Reconstruction theorem for groupoids and principal fiber bundles

The loop space formulation of 3+1 canonical quantum gravity premises that all physical information is contained within the holonomy loop functionals. This assumption is the result of the reconstruction theorem for a principla fiber bundle on a base loop space. The gauge connection for interacting gauge theories is more appropriately and readily reconstructed on a path space as opposed to a loop space. We generalize the reconstruction theorem to a base path space. Employing a holonomy groupoid map and a path connection, we trivially construct an abstract Lie groupoid from which a principal fiber bundle and gauge connection can be derived as distinctive examples. The groupoid reconstruction theorem is valid on both connected and nonconnected base manifolds, unlike the holonomy group reconstruction theorem, which can only be utilized for connected manifolds.     

Canonical transformations of local functionals and sh-Lie structures

In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector bundle. We consider those automorphisms of the vector bundle which induce mappings on the space of functionals preserving the Poisson bracket and refer to such automorphisms as canonical automorphisms.We determine how such automorphisms relate to the corresponding sh-Lie structure. If a Lie group acts on the bundle via canonical automorphisms, there are induced actions on the space of local functionals and consequently on the corresponding sh-Lie algebra. We determine conditions under which the sh-Lie structure induces an sh-Lie structure on a corresponding reduced space where the reduction is determined by the action of the group. These results are not directly a consequence of the corresponding theorems on Poisson manifolds as none of the algebraic structures are Poisson algebras.     

The Volume of a Differentiable Stack

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the base of the groupoid. Invariant sections of a square root of this line bundle constitute an “intrinsic Hilbert space” of the stack.     

BRST Cohomology and Phase Space Reduction in Deformation Quantization

In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a "quantum" Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is "sufficiently nice", e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counterexamples are discussed.     

Differential Groupoids and Their Application to the Theory of Spacetime Singularities

The transformation groupoid = × G, where is the total space of the generalized frame G-bundle over spacetime with a singular boundary, is not a Lie groupoid but a differential groupoid, i.e., a smooth groupoid in the category of structured spaces. We define this concept and use it to investigate spacetimes with various kinds of singularities. Any differential transformation groupoid can be represented by an algebra of operators on a bundle of Hilbert spaces defined on the groupoid fibers. This algebra reflects the structure of a given fiber even if it is a fiber over a singularity. It is also shown that any spacetime with singularities can be regarded as a noncommutative space. Its geometry is done in terms of a noncommutative algebra defined on the corresponding differential transformation groupoid. We focus on the structure of malicious singularities such as the ones appearing in the beginning and in the end of the closed Friedman universe.     

On the integration of Poisson homogeneous spaces

We study a reduction procedure for describing the symplectic groupoid of a Poisson homogeneous space obtained by quotient of a coisotropic subgroup. We perform it as a reduction of the Lu–Weinstein symplectic groupoid integrating Poisson Lie groups, that is suitable even for the non-complete case.     

Generalized Lie bialgebroids and Jacobi structures

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialgebroid. We also show that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce a duality theorem. Finally, some special classes of generalized Lie bialgebroids are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.     

A Unified Approach to Poisson Reduction

Given any Poisson action G×PP of a Poisson–Lie group G we construct an object =T ^*G*T^* P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids.     

q → ∞ limit of the quasitriangular WZW model

Abstract

We study the q → ∞ limit of the q-deformation of the WZW model on a compact simple and simply connected target Lie group. We show that the commutation relations of the q → ∞ current algebra are underlied by certain affine Poisson structure on the group of holomorphic maps from the disc into the complexification of the target group. The Lie algebroid corresponding to this affine Poisson structure can be integrated to a global symplectic groupoid which turns out to be nothing but the phase space of the q → ∞ limit of the q-WZW model. We also show that this symplectic grupoid admits a chiral decomposition compatible with its (anomalous) Poisson-Lie symmetries. Finally, we dualize the chiral theory in a remarkable way and we evaluate the exchange relations for the q → ∞ chiral WZW fields in both the original and the dual pictures.     

Application of the cohomology of graded Lie algebras to formal deformations of Lie Algebras

The purpose of the Letter is to show how to use the cohomology of the Nijenhuis-Richardson graded Lie algebra of a vector space to construct formal deformations of each Lie algebra structure of that space. One then shows that the de Rham cohomology of a smooth manifold produces a family of cohomology classes of the graded Lie algebra of the space of smooth functions on the manifold. One uses these classes and the general construction above to provide one-differential formal deformations of the Poisson Lie algebra of the Poisson manifolds and to classify all these deformations in the symplectic case.     

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.     

Lie algebroids,Lie groupoids and TFT

We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.     

A Berger-type theorem for metric connections with skew-symmetric torsion

We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. In particular, we get a refinement of a well-known theorem of Cartan and Schouten. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy.     

Semidirect products and the Pukanszky condition

We study the general geometrical structure of the coadjoint orbits of a semidirect product formed by a Lie group and a representation of this group on a vector space. The use of symplectic induction methods gives new insight into the structure of these orbits. In fact, each coadjoint orbit of such a group is obtained by symplectic induction on some coadjoint orbit of a “smaller” Lie group. We study also a special class of polarizations related to a semidirect product and the validity of Pukanszky's condition for these polarizations. Some examples of physical interest are discussed using the previous methods