Spin and Abelian Electromagnetic Duality¶on Four-Manifolds

We investigate the electromagnetic duality properties of an Abelian gauge theory on a compact oriented four-manifold by analysing the behaviour of a generalised partition function under modular transformations of the dimensionless coupling constants. The true partition function is invariant under the full modular group but the generalised partition function exhibits more complicated behaviour depending on topological properties of the four-manifold concerned. It is already known that there may be “modular weights” which are linear combinations of the Euler number and Hirzebruch signature of the four-manifold. But sometimes the partition function transforms only under a subgroup of the modular group (the Hecke subgroup). In this case it is impossible to define real spinor wave-functions on the four-manifold. But complex spinors are possible provided the background magnetic fluxes are appropriately fractional rather than integral. This gives rise to a second partition function which enables the full modular group to be realised by permuting the two partition functions, together with a third. Thus the full modular group is realised in all cases. The demonstration makes use of various constructions concerning integral lattices and theta functions that seem to be of intrinsic interest. Received: 5 June 2000 / Accepted: 9 October 2000     

Singular Dimensions of the¶N= 2 Superconformal Algebras. I

Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N= 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of singular vector. As a consequence we prove the conjecture of Gato-Rivera and Rosado on the possible existing types of topological singular vectors (4 in chiral Verma modules and 29 in complete Verma modules). Interestingly, we have found two-dimensional spaces of singular vectors at level 1. Finally, by using the topological twists and the spectral flows, we also obtain the maximal dimensions of the singular vector spaces for the Neveu–Schwarz N= 2 algebra (0, 1 or 2) and for the Ramond N= 2 algebra (0, 1, 2 or 3). Received: 19 August 1998 / Accepted: 15 March 1999     

Singular Dimensions¶of the N= 2 Superconformal Algebras II:¶The Twisted N= 2 Algebra

We introduce a suitable adapted ordering for the twisted N= 2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels , 1, and for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N= 2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels , 1, and . Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N= 2 embedding diagrams. Received: Received: 15 March 1999 / Accepted: 12 November 2000     

Form Invariance and Lie Symmetries of the Rotational Relativistic Birkhoff System

For a rotational relativistic Birkhoff system,the relation between the form invariance and the Lie symmetries are given Under infinitesimal transformations of groups.If the infinitesimal transformation generators ξ0 and ξμ satisfy the conditions of the form invariance,and the determining equation of Lie symmetries holds,the form invariance leads to a Lie symmetry of the system.Furthermore,if the infinitesimal transformations generators ξ0 and ξμ satisfy the conditions of the form invariance and the determining equation of Lie symmetry holds,and if there is a gauge function G satisfying the structure equation of Lie symmetry,then the form invariance will lead to the Lie symmetrical conserved quantity of the system.An example is given to illustrate the application of the results.     

Breit–Wigner Approximation and the Distribution¶of Resonances

For operators with a discrete spectrum, {λ _j ²}, the counting function of λ _j 's, N (λ), trivially satisfies N ( λ+δ ) −N ( λ−δ ) =∑ _j δ_{λ j} ((λ−δ,λ+δ]). In scattering situations the natural analogue of the discrete spectrum is given by resonances, λ _j ∈ℂ₊, and of N (λ), by the scattering phase, s(λ). The relation between the two is now non-trivial and we prove that

The Split Property and the Symmetry Breaking¶of the Quantum Spin Chain

We consider the relationship between the symmetry breaking and the split property of pure states of quantum spin chains. We obtain a representation theoretic condition implying that the half-sided uniform mixing condition leads to symmetry breaking of translationally invariant pure states. This is a mathematical generalization of Dichotomy previously found by I. Affleck and E. Lieb and M. Aizenman and B. Nachtergaele for ground states of a special class of Hamiltonians. Received: 1 February 1999 / Accepted: 5 December 2000     

Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra

We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone. Received: 28 October 1998 / Accepted: 7 March 1999     

Picard–Fuchs Uniformization and Modularity¶of the Mirror Map

Arithmetic properties of mirror symmetry (type IIA-IIB string duality) are studied. We give criteria for the mirror map q-series of certain families of Calabi–Yau manifolds to be automorphic functions. For families of elliptic curves and lattice polarized K3 surfaces with surjective period mappings, global Torelli theorems allow one to present these criteria in terms of the ramification behavior of natural algebraic invariants – the functional and generalized functional invariants respectively. In particular, when applied to one parameter families of rank 19 lattice polarized K3 surfaces, our criterion demystifies the Mirror-Moonshine phenomenon of Lian and Yau and highlights its non-monstrous nature. The lack of global Torelli theorems and presence of instanton corrections makes Calabi–Yau threefold families more complicated. Via the constraints of special geometry, the Picard–Fuchs equations for one parameter families of Calabi–Yau threefolds imply a differential equation criterion for automorphicity of the mirror map in terms of the Yukawa coupling. In the absence of instanton corrections, the projective periods map to a twisted cubic space curve. A hierarchy of “algebraic” instanton corrections correlated with the differential Galois group of the Picard–Fuchs equation is proposed. Received: 14 August 1999 / Accepted: 30 January 2000     

Long Time Behavior of the Continuum Limit¶of the Toda Lattice, and the Generation¶of Infinitely Many Gaps from C∞ Initial Data

We analyze a continuum limit of the finite non-periodic Toda lattice through an associated constrained maximization problem over spectral density functions. The maximization problem was derived by Deift and McLaughlin using the Lax–Levermore approach, initially developed for the zero dispersion limit of the KdV equation. It encodes the evolution of the continuum limit for all times, including evolution through shocks. The formation of gaps in the support of the maximizer is indicative of oscillations in the Toda lattice and the lack of strong convergence of the continuum limit. For large times, the maximizer tends to have zero gaps, which is the continuum analogue of the sorting property of the finite lattice. Using methods from logarithmic potential theory, we show that this behavior depends crucially on the initial data. We exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter (at large times), and other initial data for which this uniformity fails at all times. We then construct an example of C ^∞ smooth initial data generating, at a later time, infinitely many gaps in the support of the maximizer, while for even larger times, all gaps have closed. Received: 8 May 2000 / Accepted: 27 March 2001     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

Rational Solutions of the Master Symmetries¶of the KdV Equation

In this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for KdV. The result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master-symmetry KdV hierarchy are bispectral potentials for the Schr?dinger operator. By bispectral potentials we mean that the corresponding Schr?dinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable. This complements certain results of Airault–McKean–Moser [4], Duistermaat–Grünbaum [10], and Magri–Zubelli [40]. As a consequence of bispectrality, the rational solutions of the master symmetries turn out to be solutions of a (generalized) string equation. Received: 28 January 1999 / Accepted: 22 October 1999     

Solutions of the Dirac–Fock Equations for Atoms¶and Molecules

The Dirac-Fock equations are the relativistic analogue of the well-known Hartree-Fock equations. They are used in computational chemistry, and yield results on the inner-shell electrons of heavy atoms that are in very good agreement with experimental data. By a variational method, we prove the existence of infinitely many solutions of the Dirac-Fock equations "without projector", for Coulomb systems of electrons in atoms, ions or molecules, with Z h 124, N h 41, N h Z. Here, Z is the sum of the nuclear charges in the molecule, N is the number of electrons.     

Generalized Edwards Transformation and Principle of Permutation Invariance

The relativity of simultaneity is generalized by using the concept of synchronization gauge. The Lorentz transformation is derived without the postulate of the universal limiting speed, and a generalized Edwards transformation is obtained by using the principle of permutation invariance (covariance). It is shown that the existences of the one-way universal limiting speed (in the Lorentz transformation) and the constancy of the two-way average speed of light (in the Edwards transformation) are the necessary consequences of the principle of permutation invariance that is consistent with the postulate of relativity. The connection between the Edward transformation and the general coordinate transformation is discussed, and based on this, we find that the physical meaning of the Edwards parameter, which indicates anisotropy of the speed of light, is a gravitomagnetic potential of the spacetime.     

Bosonization and Integral Representation of Solutions¶of the Knizhnik–Zamolodchikov–Bernard Equations

We construct an integral representation of solutions of the Knizhnik–Zamolodchikov–Bernard equations, using the Wakimoto modules. Received: 5 October 1998 / Accepted: 8 February 1999     

Supersymmetry and Shape Invariance of Hartmann Potential and Ring-Shaped Oscillator Potential in the γ and θ Dimensions of Spherical Polar Coordinates

This article shows that in spherical polar coordinates, some noncentral separable potentials have super-symmetry and shape invariance in the r and θ dimensions, we choose Hartmann potential and ring-shaped oscillator astwo important examples, thus in principle the energy eigenvalues and energy eigenfunctions of such the potentials in ther and θ dimensions can be obtained by the method of supersymmetric quantum mechanics. Here we use an alternativemethod to get the required results.     

Renormalization Group¶and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions. Received: 29 January 2001 / Accepted: 8 March 2001     

Form Invariance and Noether Symmetries of Rotational Relativistic Birkhoff Systems

Under the infinitesimal transformations of groups,a form invariance of rotational relativistic Birkhoff systems is studied and the definition and criteria are given,In view of the invariance of rotational relativistic Pfaff-Birkhoff-D‘ Alembert principle under the infinitesimal transformations of groups,the theory of Noether symmetries of rotational relativistic Birkhoff systems are constructed.The relation between the form invariance and the Noether symmetries is studied ,and the conserved quantities of rotational relativistic Birkhoff systems are obtained.     

Strange Attractors Across the Boundary¶of Hyperbolic Systems

We study a bifurcation of Axiom A (hyperbolic) vector fields in dimension three leading to robust strange attractors with singularities. The Axiom A vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift. The attractors arising from this kind of bifurcation are not equivalent to the geometric Lorenz attractors. Received: 4 December 1998 / Accepted: 5 October 1999     

Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M ₂(ℂ)=ℂℤ₂·ℂℤ₂. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct q-monopoles on all the Podleś quantum spheres S ² _q,s . Received: 25 September 1998 / Accepted: 23 February 2000