Geometry of Hyper-Kähler Connections with Torsion

The internal space of a N = 4 supersymmetric model with Wess–Zumino term has a connection with totally skew-symmetric torsion and holonomy in SP(n). We study the mathematical background of this type of connection. In particular, we relate it to classical Hermitian geometry, construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory. Received: 1 October 1999 / Accepted: 30 January 2000     

Calibrated geometries and non-perturbative superpotentials in M-theory

We consider non-perturbative effects in M-theory compactifications on a seven-manifold of holonomy arising from membranes wrapped on supersymmetric three-cycles. When membranes are wrapped on associative submanifolds they induce a superpotential that can be calculated using calibrated geometry. This superpotential is also derived from compactification on a seven-manifold, to four dimensional anti-de Sitter spacetime, of eleven dimensional supergravity with non-vanishing expectation value of the four-form field strength. Received: 28 June 2000 / Published online: 21 December 2000     

Euclidean D-branes and higher-dimensional gauge theory

We consider euclidean D-branes wrapping around manifolds of exceptional holonomy in dimensions seven and eight. The resulting theory on the D-brane—that is, the dimensional reduction of 10-dimensional supersymmetric Yang-Mills theory—is a cohomological field theory which describes the topology of the moduli space of instantons. The 7-dimensional theory is an N_T = 2 (or balanced) cohomological theory given by an action potential of Chern-Simons type. As a by-product of this method, we construct a related cohomological field theory which describes the monopole moduli space on a 7-manifold of G₂ holonomy.     

Induced quantum curvature and three-dimensional gauge theories

The effects of quantum holonomy in three-dimensional gauge theories with massless fermions is examined and different definitions of the fermion determinant are discussed. The source of a global gauge and parity anomaly is identified in Schrödinger picture quantization as an induced holonomy that arises from the fermionic sector of the theory. In certain fermion representations this holonomy leads to a global obstruction to imposing either gauge or parity invariance through the implementation of Gauss' law constraint. However, such obstructions can be removed by exploiting renormalization ambiguities inherent in the definition of composite operators.     

Definite signature conformal holonomy: A complete classification

This paper aims to classify the holonomy of the conformal Tractor connection, and relate these holonomies to the geometry of the underlying manifold. The conformally Einstein case is dealt with through the construction of metric cones, whose Riemannian holonomy is the same as the Tractor holonomy of the underlying manifold. Direct calculations in the Ricci-flat case and an important decomposition theorem complete the classification for definitive signature.     

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.     

Another construction of the central extension of the loop group

By considering the geometry of the central extension of the loop group as a principal bundle it is shown that it must be the quotient of a larger group. This group is a central extension of the group of paths in the loop group and its cocycle is constructed as the holonomy around a certain path. Conversely it is shown that this definition of a cocycle gives a method of constructing the central extension. The Wess-Zumino term plays an important role in these constructions.     

Eight-dimensional quantum Hall effect and "Octonions"

We construct a generalization of the quantum Hall effect where particles move in an eight-dimensional space under an SO(8) gauge field. The underlying mathematics of this particle liquid is that of the last normed division algebra, the octonions. Two fundamentally different liquids with distinct configuration spaces can be constructed, depending on whether the particles carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20-dimensional manifold with an internal component of SO(7) holonomy, whereas the second liquid lives on a 14-dimensional manifold with an internal component of G2 holonomy.     

Anyons associated with the generalized braid groups

We discuss the inequivalent quantization of a physical system with a configuration space which is a certain orbit space of the Coxeter group. The framework for the generalization of the anyon is given. Also, we construct a gauge field whose holonomy gives rise to the statistical factor of the corresponding anyon.     

The phase and critical point of quantum Einstein–Cartan gravity

By introducing diffeomorphism and local Lorentz gauge invariant holonomy fields, we study in the recent article [S.-S. Xue, Phys. Rev. D 82 (2010) 064039] the quantum Einstein–Cartan gravity in the framework of Regge calculus. On the basis of strong coupling expansion, mean-field approximation and dynamical equations satisfied by holonomy fields, we present in this Letter calculations and discussions to show the phase structure of the quantum Einstein–Cartan gravity, (i) the order phase: long-range condensations of holonomy fields in strong gauge couplings; (ii) the disorder phase: short-range fluctuations of holonomy fields in weak gauge couplings. According to the competition of the activation energy of holonomy fields and their entropy, we give a simple estimate of the possible ultra-violet critical point and correlation length for the second-order phase transition from the order phase to disorder one. At this critical point, we discuss whether the continuum field theory of quantum Einstein–Cartan gravity can be possibly approached when the macroscopic correlation length of holonomy field condensations is much larger than the Planck length.     

Harmonic Functions and Instanton Moduli Spaces on the Multi-Taub–NUT Space

Explicit construction of the basic SU(2) anti-instantons over the multi-Taub–NUT geometry via the classical conformal rescaling method is exhibited. These anti-instantons satisfy the so-called weak holonomy condition at infinity with respect to the trivial flat connection and decay rapidly. The resulting unit energy anti-instantons have trivial holonomy at infinity.     

Off-diagonal quantum holonomy along density operators

Uhlmann's concept of quantum holonomy for paths of density operators is generalised to the off-diagonal case providing insight into the geometry of state space when the Uhlmann holonomy is undefined. Comparison with previous off-diagonal geometric phase definitions is carried out and an example comprising the transport of a Bell-state mixture is given.     

Degeneracy in loop variables

The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekar's connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.     

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom.     

Time-delay in potential scattering theory

We show that a gauge field uniquely determines its potential if and only if its holonomy group coincides with the gauge group on every open set in spacetime, provided that the field is not degenerate as a 2-form over spacetime. In other words, there is no potential ambiguity whenever such a field is irreducible everywhere in spacetime. We then show that the ambiguous potentials for those gauge fields are partitioned into gauge-equivalence classes (modulo certain homotopy classes) as a consequence of the nontrivial connectivity of spacetime. These homotopy classes depend on the gauge group, on the holonomy group and on this last group's centralizer in the gauge group.To the Memory of Jorge André SwiecaResearch supported by C.N.Pq. and M.E.C. (Brazil)