The b-boundary of S 1 with a constant connection

On a manifold with connection the frame bundle admits a Riemannian metric which Schmidt introduced to construct the b-boundary of the underlying manifold. Here we study the metrics that arise when the manifold is R or S ¹ with a constant connection.     

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.     

Spacetime tangent bundle with torsion

It is demonstrated explicitly that the bundle connection of the Finslerspacetime tangent bundle can be made compatible with Cartan's theory of Finsler space by the inclusion of bundle torsion, and without the restriction that the gauge curvature field be vanishing. A component of the contorsion is made to cancel the contribution of the gauge curvature field to the relevant component of the bundle connection. Also, it is shown that the bundle manifold remains almost complex, and that the almost complex structure can be made to have a vanishing covariant derivative if additional conditions on the torsion are satisfied. However, the Finsler-spacetime tangent bundle remains complex only if the gauge curvature field vanishes.     

Asymptotics of the Eta Invariant

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.     

Finsler-spacetime tangent bundle

The Levi-Civita connection coefficients of the spacetime tangent bundle, for the case of a Finsler spacetime, are reduced to the form given by Yano and Davies for a generic tangent bundle of a Finsler manifold. A useful expression is also obtained for the Riemann curvature scalar of a Finsler-spacetime tangent bundle.     

Projective connections

We compare the approaches of E. Cartan and of T.Y. Thomas and J.H.C. Whitehead to the study of ‘projective connections’. Although the quoted phrase has quite different meanings in the two contexts considered, we show that a class of projectively equivalent symmetric affine connections–or, more generally, sprays–on a manifold (the latter meaning) gives rise, in a global way, to a unique Cartan connection on a principal bundle over the manifold (the former meaning). The principal bundle on which the Cartan connection is defined is itself a geometric object, and exists independently of any particular connection. In the course of the discussion we derive a Cartan normal projective connection for a system of second-order ordinary differential equations (extending the results of Cartan from a single equation to many) and we generalize the concept of a normal Thomas–Whitehead connection from affine to general sprays.     

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?^* _ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?^* _ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables. Received: 28 April 2000 / Accepted: 6 December 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.     

Curvature on determinant bundles and first Chern forms

The Quillen–Bismut–Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut–Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues.

We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut–Freed superconnection is identified with classical Teichmüller theory connections, and its curvature and regularized first Chern form are computed.     

A geometrical setting for the Lax representation

A geometrical interpretation of the Lax representation is suggested; it is read as the condition on a section of a fiber bundle based on the phase manifold to be covariantly constant, with respect to a suitable connection, along the dynamics.     

Differential geometry of spacetime tangent bundle

Conditions are investigated under which the Levi-Civita connection of the spacetime tangent bundle corresponds to that of a generic tangent bundle of a Finsler manifold. Also, requirements are specified for the spacetime tangent bundle to be almost complex or Kählerian.This paper is an expanded version of an invited paper presented at the Second International Wigner Symposium, Goslar, Germany, July 1991.     

Geometric Quantization of Vector Bundles and the Correspondence with Deformation Quantization

I repeat my definition for quantization of a vector bundle. For the cases of the Toeplitz and geometric quantizations of a compact K?hler manifold, I give a construction for quantizing any smooth vector bundle, which depends functorially on a choice of connection on the bundle. Using this, the classification of formal deformation quantizations, and the formal, algebraic index theorem, I give a simple proof as to which formal deformation quantization (modulo isomorphism) is derived from a given geometric quantization. Received: 16 November 1998 / Accepted: 29 June 2000     

A direct method for minimizing the Yang-Mills functional over 4-manifolds

A direct method is employed to minimize the Yang-Mills functional over a 4-dimensional manifold. The limiting connection is shown to be Yang-Mills, but in a possibly new bundle. We show that a topological invariant of the bundle is preserved by the minimizing process. This implies the existence of an absolute minimum of the Yang-Mills functional in a wide class of bundles.     

Morita Equivalence of Fedosov Star Products and Deformed Hermitian Vector Bundles

Based on the usual Fedosov construction of star products for a symplectic manifold M, we give a simple geometric construction of a bimodule deformation for the sections of a vector bundle over M starting with a symplectic connection on M and a connection for E. In the case of a line bundle, this gives a Morita equivalence bimodule, and the relation between the characteristic classes of the Morita equivalent star products can be found very easily within this framework. Moreover, we also discuss the case of a Hermitian vector bundle and give a Fedosov construction of the deformation of the Hermitian fiber metric.     

Warps,grids and curvature in triple vector bundles

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle \(T^3M\) of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle \(T^2A\) for a vector bundle A using a connection in A to define a grid in \(T^2A\). In this case the curvature emerges from the warp theorem.

     

Non-holonomic and semi-holonomic frames in terms of Stiefel and Grassmann tangent bundles

We prove that the bundles of non-holonomic and semi-holonomic second-order frames of a real or complex manifold M can be obtained as extensions of the bundle F²(M) of second-order jets of (holomorphic) diffeomorphisms of into M, where or . If and is the bundle of -linear frames of M we will associate to the tangent bundle two new bundles and with fibers of type the Stiefel manifold and the Grassmann manifold , respectively, where . The natural projection of onto defines a -principal bundle. We have found that the subset of given by the horizontal n-planes is an open sub-bundle isomorphic to the bundle of semi-holonomic frames of second-order of M. Analogously, the subset of given by the horizontal n-bases is an open sub-bundle which is isomorphic to the bundle of non-holonomic frames of second-order of M. Moreover the restriction of the former projection still defines a -principal bundle. Since a linear connection is a horizontal distribution of n-planes invariant under the action of it therefore determines a -reduction of the bundle , in a bijective way. This is a new proof of a theorem of Libermann.     

Yang-Mills Detour Complexes and Conformal Geometry

Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation theory of Yang-Mills connections. In the case of Riemannian signature the complex is elliptic. If the connection respects a metric on the bundle then the complex is formally self-adjoint. In dimension 4 the complex is conformally invariant and generalises, to the full Yang-Mills setting, the composition of (two operator) Yang-Mills complexes for (anti-)self-dual Yang-Mills connections. Via a prolonged system and tractor connection a diagram of differential operators is constructed which, when commutative, generates differential complexes of natural operators from the Yang-Mills detour complex. In dimension 4 this construction is conformally invariant and is used to yield two new sequences of conformal operators which are complexes if and only if the Bach tensor vanishes everywhere. In Riemannian signature these complexes are elliptic. In one case the first operator is the twistor operator and in the other sequence it is the operator for Einstein scales. The sequences are detour sequences associated to certain Bernstein-Gelfand-Gelfand sequences.     

Generalized linear connections on real manifolds

The concept of the Ötsuki generalized connection is explained using a direct approach based only on the bundle of linear frames of a manifold. Simple derivations of the generalized torsion and curvature forms are given. Covariant differentiation is also derived as a generalization of the classical one. A way to possible further generalizations is indicated.     

The Pfaffian line bundle

We analyze the holomorphic Pfaffian line bundle defined over an infinite dimensional isotropic Grassmannian manifold. Using the infinite dimensional relative Pfaffian, we produce a Fock space structure on the space of holomorphic sections of the dual of this bundle. On this Fock space, an explicit and rigorous construction of the spin representations of the loop groupsLO _n is given. We also discuss and prove some facts about the connection between the Pfaffian line bundle over the Grassmannian and the Pfaffian line bundle of a Dirac operator.Supported by a National Science Foundation Graduate Fellowship     

Riemann curvature scalar of spacetime tangent bundle

The maximum possible proper acceleration relative to the vacuum determines much of the differential geometric structure of the space-time tangent bundle. By working in an anholonomic basis adapted to the spacetime affine connection, one derives a useful expression for the Riemann curvature scalar of the bundle manifold. The explicit documentation of the proof is important because of the central role of the curvature scalar in the formulation of an action with resulting field equations and associated solutions to physical problems.