Roger Picken

AN AXIOMATIC DEFINITION OF HOLONOMY

A group of loops is associated to every smooth pointed manifold M using a strong homotopy relation. It is shown that the holonomy of a connection on a principal G-bundle may be presented as a group morphism and that every such morphism satisfying a natural smoothness condition is the holonomy of some unique connection up to isomorphism.

The Duistermaat–Heckman integration formula on flag manifolds

An exposition is given of various geometrical properties of flag manifolds and of the Duistermaat–Heckman integration formula as applied to flag manifolds.

On two-dimensional holonomy

We define the thin fundamental categorical group P2(M,�) of a based smooth manifold (M,�) as the categorical group whose objects are rank-1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂: E ! G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M,�) ! C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1-form on P, a principal G bundle over M, and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module

We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M) ! C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.

TQFT’s and gerbes

We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a one-to-one correspondence between their local description in terms of locally-defined functions and forms and their non-local description in terms of a suitable class of embedded TQFTs.

Quantum Holonomies in (2+1)-Dimensional Gravity

We describe an approach to the quantisation of (2+1)-dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. Representations are constructed and a group of transformations - a quasi-modular group - which preserves this structure, is presented.Abstract We describe an approach to the quantisation of (2+1)-dimensional gravity with topology R ×T 2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. Representations are constructed and a group of transformations – a quasi-modular group – which preserves this structure, is presented.

Quantum Matrix Pairs

The notion of quantum matrix pairs is defined. These are pairs of matrices with noncommuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. Such matrices appear in an approach by the authors to quantizing gravity in 2 space and 1 time dimensions with negative cosmological constant on the torus. Explicit examples and transformations which generate new pairs from a given pair are presented.

Parametrization of the Moduli Space of Flat SL(2, R) Connections on the Torus

The moduli space of flat SL(2, R)-connections modulo gauge transformations on the torus may be described by ordered pairs of commuting SL(2, R) matrices modulo simultaneous conjugation by SL(2, R) matrices. Their spectral properties allow a classification of the equivalence classes, and a unique canonical form is given for each of these. In this way the moduli space becomes explicitly parametrized, and has a simple structure, resembling that of a cell complex, allowing it to be depicted. Finally, a Hausdorff topology based on this classification and parametrization is proposed.

A quantum Goldman bracket in 2+1 quantum gravity

In the context of quantum gravity for spacetimes of dimension (2 + 1), we describe progress in the construction of a quantum Goldman bracket for intersecting loops on surfaces. Using piecewise linear paths in (representing loops on the spatial manifold, i.e. the torus) and a quantum connection with noncommuting components, we review how holonomies and Wilson loops for two homotopic paths are related by phases in terms of the signed area between them. Paths rerouted at intersection points with other paths occur on the rhs of the Goldman bracket. To better understand their nature we introduce the concept of integer points inside the parallelogram spanned by two intersecting paths, and show that the rerouted paths must necessarily pass through these integer points.

Integration on supermanifolds and a generalized Cartan calculus

A suggestion by Berezin for a method of integration on supermanifolds is given a precise differential geometric meaning by assuming that a supermanifold is the total space of a fibre bundle with connection. The relevant objects for integration are identified as suitable horizontal/vertical projections of hyperforms. The latter are generalizations of differential forms having both covariant and contravariant indices. The exterior calculus of these projected hyperforms is developed, analogously to the Cartan calculus, by introducing appropriate derivations and determining their commutators, respectively anticommutators.