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arXiv1996

The Multidimensional Darboux Transformation

A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n greater than one. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results.

hep-th↗
arXiv1997

Sequences of Levy Transformations and Multi-Wroński Determinant Solutions of the Darboux System

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multi-Wroński determinants. These formulae are for the tangent vectors, Lamè coefficients, rotation coefficients and points of the surface.

dg-ga↗
arXiv1997

On the Spinor Representation of Surfaces in Euclidean 3-Space

The aim of the present paper is to clarify the relationship between immersions of surfaces and solutions of the inhomogeneous Dirac equation. The main idea leading to the description of a surface M^2 by a spinor field is the observation that the restriction to M^2 of any parallel spinor phi on R^3 is (with respect to the inner geometry of M^2) a non-trivial spinor field on M^2 of constant length which is a solution of the inhomogeneous Dirac equation and vice versa.

dg-ga↗
arXiv1997

Gauge theory and the division algebras

We present a novel formulation of the instanton equations in 8-dimensional Yang-Mills theory. This formulation reveals these equations as the last member of a series of gauge-theoretical equations associated with the real division algebras, including flatness in dimension 2 and (anti-)self-duality in 4. Using this formulation we prove that (in flat space) these equations can be understood in terms of moment maps on the space of connections and the moduli space of solutions is obtained via a generalised symplectic quotient: a Kaehler quotient in dimension 2, a hyperkaehler quotient in dimension 4 and an octonionic Kaehler quotient in dimension 8. One can extend these equations to curved space: whereas the 2-dimensional equations make sense on any surface, and the 4-dimensional equations make sense on an arbitrary oriented manifold, the 8-dimensional equations only make sense for manifolds whose holonomy is contained in Spin(7). The interpretation of the equations in terms of moment maps further constraints the manifolds: the surface must be orientable, the 4-manifold must be hyperkaehler and the 8-manifold must be flat.

hep-th↗
arXiv1996

Superstrings in Higher Order Extensions of Finsler Superspaces

The work proposes a geometric background of the theory of field interactions and strings in spaces with higher order anisotropy. Our approach proceeds by developing the concept of higher order anisotropic superspace which unifies the logical and mathematical aspects of modern Kaluza-Klein theories and generalized Lagrange and Finsler geometry and leads to modelling of physical processes on higher order fiber bundles provided with nonlinear and distingushed connections and metric structures. The view adopted is that a general field theory should incorporate all possible anisotropic and stochastic manifestations of classical and quantum interactions and, in consequence, a corresponding modification of basic principles and mathematical methods in formulation of physical theories. The presentation is divided into two parts. The first five sections cover the higher order anisotropic superspaces. We focus on the geometry of distinguished by nonlinear connection vector superbundles, consider different supersymmetric extensions of Finsler and Lagrange spaces and analyze the structure of basic geometric objects on such superspaces. The remaining five sections are devoted to the theory of higher order anisotropic superstrings. In the framework of supersymmetric nonlinear sigma models in Finser extended backgrounds we prove that the low-energy dynamics of such strings contains motion equations for locally anisotropic field interactions.

hep-th↗
arXiv1997

Supersymmetry and the cohomology of (hyper)Kaehler manifolds

The cohomology of a compact Kaehler (resp. hyperKaehler) manifold admits the action of the Lie algebra so(2,1) (resp. so(4,1)). In this paper we show, following an idea of Witten, how this action follows from supersymmetry, in particular from the symmetries of certain supersymmetric sigma models. In addition, many of the fundamental identities in Hodge-Lefschetz theory are also naturally derived from supersymmetry.

hep-th↗
arXiv1997

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 G_2 holonomy.

hep-th↗
arXiv1997

Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyperkaehler (but otherwise arbitrary) target X to a four-dimensional hyperkaehler manifold M is a topological sigma model localising on the space of triholomorphic maps M -> X (or hyperinstantons). When X is the moduli space M_K of instantons on a four-dimensional hyperkaehler manifold K, this theory has an interpretation in terms of supersymmetric gauge theory. In this case, the topological sigma model can be understood as an adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric Yang-Mills on the eight-dimensional manifold M x K of holonomy Sp(1) x Sp(1) in Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons.

hep-th↗
arXiv1997

Can We Look at The Quantisation Rules as Constraints?

In this paper we explore the idea of looking at the Dirac quantisation conditions as $\hbar$-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phase- space, we handle the above constraints using the standard theory of Dirac for constrained systems. The hope is to obtain, as total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets the Moyal ones. Unfortunately the program fails indicating that something is missing. We put forward at the end some ideas for future work which may overcome this failure.

dg-ga↗
arXiv1997

Scalar Curvature Rigidity for asymptotically locally hyperbolic manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the 4-dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.

dg-ga↗
arXiv1997

A class of anisotropic (Finsler-) space-time geometries

A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to their group of isometries. These turn out to be isomorphic to subgroups of the Poincaré (Lorentz-) group complemented by the generator of a dilatation. The arising Finsler geometries may be used for the construction of relativistic theories testing the isotropy of space. It is shown that the Finsler space with the only preferred null direction is the anisotropic space closest to isotropic Minkowski-space of the full class discussed.

gr-qc↗
arXiv1994

The Patchwork Divergence Theorem

The divergence theorem in its usual form applies only to suitably smooth vector fields. For vector fields which are merely piecewise smooth, as is natural at a boundary between regions with different physical properties, one must patch together the divergence theorem applied separately in each region. We give an elegant derivation of the resulting "patchwork divergence theorem" which is independent of the metric signature in either region, and which is thus valid if the signature changes. (PACS numbers 4.20.Cv, 04.20.Me, 11.30.-j, 02.40.Hw)

gr-qc↗
arXiv1994

Eta-Invariants and Determinant Lines

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, simpler proof of the holonomy formula for the determinant line bundle of a family of Dirac operators, also known as the ``global anomaly'' formula. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). A postscript file with figures was submitted separately in uuencoded tar-compressed format.

hep-th↗
arXiv1995

Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory.

hep-th↗
arXiv1994

Parametric Manifolds I: Extrinsic Approach

A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter family of hypersurfaces orthogonal to the curves, each of which inherits a metric and connection from the original manifold via orthogonal projections; this is the well-known Gauss-Codazzi formalism. We generalize this formalism to the case where the foliation is not hypersurface orthogonal. Crucial to this generalization is the notion of deficiency, which measures the failure of the orthogonal tangent spaces to be surface-forming, and which behaves very much like torsion. Some applications to initial value problems in general relativity will be briefly discussed.

gr-qc↗
arXiv1994

Parametric Manifolds II: Intrinsic Approach

A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, all based on a nonstandard action of vector fields on functions. There is a new geometric object, called the deficiency, which behaves much like torsion, and which measures whether a parametric manifold can be viewed as a 1-parameter family of orthogonal hypersurfaces.

gr-qc↗
arXiv1996

Spinors in Higher Dimensional and Locally Anisotropic Spaces

The theory of spinors is developed for locally anisotropic (la) spaces, in brief la-spaces, which in general are modeled as vector bundles provided with nonlinear and distinguished connections and metric structures (such la-spaces contain as particular cases the Lagrange, Finsler and, for trivial nonlinear connections, Kaluza-Klein spaces). The la-spinor differential geometry is constructed. The distinguished spinor connections are studied and compared with similar ones on la-spaces. We derive the la-spinor expressions of curvatures and torsions and analyze the conditions when the distinguished torsion and nonmetricity tensors can be generated from distinguished spinor connections. The dynamical equations for gravitational and matter field la-interactions are formulated.

gr-qc↗