The field copy problem: to what extent do curvature (gauge field) and its covariant derivatives determine connection (gauge potential)?

We show that a connection of a principal bundle is determined up to (global) gauge equivalence by the curvature and its covariant derivatives provided that the infinitesimal holonomy group is of constant dimension and the base space is simply connected. If the dimension of the infinitesimal holonomy group varies, there may be obstructions of a topological nature to the existence of a global or even local gauge equivalence between two connections whose curvatures and covariant derivatives of curvature agree everywhere. These obstructions are analyzed and illustrated by examples.     

Classification of singular Sobolev connections by their holonomy

For a connection on a principalSU(2) bundle over a base space with a codimension two singular set, a limit holonomy condition is stated. In dimension four, finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy. If there is no holonomy, then a codimension two removable singularity theorem is obtained.Research partially supported by NSF Grant DMS-8701813Research partially supported by NSF Grant INT-8511481     

Stratification of the Generalized Gauge Orbit Space

Different versions for defining Ashtekar's generalized connections are investigated depending on the chosen smoothness category for the paths and graphs – the label set for the projective limit. Our definition covers the analytic case as well as the case of webs. Then the action of Ashtekar's generalized gauge group on the space of generalized connections is investigated for compact structure groups G. Here, first, the orbit types of the generalized connections are determined. The stabilizer of a connection is homeomorphic to the holonomy centralizer, i.e. the centralizer of its holonomy group. It is proven that the gauge orbit type of a connection can be defined by the G-conjugacy class of its holonomy centralizer equivalently to the standard definition via -stabilizers. The connections of one and the same gauge orbit type form a so-called stratum. As the main result of this article a slice theorem is proven on . This yields the openness of the strata. Afterwards, a denseness theorem is proven for the strata. Hence, is topologically regularly stratified by . These results coincide with those of Kondracki and Rogulski for Sobolev connections. Furthermore, the set of all gauge orbit types equals the set of all (conjugacy classes of) Howe subgroups of G. Finally, it is shown that the set of all gauge orbits with maximal type has the full induced Haar measure 1. Received: 12 January 2000 / Accepted: 8 May 2000     

Special Symplectic Connections and Poisson Geometry

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure. Moreover, we show that these Poisson structures cannot be symplectically integrated by a Hausdorff groupoid. As a consequence, we obtain a canonical principal line bundle over any special symplectic manifold or orbifold, and we deduce numerous global consequences.     

Curved Space-Times by Crystallization of Liquid Fiber Bundles

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \(((\alpha ,\omega ),\varpi )\), where \((\alpha ,\omega )\) is a 1-form with coefficients in the Lie algebra of the Poincaré group and \(\varpi \) is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations.     

The internal symmetry group of a connection on a principal fiber bundle with applications to gauge field theories

The internal symmetry group of a connection on a principal fiber bundleP is studied. It is shown that this group is a smooth proper Lie transformation group ofP, which, ifP is connected, is also free. Moreover, this group is shown to be isomorphic to the centralizer of the holonomy group of the connection. Several examples and applications of these results to gauge field theories are given.     

Gauge theory in Witten's approach to the generation problem

In Witten's topological theory of the generation problem, gauge groups are identified with theE ₈ centraliser of the holonomy group of the internal manifold. Here we show that this amounts to interpreting gauge groups as generalised symmetry groups of the (internal) Levi-Civitá connection. We then give techniques for computing centralisers in exceptional groups, taking into account the fact that holonomy groups are frequently disconnected. These techniques allow us to deal with compact locally irreducible Ricci-flat Riemannian manifolds of all holonomy types and dimensions.     

Axiomatic holonomy maps and generalized Yang-Mills moduli space

This Letter is a follow-up of Barrett, J. W.,Internat. J. Theoret. Phys. 30(9), (1991). Its main goal is to provide an alternative proof of that part of the reconstruction theorem which concerns the existence of a connection. A construction of a connection 1-form is presented. The formula expressing the local coefficients of the connection in terms of the holonomy map is obtained as an immediate consequence of that construction. Thus, the derived formula coincides with that used in Chan, H.-M., Scharbach, P., and Tsou, S. T.,Ann. Physics 166, 396–421 (1986). The reconstruction and representation theorems form a generalization of the fact that the pointed configuration space of the classical Yang-Mills theory is equivalent to the set of all holonomy maps. The point of this generalization is that there is a one-to-one correspondence not only between the holonomy maps and the orbits in the space of connections, but also between all maps M G fulfilling some axioms and all possible equivalence classes ofP(M, G) bundles with connections, where the equivalence relation is defined by a bundle isomorphism in a natural way.     

Carnot-Carathéodory Metric and Gauge Fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Carathéodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes’s distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Carathéodory distance dh defined by A. In this paper we make precise this link, showing that the equality of d and d _H strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).     

String structures and the path fibration of a group

We use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class, that is the obstruction to a loop group bundle lifting to a Kac-Moody group bundle. In the case that the loop group bundle is constructed by taking loops into a principal bundle we show that the classifying map is the holonomy around loops and give an explicit formula for the string class relating it to the Pontrjangin class of the principal bunble.     

On geometric interpretation of the Aharonov–Bohm effect

A geometric interpretation of the Aharonov–Bohm effect is given in terms of connections on principal fiber bundles. It is demonstrated that the principal fiber bundle can be trivial while the connection and its holonomy group are nontrivial. Therefore, the main role is played by geometric rather than topological effects.     

On geometric interpretation of the berry phase

A geometric interpretation of the Berry phase and its Wilczek–Zee non-Abelian generalization are given in terms of connections on principal fiber bundles. It is demonstrated that a principal fiber bundle can be trivial in all cases, while the connection and its holonomy group are nontrivial. Therefore, the main role is played by geometric rather than topological effects.     

A unified theory of quantum holonomies

A periodic change of slow environmental parameters of a quantum system induces quantum holonomy. The phase holonomy is a well-known example. Another is a more exotic kind that exhibits eigenvalue and eigenspace holonomies. We introduce a theoretical formulation that describes the phase and eigenspace holonomies on an equal footing. The key concept of the theory is a gauge connection for an ordered basis, which is conceptually distinct from Mead-Truhlar-Berry’s connection and its Wilczek-Zee extension. A gauge invariant treatment of eigenspace holonomy based on Fujikawa’s formalism is developed. Example of adiabatic quantum holonomy, including the exotic kind with spectral degeneracy, are shown.     

Teichmüller Groupoids,and Monodromy in Conformal Field Theory

We study monodromy representations of the Teichmüller groupoid for the moduli space of pointed compact Riemann surfaces of any genus with first-order infinitesimal structure. To calculate these representations, using arithmetic Schottky-Mumford uniformization theory we construct a real orbifold in the moduli space consisting of fusing and simple moves which gives tangential base points. For a certain vector bundle on the moduli space with projectively flat connection, we show that the monodromy of each fusing move can be expressed as a connection matrix, and give the relations to the monodromy of simple moves. Furthermore, we describe the monodromy representation associated with Tsuchiya-Ueno-Yamadas conformal field theory, and show that this representation can be expressed as the monodromy of the Wess-Zumino-Witten model.     

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.     

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.     

Covariant derivative on non-linear fiber bundles

A gauge field is usually described as a connection on a principal bundle. It induces a covariant derivative on associated vector bundles, sections of which represent matter fields. In general, however, it is not possible to define a covariant derivative on non-linear fiber bundles, i.e. on those which are not vector bundles. We definelogarithmic covariant derivatives acting on two special non-linear fiber bundles — on the principal bundle and on the local gauge group bundle. The logarithmic derivatives map from sections of these bundles to the sections of the local gauge algebra bundle. Some properties of the logarithmic derivatives are formulated.