate Hamilton equation of motions.     

Quantum Dynamical R-Matrices and Quantum Frobenius Group

We propose an algebraic scheme for quantizing the rational Ruijsenaars-Schneider model in the R-matrix formalism. We introduce a special parametrization of the cotangent bundle over . In new variables the standard symplectic structure is described by a classical (Frobenius) r-matrix and by a new dynamical -matrix. Quantizing both of them we find the quantum L-operator algebra and construct its particular representation corresponding to the rational Ruijsenaars-Schneider system. Using the dual parametrization of the cotangent bundle we also derive the algebra for the L-operator of the hyperbolic Calogero-Moser system. Received: 24 January 1997 / Accepted: 17 March 1997     

Getzler symbol calculus and deformation quantization

In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler’s pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.